Mechanical systems, classical models. Analytical mechanics

Mechanical Systems, Classical Models

MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING Series Editor

Alan Jeffrey

The importance of mathematics in the study of problems arising from the real world, and the increasing success with which it has been used to model situations ranging from the purely deterministic to the stochastic, in all areas of today's Physical Sciences and Engineering, is well established. The progress in applicable mathematics has been brought about by the extension and development of many important analytical approaches and techniques, in areas both old and new, frequently aided by the use of computers without which the solution of realistic problems in modern Physical Sciences and Engineering would otherwise have been impossible. The purpose of the series is to make available authoritative, up to date, and self-contained accounts of some of the most important and useful of these analytical approaches and techniques. Each volume in the series will provide a detailed introduction to a specific subject area of current importance, and then will go beyond this by reviewing recent contributions, thereby serving as a valuable reference source.

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Mechanical Systems, Classical Models Volume III: Analytical Mechanics

by

Petre P. Teodorescu Faculty of Mathematics, University of Bucharest, Romania

123

Prof. Dr. Petre P. Teodorescu Str. Popa Soare 38 023984 Bucuresti 20 Romania

Translated into English, revised and extended by Petre P. Teodorescu All rights reserved © EDITURA TEHNICĂ, 2002 This translation of “Mechanical Systems, Classical Models” (original title: Sisteme mecanice.Modele clasice, Published by: Ed. Tehnicá, Bucuresti, Bucharest, Romania, 1984-2002), First Edition, is published by arrangement with EDITURA TEHNICĂ, Bucharest, ROMANIA

ISSN 1559-7458 e-ISSN 1559-7466 ISBN 978-90-481-2763-4 e-ISBN 978-90-481-2764-1 DOI 10.1007/978-90-481-2764-1 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2007943834 c Springer Science+Business Media B.V. 2009  No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Contents

Preface

ix

18.

Lagrangian Mechanics 1. Preliminary Results 1.1 Introductory Notions 1.2 Differential Principles of Mechanics 2. Lagrange’s Equations 2.1 Space of Configurations 2.2 Lagrange’s Equations of Second Kind 2.3 Transformations. First Integrals 3. Other Problems Concerning Lagrange’s Equations 3.1 New Forms of Lagrange’s Equations 3.2 Applications

1 2 2 20 45 46 57 65 83 83 98

19.

Hamiltonian Mechanics 1. Hamilton’s Equations 1.1 General Results 1.2 Lagrange’s Brackets. Poisson’s Brackets 1.3 Applications 2. The Hamilton–Jacobi Method 2.1 General Results 2.2 Systems of Equations with Separate Variables 2.3 Applications

115 115 115 138 151 160 160 172 191

20.

Variational Principles. Canonical Transformations 1. Variational Principles 1.1 Mathematical Preliminaries 1.2 The General Integral Principle 1.3 Hamilton’s Principle 1.4 Maupertuis’s Principle. Other Variational Principles 1.5 Continuous Mechanical Systems 2. Canonical Transformations 2.1 General Considerations. Conditions of Canonicity 2.2 Structure of Canonical Transformations. Properties 3. Symmetry Transformations. Noether’s Theorem. Conservation Laws 3.1 Symmetry Transformations. Noether’s Theorem

213 213 214 225 231 242 254 265 265 289 300 301 v

MECHANICAL SYSTEMS, CLASSICAL MODELS

vi 3.2 3.3 21.

22.

Lie Groups Space-Time Symmetries. Conservation Laws

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems 1. Integral Invariants. Ergodic Theorems 1.1 Integral Invariants of Order 2s 1.2 Invariants of First Order 1.3 Ergodic Theorems 2. Periodic Motions. Action-Angle Variables 2.1 Periodic Motions. Quasi-Periodic Motions 2.2 Action-Angle Variables 2.3 Adiabatic Invariance 3. Methods of Exterior Differential Calculus. Elements of Invariantive Mechanics 3.1 Methods of Exterior Differential Calculus 3.2 Elements of Invariantive Mechanics 3.3 Applications 4. Formalisms in the Dynamics of Mechanical Systems 4.1 Formalisms in Spaces with s + 1 Dimensions 4.2 Formalism in Spaces with 2s + 1 or with 2s + 2 Dimensions 4.3 Notions on the Inverse Problem of Mechanics and the Birkhoffian Formalism 5. Control Systems 5.1 Control Systems 5.2 Optimal Trajectories Dynamics of Non-holonomic Mechanical Systems 1. Kinematics of Non-holonomic Mechanical Systems 1.1 General Considerations 1.2 Conditions of Holonomy. Quasi-co-ordinates. Non-holonomic Spaces 2. Lagrange’s Equations. Other Equations of Motion 2.1 Motion of a Rigid Solid on a Fixed Surface 2.2 Lagrange’s Equations 2.3 Applications 2.4 Other Equations of Motion 3. Gibbs–Appell Equations 3.1 Gibbs–Appell Equations of Motion 3.2 Applications 4. Other Problems on the Dynamics of Non-holonomic Mechanical Systems 4.1 Collisions 4.2 First Integrals of the Equations of Motion

311 319 335 335 335 341 354 356 356 361 365 368 368 373 385 390 390 394 397 402 402 407 411 411 411 421 430 430 436 443 467 478 478 482 491 491 498

Contents

23.

24.

Stability and Vibrations 1. Stability of Mechanical Systems 1.1 Stability of Equilibrium 1.2 Stability of Motion 1.3 Applications 2. Vibrations of Mechanical Systems 2.1 Small Free Oscillations About a Stable Position of Equilibrium 2.2 Small Forced Oscillations 2.3 Non-linear Vibrations 2.4 Applications Dynamical Systems. Catastrophes and Chaos 1. Continuous and Discrete Dynamical Systems 1.1 Continuous Linear Dynamical Systems 1.2 Non-linear Differential Equations and Systems of Non-linear Differential Equations 1.3 Discrete Linear Dynamical Systems 2. Elements of the Theory of Catastrophes 2.1 Ramifications 2.2 Elementary Catastrophes 3. Periodic Solutions. Global Bifurcations 3.1 Periodic Solutions 3.2 Global Bifurcations 4. Fractals. Chaotic Motions 4.1 Fractals 4.2 Chaotic Motions

vii 505 505 505 537 554 566 566 597 606 612 629 630 630 648 675 682 683 689 697 698 707 712 712 723

Bibliography

739

Subject Index

759

Name Index

765

Preface

All phenomena in nature are characterized by motion. Mechanics deals with the objective laws of mechanical motion of bodies, the simplest form of motion. In the study of a science of nature, mathematics plays an important rôle. Mechanics is the first science of nature which has been expressed in terms of mathematics, by considering various mathematical models, associated to phenomena of the surrounding nature. Thus, its development was influenced by the use of a strong mathematical tool. As it was already seen in the first two volumes of the present book, its guideline is precisely the mathematical model of mechanics. The classical models which we refer to are in fact models based on the Newtonian model of mechanics, that is on its five principles, i.e.: the inertia, the forces action, the action and reaction, the independence of the forces action and the initial conditions principle, respectively. Other models, e.g., the model of attraction forces between the particles of a discrete mechanical system, are part of the considered Newtonian model. Kepler’s laws brilliantly verify this model in case of velocities much smaller then the light velocity in vacuum. Mechanics has as object of study mechanical systems. The first two volumes of this book dealt with particle dynamics and with discrete and continuous mechanical systems, respectively. The present one deals with analytical mechanics. We put in evidence the Lagrangian and the Hamiltonian mechanics, where the study of first integrals plays a very important rôle. The Hamilton–Jacobi method is widely considered, as well as the study of systems with separate variables. We mention also a thorough study of variational principles and canonical transformations. The symmetry transformations, including Noether’s theorem, lead to conservation laws. Integral invariants and exterior differential calculus are also included. A particular attention has been given to non-holonomic mechanical systems. Problems of stability and vibrations have been also considered in the frame of Lagrangian and Hamiltonian mechanics. The study of dynamical systems leads to catastrophes, bifurcations and chaos. One presents some applications connected to important phenomena of the nature and one gives also the possibility to solve problems presenting interest from technical, engineering point of view. In this form, the book becomes – we dare say – a unique outline of the literature in the field; the author wishes to present the most important aspects related with the study of mechanical systems, mechanics being regarded as a science of nature, as well as its links to other sciences of nature. Implications in technical sciences are not neglected. Concerning the mathematical tool, the five appendices in the first volume give the book an autonomy with respect to other works, special previous mathematical knowledge being not necessary. The numeration of the chapters follows that of the first ix

x

MECHANICAL SYSTEMS, CLASSICAL MODELS

two volumes, to which one makes reference for various results (theorems, formulae etc.). I am grateful to Mărgărit Baubec, Ph.D., for his valuable help in the presentation of this book. The excellent cooperation with the team of Springer, Dordrecht, is gratefully acknowledged. The book covers a wide number of problems (classical or new ones) as one can see from its contents. It uses the known literature, as well as the original results of the author and his more than 50 years experience as Professor of Mechanics at the University of Bucharest. It is devoted to a large circle of readers: mathematicians (especially those involved in applied mathematics), physicists (particularly those interested in mechanics and its connections), chemists, biologists, astronomers, engineers of various specialities (civil, mechanical engineers etc., who are scientific researchers or designers), students in various domains etc. Bucharest, Romania 7 January 2009

P.P. Teodorescu

Chapter 18 Lagrangian Mechanics The analytical methods of calculation have been introduced to can study in a systematic and unitary form the discrete mechanical systems (with a finite number of particles or with a finite number of degrees of freedom); thus, appears the denomination of “analytical mechanics”, besides that of “theoretical mechanics” (in the English, German and Russian literature) and that of “rational mechanics” (in the French or Italian literature). In “physical mechanics” stress is put on the experimental aspect, while in “technical mechanics” on the possibility to apply in technics the results obtained. In all these cases, the object of study is the same: mechanics, as a science of nature. In this order of ideas, we consider a discrete mechanical system S , formed by n particles Pi (or a finite number of rigid solids), of position vectors ri , subjected to the action of given forces Fi , i = 1, 2, ..., n (the resultants of all external and internal given forces which act upon the particle Pi ) and to certain links Lk , k = 1, 2, ..., nL . The basic problem which is put consists in the determination of the motion (position and velocity) of the mechanical system S in the time interval [t0 , t1 ] (eventually, [t0 , ∞ ) ) if Fi = Fi ( rj , rj ; t ) , Lk = Lk ( rj , rj ; t ) , ri (t0 ) = ri0

and

vi (t0 ) = vi0 , i = 1, 2, ..., n , k = 1, 2, ..., nL are given. A distinction between external and internal forces is not necessary; in exchange, the classification of the constraints and their representation in a finite or differential form, as well as their elimination from the calculation (when it is possible) have a particular importance. The elimination of the constraints and of the constraint forces from the calculation allows to obtain equations of motion which contain only the position vectors and their derivatives with respect to time. In general, one admits only frictionless constraints, which – obviously – restrains much the sphere of the problems which can by treated. Euler, in 1736, and Maclaurin, in 1742, have written the first treatises of mechanics in which they pass from purely geometric methods to methods based on differential calculus (Euler, L., 1736). The brothers Jacob I Bernoulli and Jean I Bernoulli, Daniel Bernoulli, the son of the latter one, Alexis-Claude Clairaut, Jacob Herman and PierreLouis-Moreau de Maupertuis have made important steps, considering many problems and enouncing various “principles”, which allowed to put in equation the problems of mechanics and to give their solution in a unitary form. “A special ability was always necessary to put in evidence the forces which have to be taken into consideration, so P.P. Teodorescu, Mechanical Systems, Classical Models, © Springer Science+Business Media B.V. 2009

1

2

MECHANICAL SYSTEMS, CLASSICAL MODELS

that each problem become piquant, exciting the emulation; ... the treatise on dynamics of Jean le Rond d’Alembert, appeared in 1743, has put an end to this kind of competition, offering a direct and general method to solve or – at least – to put in equation all the problems of dynamics which one could imagine”. These affirmations have been made by Joseph-Louis Lagrange in his famous treatise “Mécanique analytique”, which appeared in Paris, in 1788, in which mechanics is presented as an abstract science, as a chapter of mathematical analysis, which can be studied without using any figure (Lagrange,J.L., 1788). Thus, the theory of systems of ordinary differential equations, the variational calculus, the theory of punctual and canonical transformations, the theory of Pfaff forms, the theory of integral invariants etc. have been developed, the impulse given by Lagrange to the study of mechanical systems having a particular importance for the development of analysis, as well as of other chapters of mathematics. We consider, in the frame of this chapter, a discrete mechanical system S , formed – in general – by a finite number of particles and by a finite number of rigid solids (as it has been considered in Sect. 17.1.1.1); hence, the system has a finite number of degrees of freedom. For the sake of simplicity, we assume – so as it is made in any classical study, without loss of generality – that the system S is formed by a finite number of a particles; when it is necessary or when intervene particular data we refer to rigid solids too. Thus, after some preliminary results, we will do a detailed study of Lagrange’s equations of motion, which will be then applied to some important problems (Appell, P., 1941–1953; Goldstein, H., 1956; Lamb, H., 1929; Landau, L.D. and Lifchitz, E., 1960; La Valée-Poussin, Ch.J. de, 1925; Lurie, A.I., 2002; Mercier, A., 1955).

18.1 Preliminary Results First of all, we will present some introductory motions, which allow to pass from the representation of the discrete mechanical system S in the space E 3 to its representation by means of a representative point P in the representative space E 3 n . Passing in review the differential principles of mechanics, we can study the motion of the mechanical system S in the new considered space.

18.1.1 Introductory Notions In what follows we use the notions of displacement for a mathematical representation of the nL constraints Lk , k = 1, 2,..., nL ; this allows to introduce the notion of representative space E 3 n . A particular attention is given to the natural systems and to the particle subjected to non-holonomic constraints. 18.1.1.1 Displacements. Constraints One of the most important notions in the frame of the analytical methods is that of displacement; this notion has been introduced and considered at large in Sect. 3.2.1.1, putting in evidence the real displacements dri , the possible displacements Δri and the

Lagrangian Mechanics

3

virtual displacements δri of the particle Pi , of position vector ri , i = 1, 2,..., n . Starting from the displacements, one can define the real velocities vi = dri / dt , the possible velocities vi = Δri / Δt and the virtual velocities vi∗ = δri / Δt . In case of a particle subjected to stay on a fixed curve C or on a fixed surface S (steady case), the virtual displacement is directed along the tangent to the curve or is contained in the plane tangent to the surface at the point P , respectively; in both cases, the real displacement belongs to the (finite or infinite) set of virtual displacements. If the rigid S , e.g., is movable, then the virtual displacements take place in the tangent plane to this surface at the moment t (we consider the surface to be frozen at this moment), while the real displacement links the particle P situated at the point P of this surface at this moment t to the particle P situated at the point P of the surface at the moment t + dt ; hence, in a non-steady case, the real displacement does no more belong to the set of virtual displacements of the considered particle. This statement can be made in the case of a deformable surface S too. The constraints to which is subjected the discrete mechanical system S can be external or internal; in the following, this distinction is not useful. As well, we make distinction between unilateral (expressed by inequalities) and bilateral (represented in the form of equalities) constraints; in analytical mechanics, we will consider all the constraints to be – in general – bilateral. Another distinction is made between constraints of contact and constraints at distance; both these types of constraints can be expressed by equalities. A large study of the constraints is made in Chap. 3, Sect. 2.2. Another classification of the constraints, after Hertz, puts in evidence the holonomic (finite, of geometric nature) and the non-holonomic constraints (infinitesimal or differential, of kinematic nature). After Lothar Bolzmann, the constraints which do not depend explicitly on time are called scleronomic (stationary) constraints, those which vary in time being rheonomous (non-stationary) constraints. As a matter of fact, taking into account these classifications, one obtains a mathematical representation of the constraints. We mention also the distinction between ideal (perfect, smooth) constraints and constraints with friction (real). Analytical mechanics has been developed for discrete mechanical systems; some results can be adapted to the case with friction too. In the absence of the constraints Lk , k = 1, 2, ..., nL , the discrete mechanical system

S is a free mechanical system. The axiom of liberation from constraints states that there exists always a system of forces Ri , which is applied upon the mechanical system S subjected to constraints, so that each particle Pi can be treated as a free particle subjected to the action of the force Fi + Ri , i = 1, 2,..., n ; consequently, the mechanical system S can be studied as a free one. The forces Fi are given (known) forces, while the forces Ri are constraint (unknown) forces. The real elementary work dW of the given forces Fi is given by the relation (3.2.3) and the corresponding virtual work δW by the relation (3.2.3'); as well, the real elementary work dWR of the constraints forces Ri is given by (3.2.7), the corresponding virtual work δWR being of the form (3.2.7').

4

MECHANICAL SYSTEMS, CLASSICAL MODELS

The p holonomic constraints are expressed in the form (3.2.8) if they are rheonomous and in the form (3.2.33') if they are scleronomic, the number s = 3n − p representing the number of degrees of geometric freedom of the discrete mechanical system S ; the motion takes place only if p < 3n (hence, s > 0 ) . We notice that a free rigid solid, component of a discrete mechanical system S , introduces 6 degrees of freedom. The m non-holonomic constraints are represented – in general – by means of the Pfaff differential forms in the form (3.2.13) (as well in the form (3.2.13')), with the aid of the real displacements, or in the form (3.2.15) (analogously, in the form (3.2.15')), by means of the virtual displacements. In case of scleronomic constraints, the relations (3.2.32) must also take place; otherwise, the constraints are rheonomous. The number r = 3n − m represents the number of kinematic degrees of freedom of the discrete mechanical system S . In case of holonomic constraints, we have r = s . The total number of constraints is, in general, nL = p + m , while the number of degrees of freedom (obviously, kinematic degrees of freedom, because the holonomic constraints involve the same number of geometric and kinematic degrees of freedom) is 3n − nL . The non-holonomic constraints of the form (3.2.12), considered above, have been called non-holonomic constraints of the first kind by G. Hamel; he introduces also non-holonomic constraints of the second kind, where appear the accelerations of the particles too, in the form f (drj , d2 rj ; dt ) = 0 ,

(18.1.1)

the dimensional homogeneity being ensured. A constraint relation of the form (3.2.13) corresponds to a holonomic constraint if it is an integrable differential Pfaff form; otherwise it corresponds to a non-holonomic constraint. The Theorem 3.2.2 of Frobenius gives the necessary and sufficient conditions, so that such a differential form of first degree be complete integrable (does correspond to an exact differential or does admit an integrable factor). We notice that the two classifications of the constraints (holonomic or non-holonomic and scleronomic or rheonomous) are independent; we can have, e.g., non-holonomic and rheonomous constraints (the most general case) or holonomic and scleronomic constraints (the most particular case). A mechanical system subjected to holonomic or non-holonomic constraints is called a holonomic or non-holonomic mechanical system, respectively; as well, a mechanical system subjected to scleronomic or rheonomous constraints is called scleronomic or rheonomous mechanical system, respectively. In case of scleronomic constraints, the real displacements belong to the set of virtual displacements. As a matter of fact, to have this property it is sufficient that the conditions αk 0 = 0, k = 1,2,..., m , be fulfilled in the relations (3.2.13) (it is not necessary that the constraints by scleronomic); such constraints are called catastatic. As examples of mechanical systems subjected to holonomic and scleronomic constraints we mention the particle compelled to stay on a fixed curve or on a fixed surface, as well as the rigid solid (or a non-deformable discrete mechanical system). If the curve or the surface are movable (rigid and movable with respect to a fixed frame of

Lagrangian Mechanics

5

reference or even deformable), then the constraint is holonomic and rheonomous. We notice that a rigid solid behaves – in a certain sense – as a particle which has not three but six degrees of freedom. Non-holonomic constraints are, e.g., those corresponding to the motion of a heavy homogenous sphere (see Sect. 17.1.2.8) or of a heavy circular disc (see Sect. 17.1.2.9) on a horizontal plane; we mention the motion of a rigid skate on the ice field too, case considered in Sect. 3.2.2.6. An important class of constraints, in the frame of which analytical mechanics has been developed, is the class of ideal constraints. We call ideal constraints those ones for which the virtual work of the constraint forces vanishes δWR =

n

∑ Ri

i =1

⋅ δri = 0

(18.1.2)

for any system of virtual displacements of the considered mechanical system S . If the virtual displacements δri are arbitrary (we have not constraints), then the relation (18.1.2) is fulfilled if and only if Ri = 0, i = 1, 2, ..., n . But if p holonomic constraints of the form (3.2.8) and m non-holonomic constraints of the form (3.2.13) take place, then we can use the method of Lagrange’s multipliers, the constraint forces being of the form (3.2.37). In this case, the real elementary work of the constraint forces is given by (3.2.37'); this work vanishes if fl = 0, l = 1, 2, ..., p , and αk 0 = 0, k = 1, 2, ..., m (conditions in which the real displacements belong to the set of virtual displacements, case of catastatic constraints), in particular in case of scleronomic constraints. Among the mechanical systems subjected to ideal constraints we mention the particle subjected to constraints of contact, frictionless (compelled to stay on a fixed or movable curve or surface), or the discrete mechanical system for which the distance between two particles is function only on time (constraints at distance). The classical ideal constraints of the rigid solid are: the simple support, the hinge and the built-in mounting. Ideal constraints are also the constraints of the rigid solid which can slide without friction along an axis (or, in general, along a superposable curve), the constraints of a rigid solid which can slide frictionless on a plane (or on a given surface), as well as the constraints of the rigid solid which is rolling or pivoting without sliding on a plane (or on a arbitrary surface); in the first cases, the virtual displacement is normal to the constraint force and in the last cases the virtual displacement is zero. We mention also the systems of two rigid solids considered in Sect. 3.2.2.9. As well, a mechanism (a discrete mechanical system formed of two rigid solids linked rigidly or by hinges or by perfectly smooth or rough surfaces) can be considered as a mechanical system subjected to ideal constraints. We mention also the constraints by threads, which are not always bilateral. Among the constraints with friction, we consider that one which corresponds to the sliding friction, for which the analytical methods of mechanics could be adapted. We assume that the virtual displacement δr takes place in an interval of time δt (which can be the interval of time Δt in which take place the possible displacements, the difference of which is the virtual displacement). In this case, the virtual displacement of a point of a free rigid solid with respect to the inertial frame of reference R ′ will be given by

MECHANICAL SYSTEMS, CLASSICAL MODELS

6

δr ′ = vO′∗ δt + ( ω∗ δt ) × r ,

(18.1.3)

where vΟ′∗ δt (t ) and ω∗ δt (t ) are two arbitrary vectors, which define the virtual rototranslation at the moment t , r being the position vector with respect to the non-inertial frame R ; the position of this solid depends thus on six scalar parameters. We can express the virtual displacement δr ′ in the form δr ′ = δrO′ + δχ × r

(18.1.3')

too, where δχ represents a virtual rotation of the frame of reference R with respect to the frame R (of components δψ , δθ and δϕ , corresponding to Euler’s angles). In the case in which the rigid solid has a fixed point (e.g., the point O ), we have vO′∗ = 0 , its position depending only on three scalar parameters. As well, if the point O of the rigid solid has a motion given by rO′ = rO′ (t ) , then the real displacement will be given by dr ′ = vO′ dt + ( ωdt ) × r , with vO′ = drO′ / dt ; in exchange, for a virtual displacement (compatible with the constraints at the moment t ) the point O must have the position fixed at the moment t , while δr ′ = ( ω∗ δt ) × r , the rigid solid having three degrees of freedom too. The virtual work of the forces of inertia is expressed in the form δW i = − ∫∫∫ μ ( r ) a ′( r ; t ) ⋅ δr ′dV , V

(18.1.4)

where the virtual displacement δr ′ is given by (18.1.3), vO′∗ and ω∗ being arbitrary functions on time. Noting that a ′( r ; t ) represents the accelerations field in the real motion of the rigid solid, we can write, with respect to the frame of reference R ′ (we have r = r ′ − rO′ ), δW i = − { [ ( vO′∗ − ω∗ × rO′ ) δt ] ⋅ A ′ + ( ω∗ δt ) ⋅ DO′ ′ } ,

(18.1.4')

where A ′ is the dynamic resultant of the rigid solid, given by (14.1.27), while DO′ ′ is the dynamic moment with respect to the pole O ′ , given by (14.1.28). Analogously, in the case of the rigid solid, the virtual work of the constraints forces of torsor { R , MO } , will be expressed in the form

δWR = − [ ( vO′∗ − ω∗ × rO′ ) δt ] ⋅ R + ( ω∗ δt ) ⋅ MO ' = 0.

(18.1.4'')

18.1.1.2 The Representative Space E 3 n A basic aspect of the analytical methods of calculation consists in replacing the study of the motion of a discrete mechanical system S of geometric support Ω in the space E 3 by the study of the motion of a single geometric point, called representative point,

Lagrangian Mechanics

7

in another space, conveniently built up, called representative space. E.g., in Sects. 3.2.2.2 and 11.1.1.5 has been introduced the representative space E 3 n , by passing from the mechanical system S of n particles Pi , i = 1, 2, ..., n , to a representative point P , of generalized co-ordinates X k , k = 1, 2, ..., 3n , with the aid of the relations Xk = x j , k = 3(i − 1) + j , i = 1, 2,..., n , j = 1, 2, 3 , (i )

(18.1.5)

of the form (3.2.9), where x (j i ) is the component along the Ox j -axis of the position vector ri ; the real generalized displacements of this point are dXk , the possible generalized displacements are ΔXk and the virtual generalized displacements are δXk , k = 1, 2, ..., 3n , the respective properties corresponding to those in the space E 3 . In these conditions, the constraints of the mechanical system S become constraints of the representative point P . We use also the notations (i ) bj 0 = α j 0 , bjk = α , k = 3(i − 1) + l , jl

i = 1, 2,..., n , l = 1,2, 3 ,

(18.1.5')

of the form (3.2.9), where α(jli ) are the components along the Oxl -axis of the given vectors α ji , which specify the constraints relations (3.2.13); these relations take the form ωj ≡

corresponding

to

the

3n

∑ bjk dXk

k =1

+ bj 0 dt , j = 1, 2,..., m ,

relations (3.2.13''), where

(18.1.6)

bjk = bjk ( X1, X 2 ,..., X 3 n ; t ),

k = 0,1,..., 3n , or the form 3n

∑ bjkVk

k =1

+ bj 0 = 0, j = 1, 2,..., m ,

(18.1.6')

corresponding to the relations (3.2.13'''), where Vk = X k = dXk / dt , k = 1, 2,..., 3n , are the components of the generalized velocity of the point P in the representative space E 3n . Obviously, we assume that the constraints are linearly independent, hence that the matrices [bjk ] are of rank m . If a Pfaff differential form ω j is not completely integrable, then the respective constraint is non-holonomic; taking into account the form (18.1.6') of these relations, then the constraint is called kinematic too. We assume that, in general, the mechanical system S has m non-holonomic constraints (the representative point P is on a non-holonomic manifold with 3n − m dimensions, at

MECHANICAL SYSTEMS, CLASSICAL MODELS

8

the intersection of m non-holonomic hypersurfaces in the space E 3n ). We can express these constraint relations also in the form 3n

∑ bjk δXk

k =1

= 0, j = 1, 2,..., m ,

(18.1.7)

corresponding to the relations (3.2.15''), by means of the virtual displacements, or in the form 3n

∑ bjkVk*

k =1

= 0, j = 1, 2,..., m ,

(18.1.7')

corresponding to the relations (3.2.15'''), where Vk* = δX k / Δt are the virtual generalized velocities ( Δt is the interval of time in which takes place a possible displacement). If bj 0 = 0, j = 1, 2,..., m , then the mechanical system is called catastatic. In this case, the set of virtual generalized displacements coincides with the set of possible generalized displacements and the constraints are called catastatic; as well, the set of virtual generalized velocities (which coincides with the set of possible generalized velocities) contains also the case in which these velocities vanish, hence the case of rest with respect to an inertial frame of reference (the denomination used above being thus justified). A mechanical system which is not catastatic is called non-catastatic; in this case bj 0 ≠ 0 . If the conditions bj 0 = 0, bjk =

∂bjk ∂t

= 0, j = 1, 2,..., m , k = 1, 2,..., 3n ,

(18.1.8)

are verified, then the constraints do not depend explicitly on time and are called scleronomic; otherwise, the constraints are rheonomous. We notice that a scleronomic mechanical system is catastatic too, but a catastatic one may be rheonomous too; a noncatastatic mechanical system is always rheonomous. If a Pfaff differential form is completely integrable (for a system of the form (18.1.6) we can use the Theorem 3.2.2 of Frobenius), then the corresponding constraint is holonomic, being of the form fl ( Xk ; t ) ≡ fl ( X1 , X 2 ,..., X 3 n ; t ) = 0, l = 1, 2,..., p ,

(18.1.9)

hence of the form (3.2.8''); such a constraint is also called geometric. In general, we assume that the mechanical system S has p holonomic constraints (the representative point P is on a manifold with 3n − p dimensions, at the intersection of p hypersurfaces in the space E 3 n ). These finite relations can be written also in the differential form 3n

∂f

∑ ∂Xlk

k =1

dXk + fl dt = 0, l = 1, 2,..., p ,

(18.1.10)

Lagrangian Mechanics

9

or in the form ∂f

3n

∑ ∂Xlk Vk

k =1

+ fl = 0, l = 1, 2,..., p ,

(18.1.10')

so that ∂fl = blk , l = 1, 2,..., p , k = 1, 2,...3n . ∂Xk

Analogously, using the virtual generalized displacements, we can write 3n

∂f

∑ ∂Xlk

k =1

δXk = 0, l = 1, 2,..., p ,

(18.1.11)

or 3n

∂f

∑ ∂Xlk Vk*

k =1

= 0, l = 1, 2,..., p .

(18.1.11')

We assume that also these constraints are linearly independent, the matrices [ ∂fl / ∂Xk ] being of rank p . If fl = 0, l = 1,2,..., p , the mechanical system S is catastatic; at the same time we have ∂f ∂ ⎛⎜ l ⎞⎟ ∂ ∂f ⎝ Xk ⎠ = l =0 ∂t ∂Xk

too, so that the respective system is also scleronomic. We notice that these two properties of the constraints coincide in case of the holonomic systems, for which fl ( Xk ) ≡ fl ( X1, X 2, ..., X 3 n ) = 0 .

(18.1.12)

Let be two neighbouring, possible, simultaneous (at the moment t ) positions P ( Xk ) and P '( Xk + δXk ) , of the same representative point of the holonomic discrete mechanical system S . The constraint relations will be of the form fl ( Xk ) = 0 , fl ( Xk + δXk ) = 0 , l = 1, 2,..., p . Expanding into a Taylor series, we can write fl ( Xk ) +

3n

∂f

∑ ∂Xlk

k =1

δXk + ... = 0 .

Neglecting the terms of higher order and taking into account the constraint relations,

MECHANICAL SYSTEMS, CLASSICAL MODELS

10

we find the conditions (18.1.11). One observes thus that δXk represent those differential generalized displacements which must be effected by the representative point to pass from a position to another one, at the same moment t , being thus virtual displacements (corresponding to their definition, as difference of possible displacements). If Fi = Fj( i ) i j represents the resultant of the given forces which act upon the particle Pi , i = 1, 2,..., n , in the space E 3 , then we can introduce, in the space E 3n , the given

generalized forces (shortly, the generalized forces) Qk , k = 1, 2,..., 3n , which act upon the representative point P , in the form Qk = Fj , k = 3(i − 1) + j , i = 1,2,..., n , j = 1,2, 3, k = 1,2,..., 3n . (i )

(18.1.13)

As well, by using the axiom of liberation from constraints, one introduces the constraint forces Ri = R(j i ) i j , j = 1, 2,..., n , which act upon the particle Pi in the space E 3 ; we are thus led to constraint generalized forces Rk , k = 1, 2,....3n , which act in the space E 3n upon the representative point P in the form Rk = Rj , k = 3(i − 1) + j , i = 1,2,..., n , j = 1, 2, 3 . (i )

(18.1.13')

In what concerns the mass mi of the particle Pi , it is convenient to denote mi = mi

(1)

= mi

(2 )

= mi

(3)

, i = 1,2,..., n ,

so that M k = m j , k = 3(i − 1) + j , i = 1, 2,..., n , j = 1, 2, 3, k = 1,2,..., 3n , (i )

(18.1.14)

can be considered as 3 n generalized masses of the representative point P in the space E 3n . The real elementary work of the given generalized forces is given by dW =

n

3

∑ ∑ Fj( i ) dx (j i )

i =1 j =1

=

3n

∑ Qk dXk ,

(18.1.15)

k =1

while the real elementary work of the constraint generalized forces is expressed in the form

dWk = Analogously,

n

3

∑ ∑ R(j i ) dx (j i )

i =1 j =1

=

3n

∑ Rk dXk

k =1

.

(18.1.15')

Lagrangian Mechanics

11 n

3

∑ ∑ Fj( i ) δX (j i )

δW =

i =1 j =1

=

3n

∑ Qk δXk

(18.1.16)

k =1

is the virtual work of the given generalized forces, while δWR =

n

3

∑ ∑ R(j i ) δX (j i )

i = 1 j =1

=

3n

∑ Rk δXk

(18.1.16')

k =1

represents the virtual work of the constraint generalized forces.

18.1.1.3 Natural Systems We have seen in Sects. 1.1.1.12 and in 6.1.1.2 that a system of forces (corresponding, e.g., to a field of forces), which admits a simple potential (or quasi-potential) or a generalized potential (or quasi-potential) is called a natural system. A system of forces Fi = ∇ iU = Fj( i ) i j , ∇ i = i j

∂ ∂x (j i )

, i = 1,2,..., n ,

the components of which (i )

Fj

=

∂U ∂x (j i )

, i = 1, 2,..., n , j = 1, 2, 3 ,

(18.1.17)

derive from a simple potential, forms a system of conservative forces; to a simple quasi-potential of the form U = U ( r1 , r2 ,..., rn ; t ) corresponds a system of quasiconservative forces. Analogously, if Fj( i ) =

∂U ∂x (j i )

−

d ∂U , i = 1, 2,..., n , j = 1, 2, 3 , dt ∂x ( i ) j

(18.1.17')

where U = U ( r1 , r2 ,..., rn , r1 , r2 ,..., rn )

is a generalized potential, then the system of forces is conservative; in case of the generalized quasi-potential U = U ( r1 , r2 ,..., rn , r1 , r2 ,..., rn ; t )

the system of forces is quasi-conservative. In the representative space E 3n , the system of generalized forces is of components Qk =

∂U , k = 1, 2,..., 3n , ∂Xk

(18.1.18)

MECHANICAL SYSTEMS, CLASSICAL MODELS

12

in case of a simple potential (system of conservative generalized forces) U = U ( X1 , X 2 ,..., Xn )

or of a simple quasi-potential (system of quasi-conservative generalized forces) U = U ( X1 , X 2 ,..., X n ; t )

or of components Qk = [U ]k ≡

∂U d ∂U − , k = 1, 2,..., 3n , ∂Xk dt ∂Xk

(18.1.18')

in case of a generalized potential (system of conservative generalized forces) U = U ( X1 , X 2 ,..., X n , X 1 , X 2 ,..., X n )

or in case of a generalized quasi-potential (system of quasi-conservative generalized forces) U = U ( X1 , X 2 ,..., Xn , X 1 , X 2 ,..., X n ; t ) ;

we have used the Euler-Lagrange derivative corresponding to the index k , given by [

]k

=

∂ d ∂ − , k = 1, 2,..., 3n . ∂Xk dt ∂Xk

(18.1.19)

Noting that the generalized forces do not depend explicitly on the generalized accelerations d2 Xk Xk = , k = 1, 2,..., 3n , dt 2

it results that the generalized potential (quasi-potential) is of the form U =

3n

∑U j X j j =1

+ U0 ,

where

U j = U j ( X1 , X 2 ,..., Xn ) is a vector potential, while U 0 = U 0 ( X1 , X 2 ,..., Xn )

(18.1.20)

Lagrangian Mechanics

13

is a scalar potential, or where

U j = U j ( X1 , X 2 ,..., Xn ; t ) is a vector quasi-potential, while U 0 = U 0 ( X1 , X 2 ,..., Xn ; t )

is a scalar quasi-potential in the space E 3 n . Replacing in (18.1.18'), we obtain Qk =

3n

⎛ ∂U j

∑ ⎜⎝ ∂Xk j =1

−

∂U k ∂X j

∂U 0 ⎞  ⎟X j − U k + ∂X , k = 1, 2,..., 3n , k ⎠

(18.1.21)

in case of the quasi-conservative generalized forces; if the forces are conservative, then we have U k = 0 . Replacing the vector potential U j by the vector potential Uj = Uj +

∂ϕ , ϕ = ϕ ( X1 , X 2 ,..., Xn ; t ) , ∂X j

and taking into account that one can invert the order of application of the operators

∂ d d ∂ = ∂Xk dt dt ∂Xk for a function of class C 2 , we get the same generalized function Qk ; hence, the vector quasi-potential is determined leaving aside a field of gradients in the space E 3n . If to this transformation of vector quasi-potential U j → U j we associate the transformation of scalar quasi-potential U 0 → U 0 = U 0 + ϕ , hence if we effect the gauge transformation (of scale) U →U =U +

dϕ , dt

then the form (18.1.21) of the quasi-conservative generalized force remains invariant. A field of conservative generalized forces is stationary, while a field of quasi-conservative generalized forces is non-stationary. The elementary work of the conservative generalized forces which derive from a simple potential is an exact differential dW = dU ,

(18.1.22)

the work depending only on the extreme positions of the representative point P Wq = U ( X11 , X 21 ,..., X 31n 0 1 P P

) − U ( X10 , X20 ,..., X 30n ) .

(18.1.22')

MECHANICAL SYSTEMS, CLASSICAL MODELS

14

If the trajectory of the point P is a closed hypercurve C in E 3 n , then the corresponding work vanishes (WC = 0 ). If these generalized forces result from a generalized potential, then the elementary work is a total differential dW = dU 0

(18.1.22'')

too, the scalar potential U 0 playing the rôle of the simple potential in the frame of the generalized potential (18.1.20), while the vector potential has no contribution concerning this elementary work; all the considerations made in case of the simple potential remain thus valid. In case of quasi-conservative generalized forces, which derive from a simple quasi-potential, the elementary work is no more a total differential, being of the form dW = dU − Udt ,

(18.1.23)

where we put in evidence the partial derivatives with respect to time. If the generalized forces result from a generalized quasi-potential, then we get dW = dU 0 − U 0 dt − ⎛ 3n ⎞ = dU 0 − ⎜ ∑ U kVk + U 0 ⎟ dt = ⎝ k =1 ⎠

3n

∑Uk dXk

k =1 3n

∂U

∑ ⎛⎜⎝ ∂Xk0

k =1

⎞ − U k ⎟ dXk . ⎠

(18.1.23')

We see, easily, that the transformations mentioned above for the generalized potential have no influence on this elementary work. We notice that a system of conservative generalized forces derives always from a generalized potential or from a simple potential as it depends or not explicitly on the generalized velocities of the representative point P in E 3 n . Introducing the potential energy V = −U , in case of a simple potential, or the potential energy V = −U 0 , in case of a generalized potential, we can write Wq = V0 − V1 = − ΔV , V = − Wq + V0 , 0 1 0 1 P P

P P

(18.1.24)

where ΔV is the variation of the potential energy. Consequently, the potential energy of a discrete mechanical system, acted upon by conservative generalized forces (natural mechanical system), is equal to the work with changed sign, effected by the generalized forces, beginning with the initial moment (excepting an arbitrary constant V0 , which represents the potential energy at the initial moment; often, one chooses V0 so that the minimum of the potential energy be equal to zero). The generalized forces which do not derive from a simple quasi-potential can be expressed in the from

Qk =

∂U + Qk , U = U ( X1 , X2 ,..., Xn ; t ) , ∂Xk

the elementary work being given by

(18.1.25)

Lagrangian Mechanics

15 ⎛ 3n ⎞ dW = dU + ⎜ ∑ QkVk − U ⎟ dt . ⎝ k =1 ⎠

(18.1.25')

In the case in which U = 0 (U = U ( X1 , X 2 ,..., Xn ) is a potential), the quantity 3n

∑ QkVk ,

k =1

of the nature of a power, represents the power of the non-potential generalized forces. The non-potential generalized forces Qk of a vanishing power 3n

∑ QkVk

k =1

=0

are called gyroscopic generalized forces, depending – obviously – on the distribution of the generalized velocities. The generalized forces Qk are, in this case, conservative; if the power of the non-potential generalized forces is non-zero, then these forces are non-conservative. The non-conservative generalized forces of negative power 3n

∑ QkVk

k =1

<0

are called dissipative generalized forces, because – in this case – the energy diminishes (we take into account (18.1.25') and notice that U = −V ). The forces (18.1.21) which admit a generalized quasi-potential are, as well, of the form (18.1.25), where we take U = U 0 and where Qk =

3n

⎛ ∂U j

∑ ⎜⎝ ∂Xk j =1

−

∂U k ⎞  X − U k , k = 1, 2,..., 3n . ∂X j ⎟⎠ j

(18.1.26)

If neither the vector potential does not depend explicitly on time (Uk = 0 ), then it results 3n

∑ Qk X k

k =1

= 0,

the non-potential generalized forces being thus gyroscopic. Hence, any generalized force which admits a generalized potential is the sum of a generalized force which admits a simple potential and a gyroscopic force. If upon a mechanical system S act only gyroscopic generalized forces, then we have n

∑ Fi

i =1

⋅ vi =

3n

∑ QkVk

k =1

=0

and reciprocally. As well, the condition of dissipation becomes

(18.1.27)

MECHANICAL SYSTEMS, CLASSICAL MODELS

16 n

∑ Fi

i =1

⋅ vi =

3n

∑ QkVk

k =1

<0 .

(18.1.27')

18.1.1.4 Case of a Single Particle In case of a single particle P we have n = 1 , the representative space being just the space E 3 , of co-ordinates x1 , x 2 , x 3 . Using the notations in Sect. 18.1.1.2, we can write the constraint relations, in general, in the form ωα ≡ bα j dx j + bα 0 dt = 0, α = 1, 2 ,

(18.1.28)

ωα , α = 1, 2 , being differential forms of first degree; we consider, in general, that

bα j = bα j (x1 , x 2 , x 3 ; t ), j = 0,1, 2, 3 . We may write bα j v j + bα 0 = 0, α = 1, 2 ,

(18.1.28')

too. If these forms are integrable (see Sect. 3.2.2.6 too), then the constraints are holonomic, so that fk (x1 , x 2 , x 3 ; t ) = 0, k = 1, 2 ,

(18.1.29)

relations which can be written also in the form fk , j dx j + fk dt = 0, k = 1, 2 ,

(18.1.29')

fk , j v j + fk = 0, k = 1, 2 .

(18.1.29'')

or in the form

If bα 0 = 0, α = 1, 2 , then the constraints are catastatic, while if bα j = 0 , α = 1,2 ,

j = 1, 2, 3 , too, then the constraints are scleronomic; otherwise, they are rheonomous. In case of a holonomic constraint, the conditions to be a scleronomic one too are reduced to fk = 0, k = 1, 2 . As in the general case, the constraint relations can be expressed also by means of the virtual displacements in the form bα j δx j = 0, α = 1, 2 ,

(18.1.30)

bα j v ∗j = 0, α = 1, 2 .

(18.1.30')

or in the form

The relations (18.1.30) and (18.1.30') coincide with the relations (18.1.28) and (18.1.28'), respectively, in case of catastatic constraints (the set of virtual displacements

Lagrangian Mechanics

17

coincides with the set of possible constraints, while the real displacements belong to these sets). A simple example of particle subjected to a non-holonomic constraint is represented by a particle P2 which has a curvilinear and uniform motion of velocity v2 , collinear JJJJG with the vector P2 P1 , tangent to the trajectory, the point P1 having a rectilinear and uniform motion of velocity v1 (the problem of meeting; see Sect. 10.2.1.2). Starting from the relations x1′ = x1 , x 2′ = v1t + x 2 , which allow to pass from a movable frame of reference to a fixed one, and from the relation x1 = (v1 + x2 )

x1 , x2

one can write the constraint relation

( x 2′ − v1t ) dx1′ − x1′dx 2′ = 0 ,

(18.1.31)

which is non-holonomic. In case of scleronomic constraints bα j dx j = 0, bα j = bα j (x1 , x 2 , x 3 ; t ), α = 1, 2, j = 1, 2, 3 ,

(18.1.32)

the conditions of integrability are of the form (3.2.30'). The integrant factors λα = λα (x1 , x 2 , x 3 ), α = 1, 2 , must exist, so that λ1 b1 and λ2 b2 , bα = b α j i j ,

α = 1,2 , be gradients, hence curl (λα bα ) = λα curl bα + grad λα × bα = 0 ; a scalar product by bα leads to the conditions of integrability bα ⋅ curl bα =∈ijk bαi bαk , j = 0, α = 1, 2 .

(18.1.32')

If, in particular, curl bα = 0, α = 1, 2 , the conditions of integrability are fulfilled and we can take λα = const. Pfaff showed that, by a convenient transformation of coordinates, the non-integrable constraint relation (18.1.32) can be expressed in the form x 3 d x 1 − dx 2 = 0 .

(18.1.33)

We can see, easily, that the relation (18.1.32') does not hold. If bα = λα gradfα , α = 1, 2 , where λα is a scalar field of class C 1 , if fα is a scalar field of class C 2 and if λα and fα are functional independent, then the vector field bα is called biscalar; in particular, if λα and fα are functional dependent, then bα is a potential field. The necessary and sufficient condition that the vector field bα be locally biscalar is (18.1.32'); in this case, the scleronomic relations (18.1.32) are completely integrable (are locally exact or admit a locally integrant factor). J. Bertrand considered also the field lines

MECHANICAL SYSTEMS, CLASSICAL MODELS

18

ϕα ( x1 , x 2 , x 3 ) = cα′ , ψα (x1 , x 2 , x 3 ) = cα′′ , cα′ , cα′′ = const, α = 1, 2 ,

of the curl bα (vortex lines), expressed by means of the first integrals ϕα and ψα ; we notice that grad ϕα ⋅ curl bα = 0 , grad ψα ⋅ curl bα = 0 , α = 1,2 . Associating the relations (18.1.32') too, one sees that the vectors bα , grad ϕα , grad ψα are coplanar, so that bα = λα gradϕα + μα gradψα , λα2 + μα2 ≠ 0, α = 1, 2 .

In this case, bα ⋅ dr = bα j dx j = λα gradϕα ⋅ dr + μα gradψα ⋅ dr = λα dϕα + μα dψα , α = 1, 2 ,

and one can show that the ratio λα , μα ≠ 0 , μα

depends only on the first integral ϕα and ψα . Let be a particle P subjected to a holonomic and scleronomic constraint f (x1 , x 2 , x 3 ) = const ; in this case, starting from a given point (let be from the origin, to fix the ideas), one can reach only the points on a two-dimensional manifold (the points of the surface f (x1 , x 2 , x 3 ) = f (0, 0, 0) ). If the constraint is non-holonomic and scleronomic (e.g., of the form (18.1.33)), then we can find a path which leads from the origin to an arbitrary point (x1 , x 2 , x 3 ) . Let be thus the path x 2 = f (x1 ) , x 3 = f ′(x1 ) , f (x ) ∈ C 3 ;

the constraint relation (18.1.33) is satisfied and we must choose the function f (x1 ) with the properties: f (0) = 0 , f ′(0) = 0 , f (x1 ) = x 2 , f ′(x1 ) = x 3 .

One sees easily, geometrically, that it exists an infinity of such functions, e.g., the functions of the family x x 2 f ( x1 )= ⎢⎡ 3x 2 − x 3 x1 − ( 2x 2 − x 3 x1 ) 1 ⎥⎤ ⎛⎜ 1 ⎞⎟ x 1 ⎦ ⎝ x1 ⎠ ⎣ πx + C 1 x12 ( x1 − x1 )2 + C 2 sin ⎛⎜ 1 ⎞⎟ , ⎝ x1 ⎠

(18.1.34)

Lagrangian Mechanics

19

where C 1 and C 2 are arbitrary constants. Hence, starting from a given point, a particle P with 3 − 2 = 1 degrees of freedom can reach only the points of a surface (a twodimensional manifold), in case of a holonomic constraint, while, in case of a non-holonomic constraint, one can reach any point in E 3 (a three-dimensional

manifold). These considerations remain valid also in case of rheonomous constraints. In case of a particle P subjected to two constraints (hence, which has 3 − 2 = 1 degrees of freedom) one can make analogous considerations. If both constraints are holonomic. then one can reach – starting from a given position – only the points of a curve (a unidimensional manifold), while, if only one of the constraints is holonomic, one can reach the points of a surface (a two-dimensional manifold); if both constraints are non-holonomic, then one can reach any point E 3 (a three-dimensional manifold). The results obtained above put in evidence that a non-holonomic constraint leads to a greater possibility of motion (it is less strict) that a holonomic constraint. If the resultant F of the forces which act upon the particle P derive from a simple quasi-potential U = U (x1 , x 2 , x 3 ; t ) , then we can write F = grad U , curl F = 0 ,

(18.1.35)

while if the force F derives from a generalized quasi-potential U = U ⋅ v + U 0 , U = U (x1 , x 2 , x 3 ; t ) , U 0 = U 0 (x1 , x 2 , x 3 ; t ) ,

then one has F = gradU 0 + v × curl U − U .

(18.1.35')

In general, a force F = gradU + F , U = U ( x1 , x 2 , x 3 ; t ) , is non-conservative; if U = U (x1 , x 2 , x 3 ; t ) and F ⋅ v = 0 , then the force F is gyroscopic. E.g., a force of the

form

F = v × curl U , U = U ( x1 , x 2 , x 3 ; t ) (corresponding to the vector potential in

(18.1.35')), is a gyroscopic force. As an example of force which derives from a generalized quasi-potential we mention the force F exerted upon a particle P in an electromagnetic field { E, B } , where E is the intensity of the electric field, while B is the magnetic induction. If q is the electric charge of the particle P , then we can write F = q (E + v × B) ,

(18.1.36)

where q E is the force exerted in the electric field, while q v × B is the force exerted in the magnetic field (Lorentz’s force); the force F is called the generalized force of Lorentz. The magnetic induction is of the form B = curl A ,

(18.1.36')

MECHANICAL SYSTEMS, CLASSICAL MODELS

20

where A = A (x1 , x 2 , x 3 ; t ) is the magnetic vector quasi-potential, while the intensity of the electric field is of the form

(

)

 + grad A0 , E= − A

(18.1.36'')

where A0 = A0 (x1 , x 2 , x 3 ; t ) is a scalar quasi-potential. Replacing in (18.1.36), it results

(

)

F = −q gradA0 + q v × curl A − A ,

(18.1.37)

the force deriving from the quasi-potential U = q ( A ⋅ v − A0 ) .

(18.1.37')

We notice that Lorentz's force q v × B is a gyroscopic one.

18.1.2 Differential Principles of Mechanics In what follows, let us present some important differential principles of mechanics (Newton’s, d’Alembert’s, Gauss’s and Hertz’s principles), from which derive the equations of motion of the considered discrete mechanical system S in E 3 and of the representative point P in the representative space E 3 n , respectively; a special attention is given to the principle of virtual work. The results thus obtained will be illustrated by some applications.

18.1.2.1 Newton’s and d’Alembert’s Principles. General Theorems. Conservation Theorems As we have seen in Sect. 11.1.2.10, the equations of motion of a free discrete mechanical system S , free or subjected to constraints, are written in the form (11.1.8) or in the form (11.1.52), respectively, corresponding to the second principle of Newton, considered as the first differential principle of mechanics. Passing to the representative space E 3 n , the equation of motion of the representative point P will be of the form M k Xk = Qk , k = 1, 2,..., 3n ,

(18.1.38)

in case of a free point P , or of the form M k Xk = Qk + Rk , k = 1, 2,..., 3n ,

(18.1.38')

in case of a point P subjected to constraints, respectively. This differential principle can be expressed also in the form

Lagrangian Mechanics

21

d ( M k X k dt

) = Qk

+ Rk , k = 1, 2,..., 3n ,

(18.1.38'')

corresponding to the form given to this principle by Newton. The Cauchy–Lipschitz theorem, stated in Sect. 11.1.1.5 (where we have used adequate notations for the generalized force) ensures the existence and the uniqueness of the solution. Introducing the generalized forces of inertia Qki = − M k Xk , k = 1, 2,..., 3n ,

(18.1.39)

the law of motion becomes Qk + Qki + Rk = 0, k = 1, 2,..., 3n ,

(18.1.40)

and we can state. Theorem 18.1.1 (d’Alembert). The motion of the representative point P subjected to constraints in the space E 3 n takes place so that, at any moment, the point is in dynamic equilibrium under the action of the given and constraint generalized force and of the generalized force of inertia. We introduce the lost generalized force of d’Alembert

Φk = Qk + Qki = Qk − M k Xk , k = 1, 2,..., 3n .

(18.1.41)

In this case, the equations (18.1.40) become

Φk + Rk = 0, k = 1, 2,..., 3n ,

(18.1.42)

and we can state Theorem 18.1.1' (d’Alembert). The motion of the representative point P subjected to constraints in the space E 3 n takes place so that, at any moment, the constraint generalized forces are equilibrated by the lost generalized forces of d’Alembert. We notice that each of the Theorems 18.1.1 and 18.1.1' can be at the basis of the Newtonian model, representing thus a differential principle of mechanics. Formally, the equations (18.1.42), which represent the necessary and sufficient conditions of dynamic equilibrium (characterizing, entirely, the motion of the representative point P subjected to constraints, hence of the dynamic discrete mechanical system S subjected to constraints), are not different from the relations Qk + Rk = 0, k = 1, 2,..., 3 n ,

(18.1.42')

which represent the necessary and sufficient conditions of static equilibrium of the very same representative point P (hence, of the same mechanical system S ). Hence, all the considerations which can be made in case of problems of statics can be transposed to similar problems with a dynamic character, replacing the given generalized force Qk

MECHANICAL SYSTEMS, CLASSICAL MODELS

22

by the lost generalized force of d’Alembert Φk ; as well, one observers that the results concerning the problems of statics can be obtained by particularization, starting from the general dynamic case. If the momentum of the representative point P is defined in the form

H =

3n

∑ M k X k ,

k =1

(18.1.43)

then we can state a theorem of momentum

H =

3n

∑ (Qk

k =1

+ Rk ) .

(18.1.43')

As well, if the moment of momentum, with respect to the origin O , of the representative point P , is given by

KO =

n

3

∑ ∑

i = 1 j ,k ,l = 1

∈jkl M 3( i −1) +l X 3( i −1) +l X 3( i −1) +l ,

(18.1.44)

where we have introduced Ricci's symbol, then the theorem of moment of momentum is written in the form

KO =

n

3

∑ ∑

i = 1 j ,k ,l = 1

∈jkl X 3( i −1) +l ⎡⎣Q3( i −1) + l + R3( i −1) +l ⎤⎦ .

(18.1.44')

Starting from these two theorems, we obtain – easily – the corresponding conservative theorems too, hence two first integrals of the considered system of differential equations. Taking into account the above observations, the equations 3n

∑ (Qk

k =1

n

+ Rk ) = 0, ∑

3

∑

i =1 j ,k ,l = 1

∈jkl X 3( i −1) +l ⎡⎣Q3( i −1) +l + R3( i −1) +l ⎤⎦ = 0

(18.1.45)

represent the necessary and sufficient conditions of rest of the representative point P with respect to a given frame of reference, in the case in which the discrete mechanical systems S is non-deformable. The kinetic energy of the representative point P is defined in the form T =

1 3n M k Xk 2 k∑ =1

(18.1.46)

and we can write dT = dW + dWR ,

where the elementary work is given by (18.1.15) and (18.1.15'); we state thus

(18.1.47)

Lagrangian Mechanics

23

Theorem 18.1.2 (theorem of kinetic energy). The motion of the representative point P , subjected to constraints in the space E 3 n , takes place so that, at any moment, the differential of the kinetic energy of thus point is equal to the elementary work of the given and constraint generalized forces which act upon this point. If we introduces also the powers of the given and constraint generalized forces in the form P=

dWR dW , PR = , dt dt

(18.1.46')

then the relation (18.2.47) becomes T = P + PR .

(18.1.47')

We can state Theorem 18.1.2' (theorem of kinetic energy; second form). The motion of the representative point P , subjected to constraints in the space E 3 n , takes place so that, at any moment, the derivative of the kinetic energy with respect to time of this point is equal to the power of the given and constraint generalized forces. In case of catastatic ideal constraints, the elementary work of the constraint generalized forces vanishes ( dWR = 0 ; it will be shown in next section); hence d T = dW .

(18.1.48)

If the given generalized forces derive from a simple or a generalized potential, then dW = − dV , where V = −U in case of the simple potential and V = −U 0 in case of the generalized potential. Introducing the mechanical energy E , it results a mechanical energy of the form E = h , E = T + V , h = const .

(18.1.49)

We can state Theorem 18.1.3 (conservation theorem of mechanical energy). The mechanical energy of the representative point P , subjected to catastatic ideal constraints in the space E 3 n , is constant in time only and only if the given generalized forces derive from a simple or generalized potential. We notice that, in the manifold whit 3n dimensions, V = h represents the level of energy of the motion; because T > 0 , it results that the representative point P moves in the domain V < h . If we represent the arbitrary generalized forces in the form (18.1.25), then it results d (T + V ) =

3n

∑ Qk

k =1

⎛ 3n ⎞ dXk − Udt = ⎜ ∑ QkVk − U ⎟ dt , ⎝ k =1 ⎠

(18.1.50)

MECHANICAL SYSTEMS, CLASSICAL MODELS

24

where V = −U corresponds to a simple quasi-potential; if U = U ( X1 , X 2 ,..., Xn ) is a simple potential, then we can write 3n d (T + V ) = ∑ QkVk . dt k =1

(18.1.50')

In the case of gyroscopic generalized forces we can write further a conservation theorem of mechanical energy (of the form (18.1.49)), while if the forces are dissipative, then the mechanical energy decreases.

18.1.2.2 Principle of Virtual Work. Lagrange’s Equations of the First Kind We have seen that a discrete mechanical system S subjected to ideal constraints verifies the relation (18.1.2); hence, a representative point P in the space E 3 n is subjected to ideal constraints if the relation (considered by P. Appell as relation of definition) δWR =

3n

∑ Rk δXk

k =1

=0

(18.1.51)

is verified. Starting from d’Alembert’s principle (18.1.42), effecting a scalar product by the virtual generalized displacements δXk , summing for all indices and taking into account the relation of definition (18.1.51) of the ideal constraints, we get the relation 3n

∑ Φk δXk =

k =1

3n

∑ (Qk

k =1

− M k Xk ) δXk = 0 ,

(18.1.52)

which represents a necessary condition to describe the motion of the representative point P . Assuming now that the condition (18.1.52) is fulfilled and that p holonomic constraints of the form (18.1.11) and m non-holonomic constraints of the form (18.1.7) take place, we will use the method of Lagrange’s multipliers; we can thus write ⎛ ∑ ⎜ Φk + k =1 ⎝ 3n

p

∑ λI

I =1

∂fI + ∂Xk

m

⎞

j =1

⎠

∑ μj bjk ⎟ δXk

= 0,

where λl , l = 1, 2,..., p , μj , j = 1, 2,..., m , are non-determinate scalars (Lagrange’s multipliers) and where we took into account that one can invert the order of summation, in a finite double sum. The p + m constraints are linear and distinct, the matrix of the coefficients ∂fl / ∂Xk and bjk being of rank p + m ; we express thus the virtual generalized displacements δX1 , δX 2 ,..., δX p + m (we can always choose them so that the determinant Δp + m of the respective coefficients in the constraints relations be non-zero) by means of the other 3n − ( p + m ) virtual generalized displacements, where the latter ones can be considered as independent. We put the conditions that the

Lagrangian Mechanics

25

expressions in the brackets which multiply the first p + m virtual generalized displacements do vanish, determining thus, univocally, p + m multipliers λl and μj (these multipliers are given by a system of p + m linear equations with p + m unknowns, of determinant Δp + m ≠ 0 ). The independent (and arbitrary) virtual generalized displacements δX p + m + 1 , δX p + m + 2 , ..., δX 3 n can be all taken equal to zero, expecting only one, let be δXk ; this

non-zero virtual generalized displacement being arbitrary, it results that the bracket by which it is multiplied must vanish too. Making, successively, k = p + m + 1 , k = p + m + 2 , ..., 3n , and taking into account the preceding considerations, it results that all the brackets which multiply the virtual generalized displacements must vanish. Finally, we can write

Φk +

p

∑ λI

I =1

∂fI + ∂X k

m

∑ μj bjk j =1

= 0, k = 1, 2,..., 3n .

(18.1.53)

These equations are equivalent with the equations (18.1.42); in case of ideal constraints, we can express the constraint generalized forces in the form Rk =

p

∑ λI

I =1

∂fI + ∂Xk

m

∑ μj bjk , k j =1

= 1, 2,..., 3n ,

(18.1.54)

equivalent to (3.2.37). We can state (the relation (18.1.52) becomes a sufficient condition too) Theorem 18.1.4 (theorem of virtual work; d’Alembert-Lagrange). The motion of the representative point P , subjected to ideal constraints in the space E 3 n , takes place so that the virtual work of the lost generalized forces of d’Alembert, which act upon this point, vanishes for all the systems of virtual generalized displacements of the respective point. Taking into account the equivalence between the relation (18.1.52), which represents the form taken by Newton's equations, it results that the theorem of virtual work can be considered as being a principle (the principle of virtual work or the principle of virtual generalized displacements), because – starting from it – one can solve the fundamental problems of dynamics in case of ideal constraints. The equation (18.1.52) is called also the basic equation (the first form), while the equations (18.1.53) are known as Lagrange’s equations of the first kind. Introducing the virtual generalized velocities Vk∗ , we can write the conditions (18.1.52) in the form 3n

∑ ΦkVk∗

k =1

=0

(18.1.52')

too, the considered principle being thus called also the principle of virtual generalized velocities.

MECHANICAL SYSTEMS, CLASSICAL MODELS

26

If some elements of the discrete mechanical system S are rigid solids, then one uses the formula (14.1.59) for the corresponding part of the virtual work. In the static case, the principle of virtual work becomes 3n

∑ Qk δXk

k =1

= 0,

(18.1.52'')

being stated for the first time in 1717 by Jean I Bernoulli. Let be a mechanical system S of n particles Pi , of weight mi g , i = 1, 2,..., n ; choosing a co-ordinate axis Ox along the descendent vertical, we can write the virtual work of these given forces in the form δW =

n

∑ mi

i =1

⎛ n ⎞ g δx i = δ ⎜ g ∑ mi x i ⎟ = δ ( M gξ ) = M gδξ = 0, ⎝ i =1 ⎠

where ξ is the applicate of the mass centre of the system S . We can thus state Theorem 18.1.5 (E. Torricelli). A mechanical system subjected to ideal constraints and to the action of its own weight is in equilibrium only and only for an extremum of the applicate of its centre of mass. Also this theorem can be considered to be a principle (Torricelli’s principle), which may be applied for a certain sphere of problems (in case of a uniform gravitational field). As we have seen in Sect. 4.1.1.7 (in case of a single particle), the position of equilibrium is stable, labile or indifferent as the applicate of the mass centre has a minimum, a maximum or is constant, respectively. Using the expression (18.1.54) of the constraint generalized forces and the constraint relations (18.1.6), (18.1.10), we can write the real elementary work (18.1.15') of the constraint generalized forces, in case of ideal constraints, in the form ⎛ p dWR = − ⎜ ∑ λl fl + ⎝ l =1

m

⎞

j =1

⎠

∑ μ j b j 0 ⎟ dt .

(18.1.55)

In case of catastatic constraints (we have fl = 0 , l = 1, 2,..., p , and bj 0 = 0 , l = 1, 2,..., m ), the real elementary work of the constraint generalized forces vanishes; indeed, in this case the real generalized displacements belong to the set of virtual generalized displacements, the relation (18.1.51) implying dWR = 0 . The conditions in which take place the relations (18.1.48) and (18.1.49) are thus entirely justified. Starting from the relations (18.1.6), we can write the constraint relations with the aid of the possible generalized displacements ΔX k , k = 1, 2,..., 3n , in the form 3n

∑ bjk ΔXk

k =1

+ bj 0 = 0, j = 1, 2,..., m .

Introducing the possible generalized velocities Vk =

ΔX k , k = 1, 2,..., 3n , Δt

(18.1.56)

Lagrangian Mechanics

27

we can write 3n

∑ bjkVk

k =1

+ bj 0 = 0, j = 1, 2,..., m ,

(18.1.56')

too. Let be also other possible generalized velocities Vk + ΔVk , for the same position of the representative point P , at the same moment t , so that 3n

∑ bjk (Vk

k =1

+ ΔVk

) + bj 0

= 0, j = 1, 2,..., m .

Subtracting the last two relations one from the other, it results 3n

∑ bjk ΔVk

k =1

= 0, j = 1, 2,..., m .

(18.1.56'')

= 0, l = 1, 2,..., p .

(18.1.56''')

Analogously, we can write 3n

∂f

∑ ∂Xlk ΔVk

k =1

Hence, these finite variations of the possible generalized velocities verify the relations (18.1.7), (18.1.11) for virtual generalized velocities. One can thus write the basic equation (18.1.52) in the second form 3n

∑ Φk ΔVk

k =1

=

3n

∑ (Qk

k =1

− M k Xk ) ΔVk = 0 ,

(18.1.57)

for a given position of the representative point P , at a fixed moment t ; this form of the equation plays an important rôle in case of the phenomenon of collision. If ΔVk are differential quantities, then these variations are virtual generalized velocities (differences of possible generalized velocities) and we find again the form (18.1.52') of the basic equation, as it has been shown by P. E. B. Jourdain, in 1908. The corresponding principle is know also as Jourdain’s principle, being expressed in the form 3n

∑ Φk δX k

k =1

=

3n

∑ (Qk

k =1

− M k Xk ) ΔX k = 0 .

(18.1.57')

By a total differentiation of the constraint relation (18.1.56') with respect to time, we obtain the relations which must be verified by the possible generalized accelerations A = V , k = 1,2,..., 3n , in the form k

k

MECHANICAL SYSTEMS, CLASSICAL MODELS

28 3n

⎡

∑ ⎢⎣bjk Ak

k =1

+

dbjk ⎤ dbjk Vk ⎥ + = 0, j = 1, 2,...m , dt dt ⎦

(18.1.58)

where d = dt

∂

3n

∑Vk ∂Xk

k =1

+

∂ . ∂t

(18.1.59)

We consider also the possible generalized accelerations Ak + ΔAk for the same position and the same velocity of the representative point P , at the same moment t ; as above, we obtain the relations 3n

∑ bjk ΔAk

k =1

= 0, j = 1, 2,...m .

(18.1.58')

Analogously, it results 3n

∂f

∑ ∂Xlk ΔAk

k =1

= 0, l = 1, 2,..., p .

(18.1.58'')

We see that the finite variations of the possible generalized accelerations verify the relations (18.1.7), (18.1.11), corresponding to the virtual generalized displacements. We are thus led to the third form of the basic equation 3n

∑ Φk ΔAk

k =1

=

3n

∑ (Qk

k =1

− M k Xk ) ΔAk = 0 ,

(18.1.60)

for a given position and generalized velocity of the representative point P , at a fixed moment t . If ΔAk are differential quantities, as Gauss and Gibbs have considered, we can write this equation in the form 3n

∑ Φk δXk

k =1

=

3n

∑ (Qk

k =1

− M k Xk ) δXk = 0 ,

(18.1.60')

where these variations are virtual generalized accelerations (differences of possible generalized accelerations) .

18.1.2.3 Gauss’s Principle. Hertz’s Principle Let us suppose that a particle Pi , of mass mi , of the mechanical system S subjected to constraints is, at the moment t , at the position Pi of position vector ri , velocity vi and acceleration ai , i = 1, 2,..., n . If no force would act upon the particle Pi , then this one would effect the real displacement (Fig. 18.1)

Lagrangian Mechanics

29 JJJJJG Pi Pi ′ = vi Δt ,

(18.1.61)

during the time t , while if upon this one would act upon only the given force Fi , then JJJJJG the particle would effect the real displacement Pi Pi ′′ , so that the derivation of the particle Pi in this motion would be given by the formula (5.1.21) in the form JJJJJG 1 F i ( Δt )2 . Pi Pi ′′ = 2 mi

(18.1.62)

Fig. 18.1 Gauss’s principle. Deviations

Finally, if the particle Pi is acted upon both by the given and the constraint forces, JJJJJG then it effects the real displacement Pi Pi ′′′ , in the time interval Δt ; the corresponding deviation of the particle is JJJJJG 1 Pi Pi ′′′ = ai ( Δt )2 , 2

(18.1.62')

JJJJG collinear with the acceleration ai . Let be also a possible displacement PQ i i , different JJJJJG from Pi Pi ′′′ , effected in the same interval of time Δt . JJJJJJG JJJJJG We notice that Pi ′′ Pi ′′′ and Pi ′′Qi represent the deviation of the particle Pi from the

free motion, in the time interval Δt to its real motion (the particle subjected to JJJJJG JJJJJJG JJJJJG constraints). Starting from the relation Pi ′′Qi = Pi ′′ Pi ′′′ + Pi ′′′Qi , we can write n

∑ mi Pi ′′Qi

i =1

2

=

n

n

i =1

i =1

∑ mi Pi ′′ Pi ′′′ 2 + ∑ mi Pi ′′′Qi

2

n JJJJJJG JJJJJJG + 2 ∑ mi Pi ′′ Pi ′′′ ⋅ Pi ′′′ Qi . i =1

JJJJJG JJJJJG JJJJJJG We may also write Pi ′′′Qi = Pi Qi − Pi Pi ′′′ , hence as a difference between a JJJJJG possible and a real displacement (which is a possible displacement too); hence, Pi'''Qi

MECHANICAL SYSTEMS, CLASSICAL MODELS

30

JJJJJJG JJJJJJG JJJJJJG is a virtual displacement δri . On the other hand, Pi ′′ Pi ′′′ = Pi ′ Pi ′′′ − Pi ′ Pi ′′ ; taking into

account (18.1.62), (18.1.62') and introducing the lost forces of d’Alembert of the form (18.1.59), it results n JJJJJJG JJJJJG 2 ∑ mi Pi ′′ Pi ′′′ ⋅ Pi ′′′Qi = − ( Δt )2 i =1

n

∑ ( Fi

− mi ai ) ⋅ δri = − ( Δt )2

i =1

n

∑ Φi

i =1

⋅ δri .

Using the theorem of virtual work (in case of bilateral constraints) in the form (11.1.63), we notice that this sum vanishes; if the constraints are unilateral, then the relation (11.1.63'') takes place, the above sum being positive. As a conclusion, this sum is non-negative in case of ideal constraints, so that n

JJJJJG

∑ mi Pi ′′Qi

i =1

2

≥

JJJJJJG

n

∑ mi Pi ′′ Pi ′′′ 2

,

i =1

the equality taking place only if Qi ≡ Pi ′′′ . Gauss takes as measure of the deviation of the particle Pi from its free motion, in the real or in a possible motion, the constraint Z i = 2mi

Pi ′′ Pi ′′′ 2 ( Δt )4

,

or the constraint Z i = 2mi

Pi ′′′Qi

2

( Δt )4

,

respectively; in this case Z ≤ Z, Z =

n

n

i =1

i =1

∑ Zi , Z = ∑ Zi .

(18.1.63)

Hence, among all the possible motions of the discrete mechanical system S , the real motion is that for which the constraint Z is minimal (any other possible constraint Z is greater); but we notice that for all possible displacement must correspond the same positions and the same velocities at the moment t , the accelerations being different. JJJJJJG Taking into account the expression of the displacement Pi ′′ Pi ′′′ , we can express the constraint Z in the form Z =

1 n 1 1 n 1 Φ ι2 = ∑ ( F − mi ai )2 , ∑ 2 i =1 mi 2 i =1 mi i

(18.1.64)

Lagrangian Mechanics

31

while for the representative point P we can write 1 3n 1 1 3n 1 Φ 2κ = ∑ ( Qk − Mk Xk ∑ 2 k =1 M k 2 k =1 M k

Z =

)2 .

(18.1.65)

From the condition of minimum of the constraint in the real motion it results δZ = 0 ; because only the accelerations vary, we get δZ =

3n

1

∑ M k Φk δΦk

k =1

3n

3n

k =1

k =1

= − ∑ Φk δXk = − ∑ ( Qk − M k Xk

) δXk

= 0,

(18.1.66)

equation which can be identified with the equation (18.1.60'). Starting from this equation and from the relations (18.1.58'), (18.1.58''), written in the form 3n

∑ bjk δXk

k =1 3n

∂f

= 0, j = 1, 2,..., m ,

∑ ∂Xlk δXk

k =1

= 0, l = 1, 2,..., p ,

(18.1.67) (18.1.67')

respectively, and introducing Lagrange’s multipliers – on the same way as in the previous subsection – we find the equations (18.1.42). We can thus state Theorem 18.1.6 (theorem of the least constraint; C. F. Gauss). The motion of the representative point P , subjected to ideal constraints in the space E 3 n , takes place so that the constraint Z of this point has a minimum for virtual variations of the generalized accelerations (the generalized co-ordinates and the generalized velocities are considered constant). We can also say that, in the same conditions concerning the position, the generalized velocities and the generalized accelerations, the first variation of the constraint Z of the representative point must vanish (the constraint Z is stationary along the trajectory of the representative point P ). We have seen that the demonstration of this theorem implies the theorem of virtual work, being thus equivalent to Newton’s principle. Also the theorem of the least constraint, stated by Gauss in 1829, can be considered as a principle (the principle of the least constraint), because – starting from it – one can find again Newton’s equations in case of ideal constraints. H. Hertz considers that the given generalized forces, which act upon the mechanical system S , can be replaced by given constraints between this system and neighbouring systems which act upon it; thus, he studies only mechanical systems which are not acted upon by given generalized forces ( Qk = 0 ). As well, to obtain a geometric interpretation, he assumes that the generalized masses M k are equal to unity (hence, M 1 = M 2 = ... = M 3 n ). Thus, it results 2

Z =

1 3 n  2 1 3 n ⎛ d2 Xk ⎞ Xk = ∑ ⎜ . ∑ 2 k =1 2 k =1 ⎝ dt 2 ⎟⎠

(18.1.68)

MECHANICAL SYSTEMS, CLASSICAL MODELS

32

In case of a single particle, of co-ordinates x1 , x 2 , x 3 , we have (we can assume, in this case, that the mass m is not equal to unity) Z =

m 2 ( x + x22 + x32 2 1

)=

m 2 a , 2

(18.1.69)

where a is the acceleration in the considered motion; we obtain thus δZ = a δa . Using Frenet’s frame of reference, we can write a2 =

v4 + v 2 ; ρ2

this sum of squares is minimal if v = 0 (which takes place because the given force is equal to zero and one assumes that the constraints are catastatic, so that dWR = 0 , while the constraint force has no component along the tangent to the trajectory), hence if the motion is uniform and if the curvature 1/ ρ is minimal. Hence, the trajectory of the point must be of minimal curvature (as near as possible to a straight line). In case of a discrete mechanical system S , of representative point P , we can express the kinetic energy in the form (with the above considerations)

( )

2

T =

1 3n  2 1 3 n dX k ⎞ 1 ds Xk = ∑ ⎛⎜ ⎟ = ∑ 2 k =1 2 k = 1 ⎝ dt ⎠ 2 dt

2

=

1 2 s , 2

where ds 2 = dX12 + dX 22 + ... + dX 32n is the square of the arc element in the space E 3 n . Because there are not generalized forces and assuming that to constraints are catastatic ( dWR = 0 ), it results that T = const. As we can notice, 2 3 n dX 3 n dX d2 X dX k k ⎞ k k X k = s, ∑ ⎛⎜ = 0, ⎟ = 1, ∑ ds ds 2 k = 1 ⎝ ds ⎠ k = 1 ds

so that d2 Xk 2 dXk Xk = s + s ; ds ds 2

in the mentioned conditions, we have s = v = 0 , hence the motion of the representative point on its trajectory is uniform. We can thus replace the quantity Z by the quantity K2 =

d2 Xk = 2 k = 1 ds 3n

∑

n

∑

i =1

( ddrs )

2

i

=

n

∑ Ki2 ,

(18.1.70)

i =1

where we define by K the curvature of the path travelled through by the system (in a certain sense, a measure of the curvatures of the paths of all the particles Pi which form

Lagrangian Mechanics

33

the system S ); the minimum of the curvature K corresponds, obviously, to the minimum of the constraint Z . We can thus state Theorem 18.1.7 (theorem of the least curvature; theorem of the most straight path; H. Hetz). The motion of the discrete mechanical system S , subjected to ideal catastatic constraints, in the absence of the given generalized forces (in spontaneous motion) and assuming a mass of equal generalized components, takes place, starting from a given initial uniform motion, so that the curvature K of the path travelled through by the system be minimal, with respect to any other possible path (the path travelled trough be the most straight possible one). As in the preceding cases, we can consider this theorem as a principle (the principle of least curvature or the principle of the most straight path). We mention that, because of the conditions which are imposed, this principle is not so easy to apply; but the connections to Lagrangian mechanics and to relativistic mechanics are interesting. Starting from a constraint relation of the form ϕ j ( X1 , X 2 ,..., Xn , X 1 , X 2 ,..., X n ; t ) = 0, j = 1, 2,..., m ,

which is not necessarily linear in the generalized velocities, and effecting a total derivative with respect to time, we obtain 3n

⎛ ∂ϕj

∑ ⎜⎝ ∂Xk X k

k =1

+

∂ϕ j ⎞ Xk ⎟ + ϕ j = 0, j = 1, 2,..., m . ∂X k ⎠

Varying only the accelerations, we find the relations 3n

∂ϕ j

∑ ∂X

k =1

δXk = 0, j = 1, 2,..., m ,

(18.1.71)

k

which are of the same form as the relations (18.1.67) ( bjk = ∂ϕ j ∂X k ). We can thus state that Gauss’s principle can be applied even if the kinetic constraint relations are not linear in the generalized velocities, having thus a larger sphere of applicability than the principle of virtual work (for which the mentioned relations must be linear in the generalized velocities). We remark that Jourdain’s principle has a position somewhat intermediary between the d’Alembert-Lagrange principle and Gauss's principle.

18.1.2.4 Applications Let be a particle P subjected to move on a deformable surface of equation f (x1 , x 2 , x 3 ; t ) = 0 (we use the notations in the space E 3 ); the constraint relation can be written in the form f, j δx j = 0 . If we use the principle of virtual work in the form

Φj δx j = 0 and if we eliminate, e.g., the virtual displacement δx 3 , then we get

(18.1.72)

34

MECHANICAL SYSTEMS, CLASSICAL MODELS

f,1 f,2 ⎛ ⎞ ⎛ ⎞ ⎜ Φ1 − f Φ3 ⎟ δx1 + ⎜ Φ2 − f Φ3 ⎟ δx 2 = 0 . ,3 ,3 ⎝ ⎠ ⎝ ⎠

To satisfy this relation for any virtual displacement δx1 and δx 2 , the two brackets must vanish, hence f,1 f,2 f,3 = = . Φ1 Φ2 Φ3

(18.1.73)

This result corresponds to that obtained in Sect. 6.2.2.2 by the method of Lagrange’s multipliers.

Fig. 18.2 Motion of a particle on a fixed or movable or deformable sphere

In particular, let be a particle P in motion on a sphere S , fixed or movable and deformable ( r ′ − rO′ )2 = l 2 , of centre rO′ = rO′ (t ) and radius l = l (t ) (Fig. 18.2). Noting that grad[( r ′ − rO′ )2 − l 2 ] = 2( r ′ − rO′ ) , Lagrange’s equation of the first kind is of the form r ′ = λ ( r ′ − rO′ ) , λ indeterminate scalar (we assume that the given force is equal to zero); shifting the origin to the centre of the sphere, we can write the equation of motion in the form r = λr − rO′ , r = r ′ − rO′ , too, with the constraint r 2 = l 2 , putting thus in evidence the complementary force of transportation −rO′ of the movable frame of reference (in general, non-inertial). By a vector product by r and by a scalar product by r , respectively, we get d d 2 d ( r × r ) =  rO′ × r , r = λ r 2 − 2 rO′ ⋅ r . dt dt dt

If rO′ = 0 (hence, if the movable frame is inertial, the centre of the sphere having a JJJJJG rectilinear and uniform motion), then we have r × r = C , C = const , wherefrom C ⋅ r = 0 and dr 2 dt = 2λ ll ; hence, the trajectory is a great circle of the deformable sphere (the intersection of the plane C ⋅ r = 0 with the sphere). In particular, if the radius l of the sphere is constant, then the trajectory is a geodesic of it. Let now be a particle P which moves on a deformable curve of equations fk (x1 , x 2 , x 3 ; t ) = 0 , k = 1, 2 (in the space E 3 ); the constraint relations can be written

Lagrangian Mechanics

35

in the form fk , j δx j = 0 , k = 1,2 . Associating the condition (18.1.72) and eliminating the virtual displacements, we obtain Φ1

Φ3

Φ3

f1,1

f1,2

f1,3 = 0 ,

f2,1

f2,2

f2,3

(18.1.74)

relation which – together with the constraint relations – allows to determine the motion of the particle (see Sect. 6.2.2.1 too, where Lagrange’s multipliers are used).

Fig. 18.3 Motion of a particle on a movable circle of variable radius

Let us study, in particular, the motion of a particle P on a circle C , movable and of variable radius, represented by the intersection of the above considered sphere S with the plane α ⋅ ( r ′ − rO′ ) = 0 , α = α (t ) , which passes through rO′ and is normal to α (Fig. 18.3). As in the preceding case, in the absence of the given force, we can write the equation of motion in the form r = λ1 r + λ2 α − rO′ , λ1 , λ2 indeterminate scalars, r = r ′ − rO′ , with the constraint relations r 2 = l 2 , α ⋅ r = 0 . By a vector product by r and successive scalar products by r and α , respectively, we get

d ( r × r ) = λ2 r × α +  rO′ , dt d 2 d r = λ1 r 2 − 2 rO′ ⋅ r + 2λ2 α ⋅ r , dt dt α ⋅ r = λ2 α2 − α ⋅ rO' . JJJJJG If α = const (hence, if the circle C is in a plane of fixed direction), then we have α ⋅ r = 0 , α ⋅ r = 0 too (according to the constraint relations). Thus, it results

λ2 = α ⋅

rO′ , α2

hence the component λ2 α normal to the plane α ⋅ r = 0 , of the constraint force (the component contained in the plane is λ1 r ). If rO′ = 0 (the movable frame of reference

MECHANICAL SYSTEMS, CLASSICAL MODELS

36

is inertial), then the constraint force is contained in the plane of the circle, which has a fixed orientation. Let be two particles P1 and P2 of masses m1 and m2 and position vectors r1 and r2 , upon which act the given forces F1 and F2 , respectively; we assume a constraint relation at distance of the form ( r1 − r2 )2 = l 2 , l = l (t ) , which can be written as ( r1 − r2 ) ⋅ δr1 − ( r1 − r2 ) ⋅ δr2 = 0 . The principle of virtual work is written in the form Φ1 ⋅ δr1 + Φ2 ⋅ δr2 = 0 , being thus led to Lagrange’s equations of the first kind Φ1 + λ ( r1 − r2 ) = 0, Φ2 − λ ( r1 − r2 ) = 0 .

(18.1.75)

We obtain thus Φ1 + Φ2 = 0 . Assuming that F1 = m1 g + F12 , F2 = m2 g + F21 , g = g (t ) , where F12 and F21 are internal forces ( F12 + F21 = 0 ), it results m1 r1 + m2 r2 = (m1 + m2 )g , wherefrom t

τ

0

0

m1 r1 + m2 r2 = ( m1 + m2 ) ∫ dτ ∫ g ( τ )dτ + C1t + C2 , JJJJJG with C1 , C2 = const ; hence, the centre of mass C has a motion of acceleration g (t ) , being situated on the segment of a line P1P2 , so that

CP1 =

m2 l m1l , CP2 = , l = P1P2 . m1 + m2 m1 + m2

The particles P1 and P2 will be at any moment on two concentric spheres of radii CP1 and CP2 , respectively (hence, remains to be studied the problem of motion of a particle on a given sphere). In the particular case of two heavy particles, the mass centre C will describe a parabola situated in a vertical plane and having a vertical axis; the particles will be in a plane of fixed orientation, their motion with respect to the centre C taking place after the law of areas. The analytic methods of computation have been developed, especially, in case of discrete mechanical systems; however, they can be applied successfully also to problems concerning continuous mechanical systems. We consider thus a perfectly flexible, inextensible and non-torsionable thread, of given length l , fixed at the points P 0 and P 1 , acted upon by forces p (s ) on unit length. The principle of virtual work is written in the form ( Φ ⋅ δr ) ds ∫Pq P 1

2

= 0, Φ = p − μ a ,

(18.1.76)

where μ (s ) is the linear density, while a(s ) is the acceleration of an arbitrary point P . The relation ds 2 = (dr )2 leads to ds 2 δ(ds ) = dr2 ⋅ δ(dr ) ; because the thread is inextensible, we have δ(ds ) = 0 , so that the constraint relation (which takes place for all the elements of arc) is written in the form

Lagrangian Mechanics

37 dr ⋅ δ ( dr ) = 0 . ds

Starting from the points P of the curve C and considering a given system of virtual displacements (at any point P corresponds a virtual displacement δr at the moment t ), we obtain the points P , which form the curve C . If P ( r ) and P ′( r + dr ) are two neighbouring points on the curve C and P ( r + δr ) is a corresponding point on the JJJG JJJJG curve C , one can reach the point P ′ on the path PP + PP ′ or on the path JJJJG JJJJG PP ′ + P ′P ′ (Fig. 18.4); hence, the point P ′ can be specified by r + δr + d( r + δr ) = r + dr + δ( r + dr ) ,

wherefrom d( δr ) = δ(dr ) , so that the two operators are linked by the relation dδ = δd .

(18.1.77)

Obviously, the considerations made above involve, at the most, holonomic constraints; the relation of permutation (18.1.77) takes place just in such a case.

Fig. 18.4 Displacements of a particle laying on a curve

The constraint relation becomes dr ⋅ d ( δr ) = 0 , ds

so that Lagrange’s equation of the first kind will be ⎡

dr

⎤

Φ ⋅ δrds + λ ⋅ d ( δr ) = 0 . ∫Pq ⎥⎦ P ⎢ ds ⎣ 0

Integrating by parts

1

MECHANICAL SYSTEMS, CLASSICAL MODELS

38 dr

dr

⋅ d ( δr ) = λ λ ∫Pq P ds ds 0

1

⋅ δr

P1 P0

−

δr ⋅ d ( ∫Pq P 0

1

λ dr ds

)

and noting that δr vanishes at the fixed points P 0 and P 1 , it results Φd s + d ( ∫Pq P ⎢ ⎣

⎡

0

1

)

λ dr ⎤ ⋅ δr = 0 . ds ⎥⎦

Because this relation must take place for any virtual displacement δr , we obtain Φd s − λ

dr = 0; ds

introducing the tension in the thread T =T

dr ,T = − λ , ds

we can write Φ+

dT = 0, ds

(18.1.78)

finding thus again the equation (12.2.10) of motion of the thread. We present now some simple applications too, eventually with a practical character.

Fig. 18.5 Motion of a coach on a railway in a curved line

Let use consider a railway which, on a certain length, follows a curve of curvature radius ρ . D’Alembert’s principle, written for the mass centre C of a coach, in the normal plane to the curve (Fig. 18.5), leads to G + N + Ft = 0 , where G is the weight of the coach, N is the normal reaction of the ground, while Ft is the transportation force of a non-inertial frame of reference (the centrifugal force), of magnitude

Lagrangian Mechanics

39 mv 2 . ρ

Ft =

To can verify the equation of dynamic equilibrium, one must give to the ground an inclination of angle α , so that N sin α − Ft = 0 , N cos α − G = 0 , whence tan α =

Ft v2 = ; G ρg

the exterior rail must be superelevate to a height h = l sin α ≅ l tan α ( l is the clearance of the railway), so that h ≅

v 2l , gρ

(18.1.79)

at an arbitrary point of the curve.

Fig. 18.6 Belidor’s crane bridge

Let be now a rigid bar OA of weight G , which is rotating in a vertical plane, about a fixed point O . A perfectly flexible, inextensible and non-torsionable thread, which passes over a fixed pulley B , situated on the vertical line of O , connects the point A to a point P , where is suspended a weight Q . One must find the locus of the point P (a curve Γ ), so that the bar AO be in indifferent equilibrium (Fig. 18.6). The bar AO models Belidor’s crane bridge. We denote n = θ , OBP n =ϕ ; OA = 2l , AB = a , BP = r , OB = h , h > 0, BOA

we choose OB as Ox 2 -axis, the Ox1 -axis being horizontal, so that x C2 = l cos θ ,

x 2P = h − r cos ϕ . The constraint relations are a + r = L , r = r (ϕ ) , where L is the length of the thread. The mechanical system has only one degree of freedom and its

MECHANICAL SYSTEMS, CLASSICAL MODELS

40

position can be specified by the angle ϕ . The principle of virtual work is written in the form −G δx C2 − Q δx 2P = 0 . We have δa + δr = 0 , δr = r ′(ϕ ) δϕ , a 2 = 4l 2 + h 2 − 4lh cos θ ,

hence a δa = 2lh sin θ δθ ; in this case δx C2 = −l sin θ δθ = −

a a a δa = δr = r ′(ϕ )δϕ . 2h 2h 2h

On the other hand,

δx 2P = − cos ϕ δr + r sin ϕ δϕ = [ r (ϕ ) sin ϕ − r ′(ϕ ) cos ϕ ] δϕ . We obtain thus the condition of equilibrium −

Ga r ′(ϕ ) + Q [ r ′(ϕ ) cos ϕ − r (ϕ ) sin ϕ ] = 0 , 2h

which determines the corresponding angles ϕ . We get thus the differential equation

{

}

G [ L − r (ϕ )] − Q cos ϕ r ′(ϕ ) + Qr (ϕ ) sin ϕ = 0 . 2h

Fig. 18.7 Equilibrium of three particles acted upon by three forces in a particular case

By integration, we obtain G 2 PL r (ϕ ) − r (ϕ ) + Qr (ϕ ) cos ϕ = const. 4h 2h

Hence, the equation of the curve r is of the form

Lagrangian Mechanics

41

r2 − ( α + β cos ϕ )r + γ = 0, α =

2 Pl 4Qh , γ = arbitrary, ,β = − G G

(18.1.80)

representing – in general – ovals of Descartes; for γ = 0 it results a Pascal’s limaçon. C. Neumann considered three particles P1 , P2 , P3 , linked so that the area of the triangle P1P2 P3 be constant, and upon which act the given forces F1 , F2 , F3 , respectively (Fig. 18.7). If r1 , r2 , r3 are position vectors of the particles P1 , P2 and P3 , respectively, and if we denote JJJJJG rjk = Pj Pk = rk − rj , j , k = 1, 2, 3, r12 × r13 = C u ,

where u = u(t ) is a vector normal to the plane P1P2 P3 , while C = const , then the constraint relation is of the form ( r12 × r13 )2 = C 2 = const . Taking into account the properties of the mixed product of these vectors, we notice that the constraint relation can be written in the form

[ ( r12 × r13 ) × r23 ] ⋅ δr1 + [ ( r12 × r13 ) × r31 ] ⋅ δr2 + [ ( r12 × r13 ) × r12 ] ⋅ δr3 = 0 ; associating the principle of virtual work, given by F1 ⋅ δr1 + F2 ⋅ δr2 + F3 ⋅ δr3 = 0 ,

and introducing Lagrange’s multiplier λ , we can write the equations of equilibrium (corresponding to Lagrange’s equations of the first kind) in the form F1 = − λC u × r23 , F2 = − λC u × r31 , F3 = − λC u × r12 ,

(18.1.81)

where we have used the notation introduced above too. It results that, for the equilibrium, the forces F1 , F2 and F3 must be in a plane normal to the unit vector u , hence in the plane of the triangle P1P2 P3 ; they must be normal to the sides opposite to the vertices at which they act (hence, their supports must pierce at the orthocentre H of the triangle, while their magnitudes must be in direct proportion to these sides). As a simple application of Gauss's principle, let us consider Atwood’s machine (see Sect. 11.1.2.2 and Fig 11.1), where the weights G1 and G 2 are modeled as particles and where we neglect the influence of the wheel of weight G . The constraint will be Z =

1⎡ 1 1 ( m1g − m1a )2 + ( m2 g − m2a )2 ⎤⎥ , ⎢ 2 ⎣ m1 m2 ⎦

where we assume that m1 > m2 ; in this case δZ = [ − m1 (g − a ) + m2 (g + a ) ] δa = 0 ,

MECHANICAL SYSTEMS, CLASSICAL MODELS

42 being thus led to a =

m1 − m2 g, m1 + m2

(18.1.82)

a formula which corresponds to (11.1.21).

18.1.2.5 General Theorems. Conservation Theorems Let be discrete a mechanical system S of n particles Pi , acted upon by the given forces Fi , i = 1, 2,..., n ; assuming that the constraints allow an arbitrary virtual rigid displacement (with respect to an inertial frame of reference) δri′ = vO′∗ δt + ( ω∗ δt ) × ri , ri = ri′ − rO′ , i = 1, 2,..., n ,

(18.1.83)

where O is the pole of a non-inertial frame, rigidly linked to the mechanical system S in its rigid displacement (corresponding to the formula (18.1.3)). The principle of virtual work is thus written in the form n

∑ Φi

i =1

⋅ [ vO′∗ δt + ( ω∗ δt ) × ri ] = 0 .

Because the vectors vO′∗ and ω∗ are arbitrary, it results n

∑ Φi

i =1

n

= 0, ∑ ri′ × Φi = 0 , i =1

(18.1.84)

where we took into account the second relation (18.1.83). We can also write τO ′ { Φi } = 0 ,

(18.1.84')

the torsor of d’Alembert’s lost forces, with respect to the pole of an inertial frame of reference, vanishing. Starting from this result, we find again the general theorems of mechanics (the theorem of momentum and the theorem of moment of momentum). Comparing with the relation (11.1.61), written with respect to the same pole O ′ , where the lost forces of d’Alembert equilibrate the constraint forces, we see that in this new form of the theorem of torsor do not appear the constraint forces; this happens because the relation (11.1.61) can be written for any mechanical system, while the relation (18.1.84') takes place only for the mechanical systems which admit an arbitrary rigid displacement. In case a free rigid solid which, obviously, admits an arbitrary rigid displacement, the theorem of torsor becomes (V is the volume of the rigid solid, μ is the density, while F is a volumic force, which – eventually – can be reduced to a superficial one)

Lagrangian Mechanics

43

{

τO ' Φ dV

} = 0, Φ =

F − μ r ,

(18.1.85)

wherefrom

∫V ( F − μ r ) dV

= 0,

∫V r × ( F − μ r ) dV

= 0,

(18.1.85')

resulting the theorem of momentum and the theorem of moment of momentum, respectively. If the solid would have a fixed point, we could admit that this one is just the point O , so that rO′ = 0 and vO′∗ = 0 ; we obtain

∫V Φ ⋅ ⎡⎣ ( ω

∗

)

δt × r ⎤⎦ dV =

∫V ( r × Φ ) ⋅ ( ω δt ) dV ∗

= 0.

The virtual angular velocity ω∗ being arbitrary, it results

∫V r × Φ dV

= 0,

(18.1.86)

which leads to the theorem of moment of momentum. In the case in which the rigid solid has two fixed points, hence a fixed axis, we can take this axis as the O ′x 3′ -axis, O ′ ≡ O , after the vector ω∗ ; in projection on the axis O ′x 3′ ≡ Ox 3 , we can write

∫V ( x1 Φ2

− x 2 Φ1 ) ( ω ∗ δt ) dV = 0 ,

∫V ( x1 Φ2

− x 2 Φ1 ) dV = 0 ,

wherefrom (18.1.87)

being thus led to the projection of the theorem of moment of momentum on a fixed axis. If, for a discrete mechanical system S , the constraints admit as virtual displacement a translation of constant direction, specified by the constant unit vector u and of arbitrary magnitude δr ′ = k u δt , vO′∗ = k u , and if ω∗ = 0 , then we get

∫V Φ ⋅ u dV

= 0,

(18.1.88)

hence the theorem of momentum projected on the direction of unit vector u ; this allows to write a scalar first integral, component of a conservation theorem of momentum (corresponding to the theorem expressed by the first formula (18.1.84)). As well, if the mechanical system S has constraints which allow a rotation about an axis of fixed direction, specified by the unit vector u , then we can write δr ′ = ( ω∗ uδt ) × r , vO′ = 0 ; we obtain thus (if we take the point O on the fixed axis, then we have rO′ = 0 , O ≡ O ′ , hence r ′ = r )

44

MECHANICAL SYSTEMS, CLASSICAL MODELS

∫V ( r ′ × Φ ) ⋅ udV

= 0,

(18.1.89)

that is the theorem of moment of momentum with respect to the pole O ′ , projected on the fixed axis of unit vector u . We obtain thus a scalar first integral too, component of a first integral of moment of momentum (corresponding to the theorem expressed by the second formula (18.1.84)). As in case of the theorem (18.1.84), in these results intervene only given forces; in exchange, they take place only for the mentioned particular motions of the rigid solid. One can make analogous considerations concerning the theorem of kinetic energy for the conservation of the mechanical energy, assuming that the constraints of the mechanical system S are catastatic.

18.1.2.6 Influence of the Forces of Friction Let us consider a particle P subjected to the action of a given force F and to holonomic constraints, which lead to a constraint force R . In case of a single holonomic an rheonomous constraint ϕ (x1 , x 2 , x 3 ; t ) = 0 , we can write R = N + T , where N is the normal constraint force and T is the tangential constraint force. If the constraint would be scleronomic, hence the force T would correspond to the sliding friction on the fixed rigid surface S (of equation ϕ ( x1 , x 2 , x 3 ) = 0 ). We can consider, e.g., a hydrodynamic friction, of the form T = − k v , k = k ( r , r ; t ) > 0 , which leads to the replacement of the given force F = F ( r , r ; t ) by a new force F = F − k v , the problem remaining the same from a mathematical point of view. In case of a Coulombian friction, of the form T = fN v / v , f = f ( r , r ; t ) , the problem is put in a different from, due to the unknown factor N , corresponding to the normal constraint force; we remain now to this second case. We assume a constraint ϕ ( r ; t ) = 0 , which may correspond to a family of rigid

surfaces St in motion, depending on the parameter t . The velocity of the particle is

v = vt + vr , where vt is the velocity of transportation and vr is the sliding velocity on the surface St (on the nature of a relative velocity); in this case, in the expression of the constraint force T , the velocity v is replaced by the velocity vr . The constraints are no more ideal, so that the condition of ideal constraint takes no more place; this condition is replaced by an analogous condition of the form R × W = 0,

(18.1.90)

where the vector W plays the rôle of the virtual displacement δr . The principle of virtual work is thus replaced by Φ⋅W = 0.

(18.1.91)

Noting that R = ± N n − fN τ = N ( ± n − f τ ) , where τ and n are unit vectors in the frame of Darboux, we can write the condition (18.1.90) also in the form

Lagrangian Mechanics

45 (n ± fτ) ⋅ W = 0 ,

(18.1.90')

neglecting the scalar N ≠ 0 . To express the unit vectors τ and n as functions of the data of the problem, we use the curvilinear co-ordinates u and v on the surface St so that v = ru′u + rv′v + r ; putting in evidence the partial derivatives with respect to these co-ordinates, it results vr = ru′u + rv′v and then τ = vers vr . As well, n = vers gradϕ = vers ( ru′ × rv′ )

(see Sect. 4.1.1.4 too). Eliminating W between the relation (18.1.90'), (18.1.91), which do not contain the unknown scalar N , by means of Lagrange’s multiplier λ , we obtain Φ + λ(n ± fτ) = 0.

(18.1.92)

If we associate also the scalar constraint relation ϕ = 0 to this vector equation, then we can determine the unknowns r = r (t ) and λ = λ (t ) . In case of two holonomic constraints ϕk ( r ; t ) = 0 , k = 1, 2 , which can correspond to a family of rigid curves C t in motion, depending on the parameter t , we proceed as above and obtain the equations Φ ⋅ W = 0, ( nk ± f τ ) ⋅ W = 0, k = 1, 2 .

(18.1.93)

We are thus led to Lagrange’s equation of the first kind Φ + λ1 ( n1 ± f τ ) + λ2 ( n2 ± f τ ) = 0 .

(18.1.94)

In the scleronomic case, this equation – to which we associate the constraint relations – is sufficient to solve the problem. In the rheonomous case, we use a representation in curvilinear co-ordinates r = r (q ; t ) (see Sect. 4.1.1.3 too) and representations r = r (q , qk ; t ) , k = 1, 2 , of the surface, so that the position of an arbitrary point of these ones be specified by the same values of the parameters q and qk , no matter its position in space. We have nk = vers ( rq′ × rq′k ) and τ = vers vr = q vers rq′ , this being the unit vector of the tangent to the curve C t ; in this case too, the problem is entirely determined.

18.2 Lagrange’s Equations A decisive step ahead has been made by Lagrange, by eliminating the holonomic constraints from the computation and by introducing independent generalized co-ordinates, in case of mechanical systems subjected only to such constraints; thus one passes from the representative space E 3n to a new representative space Λs , called the

MECHANICAL SYSTEMS, CLASSICAL MODELS

46

space of configurations. The equations of motion of the representative point are thus Lagrange’s equations of second kind, which will be studied in this paragraph.

18.2.1 Space of Configurations By introducing the Lagrangian generalized co-ordinates, in a minimal parameterization, one passes from the space E 3 (or from the representative space E 3n ) to the space of configurations Λs . In what follows, we put in evidence the form taken by various mechanical quantities (displacement, velocity, acceleration, kinetic energy, work etc.), obtaining a new formulation of the fundamental problem of mechanics in the new representative space. 18.2.1.1 Generalized Co-ordinates. Representative Space Λs Let us consider a discrete mechanical system S of n a particles Pi , i = 1, 2,..., n , subjected to p holonomic constraints of the form (3.2.8), the corresponding representative point P in the space E 3n being subjected to p constraints of the form (18.1.9). In this case, the system S has 3n − p = s degrees of geometric freedom, its position being thus specified, at a moment t , by means of s independent parameters. To have motion, it is necessary that s ≥ 1 ; if s = 1 , then we say that the mechanical system is with complete constraints. Let be q1 , q2 ,..., qs a set of such parameters, which will be called Lagrangian generalized co-ordinates (shortly, generalized co-ordinates). The position of the mechanical system S will be thus given by relations of the form ri = ri (q1 , q2 ,..., qs ; t ) ≡ ri (q j ; t ), i = 1, 2,..., n ;

(18.2.1)

correspondingly, the position of the representative point P in the space E 3n is specified by Xk ≡ Xk (q j ; t ), k = 1,2,..., 3n .

(18.2.2)

Obviously, the relations (18.2.1) verify identically the constraint relations (3.2.8), while the relations (18.2.2) verify identically the constraint relations (18.1.9). If the constraints are scleronomic, hence of the form (3.2.33'), then the relations (18.2.1) take the form ri = ri (q1 , q1 ,..., qs ), i = 1, 2,..., n .

(18.2.1')

As well, for constraints of the form (18.1.12), the relations (18.2.2) become Xk ≡ Xk (q1 , q2 ,..., qs ), k = 1, 2,..., 3n .

(18.2.2')

Indeed, if for the relations (18.1.12), e.g., we assume that det ⎡⎣ ∂fl ∂Xk ⎤⎦ ≠ 0 , l , k = 1, 2,..., p (the matrix ⎡⎣ ∂fl ∂Xk ⎤⎦ is of rank p ), then it results, from the theorem of implicit functions, that Xk ≡ Xk ( X p + 1 , X p + 2 ,..., X 3 n ), k = 1, 2,..., p . We take

Lagrangian Mechanics

47

X p + 1 = q1 , X p + 2 = q2 ,..., X 3 n = qs , s = 3n − p ; one obtains thus relations of the

form (18.2.2'). Obviously, in case of relations of the form (18.2.1') or of the form (18.2.2') one can have only holonomic and scleronomic constraints. In particular, if no one holonomic constraint takes place (the mechanical system S is free) or if we ignore these constraints, we can take Xk = qk , k = 1, 2,..., 3n , corresponding to the representative space E 3n . But if we take into account all p holonomic constraints, the mechanical system S having only s = 3n − p degrees of freedom, then the parameterization qk , k = 1,2,..., s considered above is minimal. Otherwise, if we do not take into account all the holonomic constraints or if, during the motion, appear supplementary holonomic constraints of the form fl (qk ; t ) ≡ fl (q1 , q2 ,..., qs ; t ) = 0, l = 1,2,..., h ,

(18.2.3)

then the mechanical system S remains with s − h degrees of freedom. Obviously, these constraints are independent too, so that the matrix ⎡⎣ ∂fl ∂q j ⎤⎦ is of rank h ; as well, we suppose that no one of these constraints is a consequence of the equations of motion (it is not a first integral of these equations). The constraints of the form (18.2.3) are rheonomous; we can consider, in particular, also scleronomic constraints of the form fl (q j ) ≡ fl (q1 , q2 ,..., qs ), l = 1,2,..., p .

(18.2.3')

We mention that one can have also unilateral constraints, expressed by means of inequalities; but, in what follows, we will consider only bilateral constraints. The space of generalized co-ordinates q1 , q2 ,..., qs is, as well, a representative space with s dimensions, called the space of configurations (figurative space or Lagrange’s space), which will be denoted by Λs (in the honour of Lagrange). A point P (q1 , q2 ,..., qs ) is a representative point, its position in the space Λs specifying univocally the position of the representative point P in the space E 3 n , as well as the position of the discrete mechanical system S in the space E 3 ; obviously, we must assume that the matrix ⎡⎣ ∂Xk ∂q j ⎤⎦ is of rank s , so that – on the basis of the theorem of implicit functions – to a given discrete mechanical system S corresponds only one representative point P (we notice that one can have several non-zero determinants of order s , so that we can express the minimal parameterization q j in various modes, depending on the generalized co-ordinates Xk ; if one takes into account the constraint relations (18.1.9) too, one sees that the respective expression are equivalent). The succession of value taken by the generalized co-ordinates during time leads to the functions qk = qk (t ), t ∈ [t0 , t1 ], k = 1,2,..., s , which represent the parametric equations of the trajectory of the representative point P , characterizing – at the same time – the motion of the mechanical system S .

MECHANICAL SYSTEMS, CLASSICAL MODELS

48

In case of a particle constrained to stay on a curve one uses only one generalized co-ordinates q ( x i = x i (q , t ), i = 1,2, 3 ; see Sect. 6.2.2.1) in the space Λ1 ; e.g., in case of the simple pendulum this co-ordinate is the angle at the centre θ (see Sect. 7.1.3.1). In case of a particle constrained to stay on a surface we use two generalized co-ordinates ( r = r (q1 , q2 ; t ) ; see Sect. 6.2.2.2) in the space Λ2 ; e.g., in case of the spherical pendulum these co-ordinates are the applicate z and the angle at the centre θ (see Sect. 7.1.3.7). We notice that the particles of the mechanical system S can be replaced by rigid solids. E.g., in case of a physical pendulum we use also a generalized co-ordinate (the angle θ ; see Sect. 14.2.1.2), in the space Λ1 , while in case of a double pendulum one introduces two generalized co-ordinates (the angles θ1 and θ2 ; see Sect. 17.1.1.2), in the space Λ2 . As it can be seen, the generalized co-ordinates are not necessarily lengths; they can be angles too. Moreover, in the frame of the same Lagrangian space, some co-ordinates can be lengths, the other ones can be angles or even other quantities; it is not necessary that all generalized co-ordinates have all the same physical dimension. Practically, one chooses as generalized co-ordinates the parameters which can specify the configuration (position) of the discrete mechanical system S at any moment t . 18.2.1.2 Generalized Displacements. Generalized Velocities. Constraints. Generalized Accelerations Starting from the relation (18.2.1), we can express the real displacements in the form (we denote ri = ∂ri ∂t ) dri =

∂ri dq j + ri dt , i = 1, 2,..., n , ∂q j

(18.2.4)

putting thus in evidence the real generalized displacements dq j . Here and in what follows we use the convention of summation of dummy indices (from 1 to s ) in the space Λs ; analogously, we express the possible displacements in the form Δri =

∂ri Δq j + ri Δt , i = 1, 2,..., n , ∂q j

(18.2.5)

the virtual displacements being given by δri =

∂ri δq , i = 1, 2,..., n . ∂q j j

(18.2.6)

One introduces thus the possible generalized displacements Δq j and the virtual generalized displacements δq j , which have the same properties as those corresponding to the spaces E 3 and E 3 n . E.g., the virtual generalized displacements are differences of possible generalized displacements (as one sees from (18.2.5), (18.2.6)); as well, the

Lagrangian Mechanics

49

real generalized displacements belong to the set of possible generalized displacements, which – in general – does not coincide with the set of virtual generalized displacements. Let be two virtual displacements δ1 ri and δ2 ri of the same particle Pi , written in the form δ1 ri =

∂ri ∂r δ1q j , δ2 ri = i δ2q j , ∂q j ∂q j

where δ1q j and δ2q j are corresponding virtual generalized displacements; we may write δ1 ri + δ2 ri =

∂ri ( δ q + δ2q j ∂q j 1 j

)

too. Hence, by composition of two virtual displacements one obtains a new virtual displacement (the same property for the virtual generalized displacements). Let be the virtual generalized displacements δq j ; if −δq j is, as well, a virtual generalized displacement of the representative point P at a given moment t , then the displacement δq j is a reversible virtual generalized displacements. From the relations (18.2.6), it results that to reversible virtual displacements correspond reversible virtual generalized displacements. The velocities of the particles of the mechanical system S will be given by (we denote q j = dq j dt ) vi =

∂ri q + ri , i = 1, 2,..., n , ∂q j j

(18.2.4')

putting in evidence the real generalized velocities q j (shortly, the generalized velocities). As well, we express the possible velocities in the form (we denote q j = Δq j Δt ) vi =

∂ri q + ri , i = 1, 2,..., n , ∂q j j

(18.2.5')

the virtual displacements being given by (we denote q∗j = δq j Δt ) vi∗ =

∂ri ∗ q , i = 1, 2,..., n . ∂q j j

(18.2.6')

We introduce thus the possible generalized velocities q j and the virtual generalized velocities q∗j . We mention that also these quantities can have different dimensions (can be linear velocities, angular velocities etc.).

MECHANICAL SYSTEMS, CLASSICAL MODELS

50

Using the generalized displacements introduced above, we can express the constraint relation (18.2.3) in one of the forms ∂fl dq j + fl dt = 0, l = 1, 2,..., p , ∂q j

(18.2.7)

∂fl δq = 0, l = 1, 2,..., p . ∂q j j

(18.2.8)

(

If fl = 0 , then it results ∂ ∂fl ∂q j

)

∂t = ∂fl ∂q j = 0, j = 1, 2,..., s , too, so that the

constraints are scleronomous; in this case, the set of virtual generalized displacements coincides with the set of possible generalized displacements, while the real generalized displacements belong to both sets (in case of holonomic constraints). Obviously, we can express the relations (18.2.7), (18.2.8) in one of the forms ∂fl q + fl = 0, l = 1, 2,..., p , ∂q j j

(18.2.7')

∂fl ∗ q = 0, l = 1, 2,..., p , ∂q j j

(18.2.8')

where we have introduced the generalized velocities and the virtual generalized velocities, respectively. If we replace in (3.2.13) the real displacements by the relations (18.2.4) or (18.2.4'), then we can express – in general – the constraint relations in one of the forms akj dq j + ak 0 dt = 0, k = 1, 2,..., m ,

(18.2.9)

akj q j + ak 0 = 0, k = 1, 2,..., m ,

(18.2.9')

where akj = ak 0 =

n

∑ αkl

i =1

n

⋅

∂ri 0 , j = 1, 2,..., s , ∂q j

∑ αkl ⋅ ri + αk 0 , k = 1,2,..., m .

(18.2.9'')

i =1

We notice, from (18.2.4') – (18.2.6'), that a necessary condition to have catastatic constraints is ri = 0, i = 1, 2,..., n ; indeed, in this case, to q j = 0 corresponds vi = 0 (hence, rest with respect to an inertial frame of reference), the inverse implication taking, as well, place (the matrix ⎡⎣ ∂Xk ∂q j ⎤⎦ is of rank s ). In case of holonomic constraints, the respective condition in sufficient too (as a mater of fact, in this case the constraints are scleronomic). In case of non-holonomic constraints one must have also αk 0 = 0 ; hence, from (18.2.9'') it results that these constraints are catastatic if and only if ak 0 = 0, k = 1,2,..., m . We notice that

Lagrangian Mechanics

51 akj =

n

∑ α ki

i =1

⋅

∂ri + ∂q j

n

∑ αki

i =1

⋅

∂ri . ∂q j

Because ri = 0 , while the matrix ⎡⎣ ∂Xk ∂q j ⎤⎦ is of rank s , it results that akj = 0 , j = 1,2,..., s , ⇔ α ki = 0, i = 1, 2,..., n , k = 1, 2,..., m . Hence, if the conditions akj = 0 are fulfilled, then the constraints (18.2.9'') are scleronomic.

We can express the non-holonomic constraint relations with the aid of the virtual generalized displacements too in one of the forms akj δq j = 0, k = 1,2,..., m ,

(18.2.10)

akj q ∗j

(18.2.10')

= 0, k = 1,2,..., m .

By means of the possible generalized displacements, it results akj Δq j = 0, k = 1, 2,..., m ,

(18.2.11)

akj q j∗ = 0, k = 1, 2,..., m ,

(18.2.11')

too. We can thus that, in case of catastatic constraints (holonomic or non-holonomic), the set of virtual generalized displacements coincides with the set of possible generalized displacements and the real generalized displacements belong to these sets. We notice, from (18.2.10), that – in case of bilateral constraints – the virtual generalized displacements are reversible; in case of non-strict unilateral constraints, only some virtual generalized displacements are reversible (these which are equated to zero in the constraint relations). In case of a scalar function ϕ (q1 , q2 ,..., qs ; t ) or of a vector function W (q1 , q2 ,..., qs ; t ) , the operator of total differentiation with respect to time is of the form d ∂ ∂ . = q j + dt ∂q j ∂t

(18.2.12)

This operator contains the operator of space differentiation q j ∂ ∂q j and the operator of time differentiation ∂ ∂t . Noting that ∂ d ∂2 ∂2 d ∂ ∂2 ∂2 = q j + , = q j + ∂qk dt ∂qk ∂q j ∂qk ∂t dt ∂qk ∂q j ∂qk ∂t ∂qk

and assuming that the considered functions are of class C 2 with respect to the corresponding variables (in this case, according to Schwartz’s theorem, the mixed derivatives are independent on the order of differentiation), we have the operator relation

MECHANICAL SYSTEMS, CLASSICAL MODELS

52

∂ d d ∂ . = ∂qk dt dt ∂qk

(18.2.13)

Starting from (18.2.4') and taking into account (18.2.12), (18.2.13), we can calculate the acceleration of the particle Pi in the form ai =

dvi d ⎛ ∂ri = dt dt ⎜⎝ ∂q j

dri ∂ri ∂ri ∂ ⎞ ⎟ q j + ∂q qj + dt = ∂q ( vi + ri ) q j + ∂q qj + ri , j j j ⎠

wherefrom

ai =

∂ri ∂2 ri ∂r qj + q j qk + 2 i q j + ri , i = 1, 2,..., n . ∂q j ∂q j ∂qk ∂q j

(18.2.14)

We have thus put in evidence the generalized accelerations qj = d2q j dt 2 , j = 1,2,..., s .

18.2.1.3 Kinetic Energy. Work. Natural Systems Starting from the expression (18.2.14') of the velocity of the particle Pi , we can calculate the kinetic energy of the discrete mechanical system S in the form T =

1 n 1 n ⎛ ∂r ⎞ ⎛ ∂r ⎞ mi vi2 = ∑ mi ⎜ i q j + ri ⎟ ⋅ ⎜ i q j + ri ⎟ . ∑ 2 i =1 2 i =1 ⎝ ∂q j ∂ q j ⎠ ⎝ ⎠

It results T = T2 + T1 + T0 ,

(18.2.15)

where T2 =

1 g q q , g = gkj = 2 jk j k jk

∂r

n

∑ mi ∂qij

i =1

⋅

∂ri , j , k = 1, 2,...s , ∂qk

(18.2.15')

is a quadratic form in the generalized velocities, T1 = g j q j , g j =

n

∂r

∑ mi ∂qij

i =1

⋅ ri , j = 1, 2,...s ,

(18.2.15'')

is a linear form in the generalized velocities, while

T0 = g 0 , g 0 =

1 n mi ri , 2 i∑ =1

is a constant with respect to these velocities.

(18.2.15''')

Lagrangian Mechanics

53

We notice that T0 ≥ 0 , being thus a non-negative quantity; the equality to zero takes place in case of catastatic constraints (if the constraints are holonomic, then they are scleronomic too). The linear form T1 can have any sign; also this form vanishes in case of catastatic constraints. Hence, in case of such constraints we have T = T2 =

1 n mi 2 i∑ =1

2

⎛ ∂ri ⎞ ⎜ ∂q q j ⎟ ⎝ j ⎠

≥ 0.

(18.2.16)

This quadratic form vanishes only if each bracket vanishes, hence for ∂ri q = 0, i = 1, 2,...n . ∂q j j

We notice that the matrix [ ∂Xk ∂q j ] of the coefficients of this system of linear equations is of rank s ; it results q j = 0, j = 1,2,..., s , the quadratic form T2 being positive definite. We remark, as well, that – in case of catastatic constraints – we have T2 = 0 too, the kinetic energy non-depending explicitly on time. One can state that g ≠ 0, g = det ⎡⎣ g jk ⎦⎤ .

(18.2.17)

Indeed, otherwise, the system of linear equations g jk qk = 0, j = 1,2,..., s , would have non-trivial solutions; thus, a linear combination of these equations could lead to 2T2 = g jk q j qk = 0 , without having all generalized velocities equal to zero. But the quadratic form T2 is positive definite, obtaining thus a contradiction. We mention also that T2 is a non-singular quadratic form (eventually, excepting the points (isolated) in which the correspondence (18.2.1) (or (18.2.2)) is not one-to-one). In general, the s (s + 1) 2 coefficients of the quadratic form T2 are of the form

g jk = g jk (q1 , q 2 ,..., qs ; t ),

j , k = 1, 2,..., s ;

as

well,

g j = g j (q1 , q 2 ,..., qs ; t ),

j = 1, 2,..., s . In case of catastatic constraints we have g jk = g jk (q1 , q 2 ,..., qs ),

j , k = 1, 2,..., s , g j = g j (q1 , q2 ,..., qs ), j = 1,2,..., s . The real elementary work of the given forces (external and internal) is given by dW =

n

∑ Fi

i =1

⋅ dri =

n

∑ Fi

i =1

⋅

∂ri dq j , ∂q j

where we have taken into account (18.2.4); we can also write d W = Q j dq j , Q j =

n

∑ Fi

i =1

⋅

∂ri , j = 1,2,..., s . ∂q j

(18.2.18)

MECHANICAL SYSTEMS, CLASSICAL MODELS

54

By analogy, Q j are called generalized forces (by convention, we can assume that they act upon the representative point P ). The virtual work of these forces is expressed in the form δW = Q j δq j .

(18.2.19)

For the practical computation of the components (co-ordinates) of the generalized force, we can fix all co-ordinates excepting one of them (let be qk , to which corresponds the virtual displacement δqk ); calculating δW in E 3 , one can obtain Qk , making then – successively – k = 1, 2,..., s . We notice that a component Q j of the generalized force has the dimension of a force only if the corresponding generalized co-ordinate q j has the dimension of a length. Supposing that the given forces Fi , i = 1, 2,..., n , derive from a simple quasi-potential U ( ri ; t ) = U ( r1 , r2 ,..., rn ; t ) , then we can write U [ ri (q j ; t ); t ] = U (q1 , q2 ,..., qs ; t ) ,

obtaining thus a simple quasi-potential in generalized co-ordinates; indeed we can write Qj =

n

∑ Fi ⋅

i =1

∂ri = ∂q j

n

∑ ∇iU ⋅

i =1

∂ri = ∂q j

(i ) ∂U ∂x k ∑ ∑ ( i ) ∂q j , i = 1 k = 1 ∂x n

3

k

wherefrom Qj =

∂U , j = 1, 2,..., s , ∂q j

(18.2.20)

the generalized forces being thus quasi-conservative. If U = U (q1 , q1 ,..., qs ) is a simple potential, then the corresponding generalized forces are conservative. But we must notice that the existence of a quasi-potential for the given generalized forces does not imply the existence of a quasi-potential for the given corresponding forces in the space E 3 . If Q j = [U ]j =

∂U d ∂U , j = 1,2,..., s , − dt ∂q j ∂q j

(18.2.21)

then we say that the generalized forces are quasi-conservative, deriving from a generalized quasi-potential U = U (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) . We notice that the generalized forces cannot depend on the generalized accelerations (because the given forces cannot depend on the accelerations), so that the quasi-potential is of the form U = U k qk + U 0 ,

where U j = U j (q1 , q2 ,..., qs ; t ), j = 0,1, 2,..., s . Hence,

(18.2.22)

Lagrangian Mechanics

55

∂U j ⎞ ∂U 0 ⎛ ∂U k  Qj = ⎜ − ⎟ qk − U j + ∂q , j = 1,2,..., s , q q ∂ ∂ j k ⎠ ⎝ j

(18.2.21')

in case of quasi-conservative generalized forces; if the generalized forces are conservative, corresponding to a generalized potential, where U j = U j (q1 , q2 ,..., qs ) , j = 0,1,2,..., s , then we have U j = 0 . By following gauge transformation U k → U k = U k + ∂ϕ ∂qk , k = 1, 2,..., s , U 0 → U 0 = U 0 + ϕ , where

have

ϕ = ϕ (q1 , q2 ,..., qs ; t ) is a function of class C 2 (to ( ∂ ∂qk )(d dt ) = (d dt )( ∂ ∂qk ) ), hence by a transformation of the form

U → U = U + dϕ dt , we obtain the same generalized forces Q j . Hence, the genera-

lized quasi-potential from which derives a quasi-conservative generalized force is determined excepting the total derivative with respect to time of a function ϕ of class C 2 . A field of quasi-conservative generalized forces is non-stationary, while a field of conservative generalized forces is stationary. The elementary work of conservative generalized forces is a total differential, being of the form (18.1.22), in case of a simple potential, or of the form (18.1.22''), in case of a generalized potential. If the generalized forces are quasi-conservative, deriving from a simple quasi-potential, then the elementary work is of the form (18.1.23); if these generalized forces derive from a generalized quasi-potential, then we get ⎛ ∂U 0 ⎞ − U j ⎟ dq j . dW = dU 0 − U 0 dt − U j dq j = dU 0 − (U j q j + U 0 ) dt = ⎜ ∂ q ⎝ j ⎠

(18.2.23)

We notice that the gauge transformation mentioned above for the generalized potential does not affect this elementary work. The mechanical systems which admit a simple or generalized potential (quasipotential) are called natural systems. The generalized forces which do not derive from a simple quasi-potential can be always expressed in the form Qj =

∂U + Q j , U = U (q1 , q2 ,..., qs ; t ) , ∂q j

(18.2.24)

the elementary work being given by dW = dU + ( Q j q j − U ) dt .

(18.2.24')

If U = U (q1 , q2 ,..., qs ) is a potential, hence if U = 0 , then the quantity P = Q j q j represents the power of the non-potential generalized forces. By analogy with the definitions given in E 3 and E 3 n , the non-potential generalized forces Q j of zero power ( P = 0 ) are called gyroscopic generalized forces and depend, obviously, on the

MECHANICAL SYSTEMS, CLASSICAL MODELS

56

distribution of the generalized velocities; in this case, the generalized forces Q j are conservative. If the power of the non-potential generalized forces does not vanish, then these forces are non-conservative. If P < 0 , then the corresponding generalized forces are dissipative. The quasi-conservative forces (18.2.21') which derive from a generalized quasi-potential are of the form (18.2.24), so that (we take U = U 0 ) ∂U j ⎛ ∂U k − Qj = ⎜ ∂qk ⎝ ∂q j

⎞  ⎟ qk − U j , j = 1, 2,..., s . ⎠

(18.2.24'')

If U j = 0, j = 1,2,..., s then it results Q j q j = 0 , the non-potential generalized forces being thus gyroscopic. Hence, any generalized force which admits a generalized potential is the sum of a generalized force which admits a simple potential and a gyroscopic generalized force. Let be, in general, a non-potential generalized force of the form Q j = − αij qi , j = 1, 2,..., s .

(18.2.25)

By a decomposition in symmetric and skew-symmetric parts of the tensor quantity αij , we observe that the non-potential generalized forces Q j = − α( ij )qi , j = 1, 2,..., s ,

(18.2.25')

where the quadratic form R =

1 α q q , 2 ( ij ) i j

(18.2.26)

called the Rayleigh dissipative function, is positive, are dissipative; indeed, we have Q j q j = − α( ij )qi q j ≤ 0 .

We notice that Qj = −

∂R , j = 1, 2,..., s . ∂q j

(18.2.26')

Analogously, the non-potential generalized forces Q j = − α[ ij ]qi = α[ ji ]qi , j = 1, 2,..., s ,

(18.2.25'')

are gyroscopic, because Q j q j = α[ ji ]qi q j = 0 . We observe that the notion of power of the non-potential generalized forces has a signification larger than the classical one, coinciding with the latter one in case of a potential (U = 0 ).

Lagrangian Mechanics

57

18.2.2 Lagrange’s Equations of Second Kind The basic problem of motion of a discrete mechanical system S in the space E 3 is reduced to the problem of motion of the corresponding representative point P , subjected to the action of the generalized forces Q j , j = 1, 2,..., s , and to m non-holonomic constraints (18.2.9) (or (18.2.9')) in the space Λs (the representative point P is on a non-holonomic manifold with s − m dimensions, at the intersection of m non-holonomic hypersurfaces). In what follows, we deduce Lagrange’s equations of second kind in case of holonomic or non-holonomic constraints; a special attention will be given to the case of quasi-conservative forces, by introducing Lagrange’s kinetic potential.

18.2.2.1 Lagrange’s Equations in Case of Holonomic Constraints Let be a discrete mechanical system S subjected to holonomic ideal constraints; the motion of this mechanical system, hence the motion of the representative point P in the space Λs will be governed by the theorem of virtual work, written in the form n

∑ mi

i =1

d2 ri ⋅ δri = δW = dt 2

n

∑ Fi

i =1

⋅ δri ,

(18.2.27)

corresponding to the formulae (11.1.59), (11.1.63) and to the formula (18.1.52), respectively. We calculate n

∑ mi

i =1

=

n

d2 ri ⋅ δri = dt 2

⎡ d ⎛ dr

∑ mi ⎢⎣ dt ⎝⎜ dti

i =1

⋅

n

∑ mi

i =1

d2 ri ∂ri ⋅ δq j dt 2 ∂q j

∂ri ⎞ dri d ⎛ ∂ri ⎞ ⎤ − ⋅ δq . ∂q j ⎠⎟ dt dt ⎝⎜ ∂q j ⎠⎟ ⎦⎥ j

The formula (18.2.4') allows to write ∂vi ∂r = i , i = 1, 2,..., n , j = 1, 2,..., s . ∂q j ∂q j

(18.2.28)

Using the operator relation (18.2.13) too, it results n

∑ mi

i =1

=

1 ⎡d ∂ ( mi vi2 ⎢ 2 i∑ = 1 ⎣ dt ∂q j n

∂vi ⎞ ∂v ⎤ − vi ⋅ i ⎥ δq j ⎟ ∂q j ⎠ ∂q j ⎦ i =1 ∂ ⎤ ⎡ d ⎛ ∂T ⎞ ∂T ⎤ − ( mi vi2 ) ⎥ δq j = ⎢ dt ⎜ ∂q ⎟ − ∂q ⎥ δq j , ∂q j j ⎦ ⎦ ⎣ ⎝ j ⎠

d2 ri ⋅ δri = dt 2

)

n

⎡d⎛

∑ mi ⎢⎣ dt ⎝⎜ vi

⋅

where we have introduced the kinetic energy T . Taking into account also the formula (18.2.19), we can write the equation (18.2.27) in the form ⎡ d ⎛ ∂T ⎞ ∂T ⎤ ⎢ dt ⎜ ∂q ⎟ − ∂q − Q j ⎥ δq j = 0 . j ⎣ ⎝ j ⎠ ⎦

(18.2.27')

58

MECHANICAL SYSTEMS, CLASSICAL MODELS

As a matter of fact, this represents the fourth form of the basic equation. Because the constraints are holonomic, it results that the virtual generalized displacements δq j are independent; hence, we may write d ⎛ ∂T ⎞ ∂T − = Q j , j = 1,2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.29)

getting thus the equation of motion of the representative point P , that is Lagrange’s equations of second kind (shortly, Lagrange’s equations), obtained be Lagrange in 1760. This system of differential equations of second order (the kinetic energy depends on the generalized co-ordinates, the generalized velocities and time) allows to determine the unknown functions, q j = q j (t ), j = 1, 2,..., s , t ∈ [ t0 , t1 ] . The fundamental problem of mechanics remains, further, deterministic; starting from the initial conditions (11.1.9), written for a discrete mechanical system, and taking into account the relations (18.2.1) or (18.2.1') and (18.2.4'), we associate the initial conditions of Cauchy type

q j (t0 ) = q j0 , q j (t0 ) = q j0 .

(18.2.30)

These conditions are univocally determinate (as we have seen in Sects. 18.2.1.1 and 18.2.1.2, in the general case). If q j0 = 0, q j0 = 0, j = 1,2,..., s , then we say that we have to do with homogeneous initial conditions. Thus, the problem is entirely formulated; the functions q j (t ) must be of class C 2 .

18.2.2.2 Case of Non-holonomic Constraints. Natural Systems. Kinetic Potential. Conditions of Equilibrium Let us suppose that the discrete mechanical system S , hence also the representative point P in the space Λs , is subjected to m < s non-holonomic constraints, which are expressed in the form (18.2.10), by means of the generalized displacements. In this case the problem is reduced to the determination of the functions q j = q j (t ), j = 1,2,..., s , so that, besides Lagrange’s equations (18.2.29), the relations (18.2.27') be also satisfied, for any virtual generalized displacements δq j , which verify the constraint relations (18.2.10); applying the method of Lagrange’s multipliers, used several times till now, we can write m ⎡ d ⎛ ∂T ⎞ ∂T ⎤ − − Q − λk akj ⎥ δq j = 0 . j ∑ ⎜ ⎟ ⎢ dt ∂q ∂ q j ⎣ ⎝ j ⎠ ⎦ k =1

(18.2.31)

Assuming that the non-holonomic constraints are distinct, the matrix [akj ] is of rank m ; we can denote the generalized co-ordinates so that det[akj ] ≠ 0, j , k = 1,2,..., m ,

which allows to determine univocally the multipliers λk , by annulling the first m

Lagrangian Mechanics

59

square brackets. From the constraint relations we obtain the virtual generalized displacements δql , l = 1, 2,..., m , as functions of the virtual generalized displacements δqm + 1 , δqm + 2 ,..., δqs , which can be considered to be independent; following the usual reasoning, it follows that all square brackets vanish, so that we obtain Lagrange’s equations in case of non-holonomic constraints (Lagrange’s equations with multipliers) in the form

d ⎛ ∂T ⎞ ∂T − = Q j + Rj , j = 1, 2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.32)

where Rj =

m

∑ λk akj , j

k =1

= 1, 2,..., s ,

(18.2.33)

are the components of the constraint generalized force. To determine the s + m unknowns (the functions q j = q j (t ), j = 1,2,..., s , and the parameters λk , k = 1,2,..., m ) we have thus at our disposal s equations of Lagrange with multipliers and m constraint relations. If the mechanical system S has h holonomic constraints (18.2.3), which have not been eliminated, then the constraint forces are Rj =

m

h

k =1

l =1

∂f

∑ λk akj + ∑ μl ∂qlj

,

(18.2.33')

where μl , l = 1, 2,..., h , are also Lagrange’s multipliers which must be determined, taking into account supplementary constraint relations too. In case of a natural system, the generalized forces derive from a simple or generalized quasi-potential, being of the form (18.2.20) or of the form (18.2.21). Introducing Lagrange’s kinetic potential (the Lagrangian, quantity of the nature of the energy)

L = T +U ,

(18.2.34)

we can write the equations (18.2.32) in the form d ⎛ ∂L ⎞ ∂L − = Rj , j = 1, 2,..., s . dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.35)

If we express the generalized forces in the form (18.2.24), the kinetic potential being, further, given by (18.2.34), then the considered Lagrange equations become

d ⎛ ∂L ⎞ ∂L − = Q j + Rj , j = 1, 2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.36)

MECHANICAL SYSTEMS, CLASSICAL MODELS

60

putting in evidence the non-potential generalized forces. If the constraints of the discrete mechanical system equations become

S are holonomic, then these

d ⎛ ∂L ⎞ ∂L − = Q j , j = 1,2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.37)

while in case of a natural system it results

d ⎛ ∂L ⎞ ∂L − = 0, j = 1, 2,..., s . dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.38)

Using the Euler–Lagrange operator, we can formally write [T ]j + Q j + Rj = 0, j = 1, 2,..., s ,

(18.2.32')

[T ]j + Q j = 0, j = 1, 2,..., s ,

(18.2.29')

[ L ]j + Rj = 0, j = 1, 2,..., s ,

(18.2.35')

[ L ]j + Q j + Rj = 0, j = 1,2,..., s ,

(18.2.36')

[ L ]j + Q j = 0, j = 1,2,..., s ,

(18.2.37')

[ L ]j = 0, j = 1, 2,..., s .

(18.2.38')

as well as

Thus, Lagrange’s equations take a simpler form, which is easier to handle in case of natural systems; moreover, one obtains – in this case – interesting results concerning the integration of the respective system of equations. We notice that

L = L2 + L1 + L0 ,

(18.2.34')

where we take into account (18.2.15), so that

L2 = T2 , L1 = T1 + U − U 0 , L0 = T0 + U 0 ,

(18.2.34'')

in case of a generalized quasi-potential; if the quasi-potential is a simple one, then U = U 0 . The mechanical systems which admit a kinetic potential are called mechanical systems of first kind. In general, we can consider a mechanical system the motion of which is specified by a Lagrangian L = L (q1 , q1 ,..., qs , q1 , q2 ,..., qs ; t ) , even in the case in which the system is non-natural, with the condition that the Hessian of this function with respect to the generalized velocities be non-zero

Lagrangian Mechanics

61 ⎡ ∂2 L ⎤ det ⎢ ⎥ ≠ 0. ⎣ ∂q j ∂qk ⎦

(18.2.34''')

As a matter of fact, if the system is natural, then we have ∂2 L ∂2T = = g jk , ∂q j ∂qk ∂q j ∂qk

so that the considered Hessian is given by det[ g jk ] = g ≠ 0 , as it has been shown in Sect. 18.2.1.3. To the state of rest of the discrete mechanical system S with respect to an inertial frame of reference corresponds a position of rest of the representative point P in the space Λs . From (18.2.32) it results that (we make q j = 0, qj = 0, j = 1,2,..., s ) Q j + Rj = 0, j = 1, 2,..., s .

(18.2.39)

Obviously, these relations must be verified also by the initial conditions, where q j0 = 0, j = 1, 2,..., s . More precisely, the conditions (18.2.39) should be written in the form Q j (q10 , q20 ,..., qs0 , 0, 0,..., 0; t ) + Rj (q10 , q20 ,..., qs0 , 0, 0,..., 0; t ) = 0, t ∈ [t0 , t1 ],

(18.2.39')

where we have put in evidence also the position of equilibrium (corresponding to the initial conditions). If the discrete mechanical system S is subjected only to holonomic constraints, then the conditions of equilibrium are written in the form Q j = 0, j = 1, 2,..., s .

(18.2.40)

If the generalized forces derive from a simple quasi-potential, we obtain ∂U = 0, j = 1, 2,..., s , ∂q j

(18.2.40')

hence a necessary condition for the quasi-potential U to admit an isolated extremum. Concerning the state of rest, one puts important problems of stability, which will be considered afterwards.

18.2.2.3 Normal Form of Lagrange’s Equations Taking into account the expression (18.2.15), (18.2.15'), (18.2.15''), and (18.2.15''') of the kinetic energy, we can find an explicit form of Lagrange’s equations (18.2.32). We calculate thus ∂T ∂q j = g jk qk + g j and then

MECHANICAL SYSTEMS, CLASSICAL MODELS

62

∂g jk d ⎛ ∂T ⎞ ⎛ ∂g j ⎞ = g jk qk + q q + + g jk ⎟ qk + g j , ⎜ ⎟ dt ⎝ ∂q j ⎠ ∂ql k l ⎜⎝ ∂qk ⎠ ∂gk ∂q 0 ∂T 1 ∂gkl q q + q + = ; ∂q j 2 ∂q j k l ∂q j k ∂q j

noting that the product qk ql is symmetric in the indices k and l , it results ∂g jk ∂ql

qk ql =

∂g jl ⎞ 1 ⎛ ∂g jk q q , + 2 ⎜⎝ ∂ql ∂qk ⎟⎠ k l

so that we write the equations (18.2.32) in the form g jk qk + [ kl , j ]qk ql + (2 γ jk + g jk )qk −

∂g 0 + g j = Q j + Rj , j = 1, 2,..., s , ∂q j

(18.2.41)

where we have introduces Christoffel’s symbols of the first kind [ kl , j ] =

∂g jl ∂g 1 ⎛ ∂g jk + − kl ⎜ ∂qk ∂q j 2 ⎝ ∂ql

⎞ ⎟ , j , k , l = 1, 2,..., s , ⎠

(18.2.42)

⎞ ⎟ , j , k = 1, 2,..., s . ⎠

(18.2.43)

as well as the skew-symmetric quantities γ jk = − γkj =

∂g 1 ⎛ ∂g j − k ⎜ ∂q j 2 ⎝ ∂qk

We have thus obtained a system of linear equations in the generalized accelerations qk , k = 1, 2,..., s . Let g jk = g kj be the algebraic complement of the element g jk in the determinant g , which is definite by (18.2.17); the normalized algebraic complement will be g jk = g kj = g jk g . We notice that the relations det[ g jk ] =

1 , g g mj = δkm , g jk

(18.2.44)

where δkm is Kronecker’s symbol (equal to unity for k = m and equal to zero for k ≠ m ), take place. Multiplying the system of equations (18.2.41) by g mj and summing with respect to j , it results (we replace then the free index m by j ) ⎧⎪ j qj + ⎨ ⎩⎪ k

⎫⎪ ∗ ∗ ⎬ q q + α jk qk + β j = Q j + Rj , j = 1, 2,..., s , l ⎭⎪ k l

(18.2.45)

Lagrangian Mechanics

63

where we have introduced Christoffel’s symbols of second kind, defined by (the relations of definitions for Christoffel’s symbols correspond to the relations of definition (A.1.44) and (A.1.45) from E 3 ) ⎧⎪ j ⎨ ⎩⎪ k

⎫⎪ mj ⎬ = [ kl , m ]g , j , k , l = 1, 2,..., s , l ⎭⎪

(18.2.42')

as well as the notations (as everywhere in this work, we make not considerations on covariance or on contravariance, the position of the indices – up or down – having no one signification) ⎛ ∂g ⎞ αmk = (2 γ jk + g jk )g mj , βm = ⎜ − 0 + q j ⎟ g mj , ∂ q j ⎝ ⎠ ∗ ∗ mj mj Qm = Q j g , Rm = Rj g , m = 1, 2,..., s .

(18.2.46)

The system of equations (18.2.45) represents thus the normal form of Lagrange’s equations, in which the generalized accelerations are expressed as functions of the generalized velocities and of the generalized co-ordinates; this form of the equations can be always obtained, because g ≠ 0 . In the case of catastatic constraints we have T = T2 , and the equations (18.2.45) take the form ⎧⎪ j qj + ⎨ ⎩⎪ k

⎫⎪ ∗ ∗ ⎬ q q = Q j + Rj , j = 1, 2,..., s . l ⎭⎪ k l

(18.2.47)

If the constraints are holonomic, then the equations (18.2.45) become ⎧⎪ j qj + ⎨ ⎪⎩ k

⎫⎪ ∗ ⎬ q q + α jk qk + β j = Q j , j = 1, 2,..., s , l ⎪⎭ k l

(18.2.45')

while the equations (18.2.47) read (in this case, the constraints are scleronomic too) ⎧⎪ j qj + ⎨ ⎪⎩ k

⎫⎪ ∗ ⎬ q q = Q j , j = 1, 2,..., s . l ⎪⎭ k l

(18.2.47')

In case of a single particle, the equations (18.2.47') become (6.1.24''').

18.2.2.4 Theorems of Existence and Uniqueness We can replace the system of s differential equations of second order (18.2.45') by a system of 2s differential equations of first order in the normal form ⎪⎧ j q j = u j , u j = ϕ j , ϕ j = Q j∗ − ⎨ ⎪⎩ k

⎪⎫ ⎬ u u − α jk uk − β j , j = 1,2,..., s , l ⎪⎭ k l

(18.2.48)

MECHANICAL SYSTEMS, CLASSICAL MODELS

64

where u j = u j (q1 , q2 ,..., qs ; t ), ϕ j = ϕ j (q1 , q2 ,..., qs , u1 , u2 ,..., us ; t ) . This system is autonomous; if the time does not intervene explicitly in u j and ϕ j , then the system is autonomous (or dynamical). We associate to this system the initial conditions (18.2.30) in the form q j (t0 ) = q j0 , u j (t0 ) = q j0 , j = 1, 2,..., s ,

(18.2.48')

the problem at the limit (18.2.48), (18.2.48') being thus a problem of Cauchy type. The problem (18.2.45'), (18.2.30) is equivalent to the problem (18.2.48), (18.2.48'); for the latter one we can state Theorem 18.2.1 (of existence and uniqueness; Cauchy–Lipschitz). If the functions u j and ϕ j , j = 1,2,..., s , are continuous on the interval (2s + 1) -dimensional D , specified

q j0 − Q j0 ≤ q j ≤ q j0 + Q j0 , q j0 − U j0 ≤ u j ≤ q j0 + U j0 ,

by

t0 − T0 ≤ t ≤ t0 + T0 , Q j0 , U j0 , T0 = const, j = 1, 2,..., s , and definite on the space

Cartesian product of the phase space (of canonical co-ordinates q1 , q2 ,..., qs , u1 , u2 ,..., us ) by the time space (of co-ordinate t ) and if Lipschitz’s conditions u j (q1 , q2 ,..., qs ; t ) − u j (q1 , q2 ,..., qs ; t ) ≤

1 λ q − qk , μj T k k

ϕ j (q1 , q2 ,..., qs , u1 , u2 ,..., us ; t ) − ϕ j (q1 , q2 ,..., qs , u1 , u2 ,..., us ; t ) ≤

1 1 λ ⎡ q − qk + uk − uk ⎤ ν j T k ⎢⎣ τ k ⎦⎥

are satisfied for j = 1,2,..., s , where T > 0 is a time constant independent on q j , u j and t , τ is a time constant equal to unity, while λk , μj , ν j are constants equal to unity (dimensionless if the corresponding generalized co-ordinate is a length and having the dimension of a length if the corresponding generalized co-ordinate is nondimensional), then there exists a unique solution q j = q j (t ), u j = u j (t ) of the system (18.2.48), which satisfies the initial conditions (18.2.48') and is definite on the interval t0 − T ≤ t ≤ t0 + T , where Q j0 U j0 ⎡ ( ! ) , τμj ( ! ), T T ≤ min ⎢T0 , λj V V ⎣

⎤ ⎥ , V = max μj u j ( ! ) , τν j ϕ j ( ! ) in D , ⎦

(

)

the sign (!) indicating “without summation”. The continuity of the functions u j and ϕ j on the interval D ensures the existence of the solution, according to Peano’s theorem. The condition of Lipschitz must be fulfilled too for the uniqueness of the solution; these latter conditions may be replaced by other ones less restrictive, in accordance with which the partial derivatives of the first order of the functions u j and ϕ j , j = 1,2,..., s , must exist and be bounded in absolute value

Lagrangian Mechanics

65

on the interval D . Besides, the conditions in the Theorem 18.2.1 are sufficient conditions of existence and uniqueness, which are not necessary too. The existence and the uniqueness of the solution have been put in evidence only on the interval [t0 − T , t0 + T ] , in the neighbourhood of the initial moment t0 (as a matter of fact, the moment t0 must not be the initial moment, but can be a moment arbitrarily chosen); taking, e.g., the moment t0 + T as initial moment, it is possible – repeating the above reasoning – to extend the solution on a interval of length 2T1 if, naturally, the sufficient conditions of existence and uniqueness of the theorem are verified in the neighbourhood of the new initial moment. Thus, we can prolong the solution for t ∈ [t1 , t2 ] , corresponding to a certain interval of time in which the considered mechanical phenomenon takes place, or even for t ∈ [t1 , ∞ ) or t ∈ ( −∞, t1 ] or t ∈ ( −∞, ∞ ) . We can state also theorems on the continuous or analytical dependence of the solution on a parameter, analogous to the Theorems 6.1.3 and 6.1.4; as well, we can state Theorem 18.2.2 (on the differentiability of the solutions). If, in the neighbourhood of a point P (q1 ,q2 ,..., qs , u1 , u2 ,..., us ; t ) ∈ D , the functions u j (q1 ,q2 ,..., qs ; t ) and ϕ j (q1 ,q2 ,..., qs , u1 , u2 ,..., us ; t ), j = 1,2,..., s , are of the class C k , then the solutions

q j (t ) and u j (t ) of the system (18.2.48), which satisfy the initial conditions (18.2.48'), are of class C k + 1 in a neighbourhood of the point P . In particular, if q j0 = 0 and Q j (t0 ) = 0 , which leads also to ϕ j (t0 ) = 0 ,

j = 1,2,..., s , the constraints being scleronomic, then we obtain q j (t ) = q j0 , j = 1,2,..., s , the solution being unique, according to the Theorem 18.2.1; hence, the representative point is at rest with respect to a given fixed frame of reference.

18.2.3 Transformations. First Integrals In the study of Lagrange’s equations an important rôle is played by the transformations of co-ordinates and by the gauge transformations which will be considered in what follows. As well, to integrate this system of equations, the problem is put to find first integrals; in this order of ideas, we will put in evidence – especially – the first integral of energy.

18.2.3.1 Point Transformations We call point transformation a transformation of generalized co-ordinates of the form q j = q j (q1 ,q2 ,..., qs ; t ), j = 1, 2,..., s ,

(18.2.49)

where q j are functions of class C 2 with respect to the new generalized co-ordinates; we assume that

MECHANICAL SYSTEMS, CLASSICAL MODELS

66

⎡ ∂q j ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂qk ⎦

(18.2.49')

the correspondence between the two systems of co-ordinates being one-to-one. In this section, we will denote dq j dt = q j and will have q j =

∂q j ∂q j qk + , j = 1, 2,..., s , ∂qk ∂t

(18.2.49'')

hence, q j = q j (q1 ,q2 ,..., qs , q1 , q2 ,..., qs ; t ) , so that ∂q j ∂q j = , j , k = 1,2,..., s .  ∂qk ∂qk

(18.2.50)

One the other hand

∂q j ∂ 2q j ∂ 2q j ∂ ⎛ ∂q j ⎞  ∂ ⎛ ∂q j ⎞ d ⎛ ∂q j ⎞ , = = = ql + q + ∂qk ∂qk ∂ql ∂qk ∂t ∂ql ⎜⎝ ∂qk ⎟⎠ l ∂t ⎜⎝ ∂qk ⎟⎠ dt ⎜⎝ ∂qk ⎟⎠ wherefrom it result the relation between operators

( ) = ddt ⎛⎜⎝ ∂∂q

∂ d ∂qk dt

k

⎞ ⎟ , k = 1,2,..., l . ⎠

(18.2.51)

By the transformation (18.2.49), (18.2.49'), and (18.2.49'') we obtain T (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) = T (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) , where T is a new function which represents the kinetic energy in the generalized co-ordinates. We can write ∂T ∂T ∂q j ∂T ∂q j ∂T ∂q j ∂T d ⎛ ∂q j = + = + ∂qk ∂q j ∂qk ∂q j ∂qk ∂q j ∂qk ∂q j dt ⎜⎝ ∂qk d ⎛ ∂T ⎞ ∂q j ∂T ∂T ∂q j ∂T ∂q j d ⎛ ∂T ⎞ ∂T , = = = + ∂q j ∂qk ∂q j ∂qk dt ⎜⎝ ∂qk ⎟⎠ dt ⎜⎝ ∂q j ⎟⎠ ∂qk ∂q j ∂qk

⎞ ⎟, ⎠

d ⎛ ∂q j dt ⎜⎝ ∂qk

⎞ ⎟, ⎠

where we took into account (18.2.50), (18.2.51); analogously, the generalized forces read Qk =

n

∑ Fi

i =1

⋅

∂ri = ∂qk

n

∑ Fi

i =1

⋅

∂ri ∂q j , ∂q j ∂qk

wherefrom Qk = Q j

∂q j , k = 1,2,..., s , ∂qk

(18.2.52)

Lagrangian Mechanics

67

Thus, it results

d ⎛ ∂T ⎞ ∂T ⎡ d ⎛ ∂T ⎞ ∂T ⎤ ∂q j − − Qk = ⎢ ⎜ − − Qj ⎥ . ⎜ ⎟ ⎟  dt ⎝ ∂qk ⎠ ∂qk ⎣ dt ⎝ ∂q j ⎠ ∂q j ⎦ ∂qk

(18.2.53)

Hence, if the equations (18.2.29) take place, then we will have d ⎛ ∂T ⎞ ∂T − = Qk , k = 1, 2,..., s , dt ⎜⎝ ∂qk ⎟⎠ ∂qk

(18.2.54)

too. On the other hand, taking into account the condition (18.2.49'), the relations (18.2.54) entail the equations (18.2.29). We can thus state Theorem 18.2.3 Lagrange’s equations are invariant to a point transformation of the form (18.2.49), (18.2.49') in case of a holonomic discrete mechanical system.

18.2.3.2 Gauge Transformations We have seen in Sect. 18.2.2.2 that the motion of a natural mechanical system is governed be Lagrange’s equations (18.2.38), for which it is necessary to determine Lagrange’s kinetic potential L . We notice that the Lagrangian K L , K = const , leads to the same equations. One puts the problem to find a kinetic potential L ′ = L + L , if this one exists, to obtain the same equations (18.2.38); in this case, one must have

d ⎛ ∂L ⎞ ∂L − ≡ 0, dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.55)

for any q j = q j (t ), j = 1, 2,..., s . For this, it is necessary that

L = a j q j + a 0 , a j = a j (q1 , q2 ,..., qs ; t ), j = 1, 2,.., s , a0 = a 0 (q1 , q2 ,..., qs ; t ), otherwise the coefficients of the generalized accelerations qj cannot vanish, Replacing in (18.2.25), it results (the point corresponds to a partial derivative with respect to time) ∂ak ⎛ ∂a j ⎜ ∂q − ∂q j ⎝ k

∂a 0 ⎞ ⎟ qk + a j − ∂q ≡ 0, j = 1,2,..., s , j ⎠

so that we obtain the conditions ∂a j ∂a ∂a − k = 0, a j − 0 = 0, j , k = 1,2,..., s . ∂qk ∂q j ∂q j

Hence, it exists a function ϕ = ϕ (q1 , q2 ,..., qs ; t ) , so that

MECHANICAL SYSTEMS, CLASSICAL MODELS

68 aj =

∂ϕ , j = 1, 2,..., s , a 0 = ϕ . ∂q j

We find thus

L =

dϕ ∂ϕ . q + ϕ = ∂q j j dt

(18.2.56)

We can state Theorem 18.2.4 In case of a discrete mechanical system which admits a kinetic potential L , this one is determined excepting the total derivative with respect to time of an arbitrary function of class C 2 , of generalized co-ordinates and time. In particular, the Lagrangian can be of the form (18.2.34). This gauge transformation corresponds to the analogous transformation put in evidence in Sect. 18.2.1.3 for a generalized quasi-potential.

18.2.3.3 First Integrals of Lagrange’s Equations A function f (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) which is reduced to a constant along the integral curves of Lagrange’s system of equations (18.2.29) is a first interval of this system. If we determine l ≤ 2s first integrals fk (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) = ck , ck = const, k = 1, 2,..., l ,

(18.2.57)

the matrix ∂ ( f1 , f2 ,..., fl ) ⎡ ⎤ M = ⎢ ⎥⎦ ( q , q ,..., q , q , q ,..., q ) ∂    s s ⎣ 1 2 1 2

being of rank l , then all these integrals are functionally independent (independent first integrals) and we can express l unknown functions of the system (18.2.29) by means of the other ones; replacing in this system of equations, the problem is reduced to the integration of a system of equations with 2s − l unknowns (hence, a smaller number of unknowns). If l = 2s , then all the first integrals are independent, so that the system (18.2.57) of first integrals determines all the unknown functions. We notice that the first integrals (18.2.57) are no more independent for l > 2s ; hence we can set up at the most 2s independent first integrals. Solving the system (18.2.57) for l = 2s , we obtain (the matrix M is a square matrix of order 2s , for which det M ≠ 0 ) q j = q j (t ; c1 , c2 ,..., c2 s ), q j = q j (t ; c1 , c2 ,..., c2 s ), j = 1,2,..., s ,

(18.2.58)

hence the general integral of the system of equations (18.2.29). Thus, one puts in evidence 2s scalar constants of integration. Imposing the initial conditions (18.2.30), we get

Lagrangian Mechanics

69

q j (t0 ; c1 , c2 ,..., c2 s ) = q j0 , q j (t0 ; c1 , c2 ,..., c2 s ) = q j0 , j = 1, 2,..., s ;

the conditions (18.2.30) being independent, we can write ⎡ ∂ (q10 , q20 ,..., qs0 , q10 , q20 ,..., qs0 ) ⎤ det ⎢ ⎥ ≠ 0 ∂ (c1 , c2 ,..., c2 s ) ⎣ ⎦

and – on the basis of the theorem of implicit functions – we deduce

ck = ck (t0 ; q10 , q20 ,..., qs0 , q10 , q20 ,..., qs0 ), k = 1, 2,..., 2s . Thus, it results, finally, q j = q j (t ; t0 , q10 , q20 ,..., qs0 , q10 , q20 ,..., qs0 ) ,

q j = q j (t ; t0 , q10 , q20 ,..., qs0 , q10 , q20 ,..., qs0 ) .

(18.2.59)

Hence, in the frame of the conditions of the theorem of existence and uniqueness, the principle of virtual work and the principle of initial conditions determine, univocally, the motion of the discrete mechanical system S , subjected to ideal and holonomic constraints, in a finite interval of time; by prolongation, the statement can become valid for any t . Thus, the deterministic aspect of mechanics is put in evidence in its Lagrangian presentation too. If f1 = c1 and f2 = c2 are two distinct first integrals, then f ( f1 , f2 ) = c will be also a first integral. Any other first integral is expressed, in general, as a function of the 2s distinct first integrals of Lagrange’s equations. From (18.2.57) it results dfk dt = 0 ; hence dfk df q j + k qj + f = 0, k = 1,2,.., l , dq j dq j

(18.2.57')

relations which must be identically satisfied if one takes into account the equations of motion (or the general integral of these equations, for any constants of integration).

18.2.3.4 First Integrals of Painlevé and Jacobi. First Integral of Mechanical Energy We can find a first integral particularly important and useful of the system of equations (18.2.29), multiplying these equations by the generalized velocities q j , j = 1, 2,..., s , and summing for all indices j ; we obtain

d ⎛ ∂T dt ⎜⎝ ∂q j We notice that

∂T ⎞ ⎟ q j − ∂q q j = Q j q j . j ⎠

MECHANICAL SYSTEMS, CLASSICAL MODELS

70

d ⎛ ∂T ⎞ d ⎛ ∂T ⎞ ∂T q = q − q . dt ⎜⎝ ∂q j ⎟⎠ j dt ⎜⎝ ∂q j j ⎟⎠ ∂q j j

Because T = T (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) , we have, as well, dT ∂T ∂T = q j + q + T , dt ∂q j ∂q j j

so that

d ⎛ ∂T ⎞ q − T ⎟ = Q j q j − T . dt ⎜⎝ ∂q j j ⎠

(18.2.60)

Taking into account (18.2.15) and noting that ∂T2 ∂T q = 2T2 , 1 q j = T1 , ∂q j j ∂q j

according to Euler’s theorem concerning the homogeneous functions, we can also write d ( T − T0 ) = Q j q j − T . dt 2

(18.2.60')

If it exists a function W = W (q1 , q2 ,..., qs ; t ) , so that Q j q j − T = dW dt (the function W cannot depend on the generalized velocities too, because dW dt would introduce also generalized accelerations), then we state Painlevé’s first integral T2 − T0 − W = h , h = const .

(18.2.61)

We notice that one must have −T0 = W , because they are terms which do not depend on the generalized velocities q j ; as well, if the given forces Fi , i = 1, 2,..., n , do not depend on the velocities, hence if the generalized forces Q j , j = 1, 2,..., s , do not depend on the generalized velocities, then cannot appear quadratic terms in the generalized velocities, resulting T2 = 0 . In these conditions, we state that Painlevé’s first integral does not contain explicitly the time. If the generalized forces admit a simple quasi-potential of the form (18.2.20), then we can introduce the kinetic potential (18.2.34), so that the relation (18.2.60) reads d ⎛ ∂L ⎞ q − L ⎟ + L = 0 ; dt ⎜⎝ ∂q j j ⎠

(18.2.62)

assuming that the kinetic potential L does not depend explicitly on time (L = 0) , we get Jacobi’s first integral (Jacobi,C.G.J., 1882)

Lagrangian Mechanics

71 ∂L q − L = h , h = const , ∂q j j

(18.2.63)

which can be expressed also in the form (hence, a particular case of Painlevé’s first integral) T2 − T0 − U = h , h = const .

(18.2.63')

In particular, if the mechanical system is catastatic, hence scleronomic (being holonomic), then we have T = 0 , so that L = 0 if U = 0 , hence if the forces are conservative, deriving from a simple potential; we obtain the first integral of mechanical energy E = T − U = h , T = T2 , h = const .

(18.2.64)

The relation (18.2.62) with L = 0 can be written in the form

⎡ d ⎛ ∂L ⎞ ∂L ⎤ ⎢ dt ⎜ ∂q ⎟ − ∂q ⎥ q j = 0 j ⎦ ⎣ ⎝ j ⎠

(18.2.62')

too. Starting from this result, some authors try to obtain Lagrange’s equations, which is not possible because the generalized velocities q j cannot by arbitrarily chosen (they correspond to the motion in the actual state). If the generalized forces admit a generalized quasi-potential of the form (18.2.21), with (18.2.22), then we can express them by the relations (18.2.3); we obtain ∂U j ⎞ ∂U 0 ⎛ ∂U k  − Q j q j = ⎜ ⎟ q j qk − U j q j + ∂q q j q q ∂ ∂ j j k ⎝ ⎠ ∂U 0 dU 0 dU 0 = − U 0 − U j q j = − U . q − U j q j = ∂q j j dt dt

The relation (18.2.60') becomes d ( T − T0 − U 0 ) + L = 0 . dt 2

(18.2.65)

Supposing that the kinetic potential L = T + U does not depend explicitly on time (L = 0) , it results a first integral of Jacobi type (particular case of the first integral of Painlevé) T2 − T0 − U 0 = h , h = const .

(18.2.66)

As well, in case of scleronomic constraints we have T0 = 0 (and T0 = 0 too), being thus sufficient that U 0 = 0 , hence that the forces be conservative, deriving from a generalized potential, to have a first integral of mechanical energy in the form

72

MECHANICAL SYSTEMS, CLASSICAL MODELS

E = T − U 0 = h , T = T2 , h = const .

(18.2.67)

The mechanical systems which admit a first integral of Jacobi or a first integral of Jacobi type are called generalized conservative systems; we have seen that these systems are natural systems for which the kinetic potential L does not depend explicitly on time (L = 0) . The quasi-potential U or the scalar potential U 0 are quantities of energetical nature; hence, the potential energy is V = −U , in case of a simple quasi-potential, or V = −U 0 , in case of a generalized quasi-potential. We introduce thus the generalized mechanical energy

E = T2 − T0 + V ,

(18.2.68)

which is constant in case of Jacobi’s first integral or of a first integral of Jacobi type. In particular, if the forces are conservative (we have V = 0 ) and the constraints are scleronomic (we have T0 = 0 and T = T2 ), then the generalized mechanical energy is equal to the mechanical energy, obtaining thus the first integral of mechanical energy. In case of generalized forces of the form (18.2.24), the relation (18.2.60) reads d ⎛ ∂L ⎞ q j − L ⎟ + L = Q j q j . ⎜ dt ⎝ ∂q j ⎠

(18.2.69)

If the kinetic potential does not depend explicitly on time ( L = 0 ), then it results

dE d ⎛ ∂L ⎞ = q − L ⎟ = Q j q j . dt dt ⎜⎝ ∂q j j ⎠

(18.2.69')

We can thus state that the mechanical system S admits a Jacobi’s first integral (written in the form (18.2.63) or in the form (18.2.63')) for generalized non-potential forces too. The relation (18.2.69') allows also to notice that, in case of dissipative nonpotential forces, the energy of the mechanical system diminishes. We observe that in none of the above considered first integrals does not intervene the component T1 of the kinetic energy; this is explained because the kinetic energy is determined excepting the total derivative with respect to time of a function ϕ (q1 , q2 ,..., qs ; t ) of class C 2 , hence excepting dϕ dt = ( ∂ϕ ∂q j )q j + ϕ . In case of non-holonomic constraints, Lagrange’s equations have the form (18.2.32), so that the relation (18.2.60) becomes d ⎛ ∂T ⎞ q − T ⎟ = ( Q j + R ) q j − T . dt ⎜⎝ ∂q j j ⎠

Taking into account (18.2.29') and (18.2.33), we can also write

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73

m d ⎛ ∂T ⎞ q j − T ⎟ = Q j q j − T − ∑ λk ak 0 . ⎜ dt ⎝ ∂q j ⎠ k =1

(18.2.70)

If the constraints are catastatic (we have ak 0 = 0, k = 1, 2,..., m ), then the relation (18.2.60) becomes (we have T = 0 too)

d ⎛ ∂T ⎞ q − T ⎟ = Q j q j , dt ⎜⎝ ∂q j j ⎠

(18.2.71)

so that the influence of these constraints is not seen; written in the form (18.2.60'), this relation becomes (we have also T0 = T1 = 0 ) dT2 = Q j q j . dt

Expressing the generalized ( E = T2 + V , V = −U )

forces

in

(18.2.71') the

form

dE = Q j q j − U , dt

(18.2.25),

we

can

write

(18.2.72)

so that, in the case in which the potential part does not depend explicitly on time (U = 0 ), while the non-potential forces are gyroscopic, we get a first integral. Hence, in general, a natural and catastatic, non-holonomic mechanical system, for which the generalized forces derive from a simple or generalized potential, has a first integral of energy of the form E = T + V = h , h = const .

(18.2.73)

Such a mechanical system is called a conservative system. If these generalized forces have only the non-potential part (U = 0 ), which is gyroscopic, even the kinetic energy T is conserved during the motion. We notice that this first integral of energy differs from that obtained for a discrete mechanical system S in E 3 because the holonomic constraint forces do not intervene any more. As a matter of fact, in case of catastatic constraints no essential difference appear, the real elementary work of the corresponding constraints forces vanishing. We notice also that, if we take into account (18.2.25'), (18.2.26) in case of a potential (U = 0 ), then the relation (18.2.72) leads to dE = −2 R , dt

(18.2.74)

where R is Rayleigh’s dissipative function, which characterizes the decrease of the mechanical energy E .

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74

18.2.3.5 Reduction of Lagrange’s System of Equations by Means of Jacobi’s First Integral One can state Theorem 18.2.5. Jacobi’s first integral allows the reduction of Lagrange’s system of equations (18.2.38), corresponding to a holonomic and natural mechanical system with s degrees of freedom, to a system of equations corresponding to a mechanical system with s − 1 degrees of freedom (Jacobi,C.G.J., 1882). In particular, in case of a mechanical system with a single degree of freedom (a mechanical system with complete constraints), the first integral ∂L q − L = h , h = const , ∂q

(18.2.75)

reduces the problem to a quadrature; indeed, from (18.2.75) we can get q = f (q ) , wherefrom

t − t0 =

q

∫q

0

dq . f (q )

(18.2.75')

In case of a mechanical system with s degrees of freedom, we make the change of generalized velocities q1 → q1 , q2 → q1q2′ ,..., qs → q1qs′ , with qk′ = dqk dq1 = qk q1 ,

k = 1,2,..., s , so that L (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) = Λ(q1 , q2 ,..., qs , q1 , q2′ ,..., qs′ ) . Differentiating, we can write too ∂L ∂Λ ∂L ∂Λ ∂qk′ 1 ∂Λ = = = , j = 1, 2,..., s , ( ! ) , k = 2, 3,..., s , ∂q j ∂q j ∂qk ∂qk′ ∂qk q1 ∂qk′ ∂Λ ∂L = + ∂q1 ∂q1

∂L ∂qk ∂L = +  ∂ ∂q1 q 1 k =2 s

∑ ∂qk

s

∂L

∑ ∂qk

k =2

qk′ =

∂L + ∂q1

∂L qk . q1 k =2 s

∑ ∂qk

By the above mentioned change of variables in the first integral (18.2.63) and by taking into account the preceding results, we can write

q1

∂Λ −Λ = h, ∂q1

(18.2.76)

wherefrom q1 = q1 (q1 , q2 ,..., qs , q2′ , q 3′ ,..., qs′ ) . We denote L ′ = ∂Λ ∂q1 , so that

L ′ = L ′(q1 , q2 ,..., qs , q2′ , q 3′ ,..., qs′ ) . Using the notations thus introduced and taking into account the form (18.2.15) of the kinetic energy, we can write T (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) = T (q1 , q2 ,..., qs , q1 , q2′ ,..., qs′ ) ,

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75

so that T = q12T , where T = T (q1 , q2 ,..., qs , q2′ , q 3′ ,..., qs′ ) ; on the other hand, assuming the existence of a simple quasi-potential, we have T = U + h , h = const , while L ′ = ∂T ∂q1 = 2q1T , whence L ′ = 2 T (U + h ) . We can calculate ∂L ′ ∂2 Λ ∂ 2 Λ ∂q = + 2 1 , j = 1, 2,..., s , ∂q j ∂q j ∂q1 ∂q1 ∂q j ∂L ′ ∂2 Λ ∂2 Λ ∂q = + 2 1 , k = 2, 3,..., s ; ∂qk′ ∂qk′ ∂q1 ∂q1 ∂qk′

as well, by a partial differentiation of the relation (18.2.76) with respect to the generalized co-ordinates and the new generalized velocities, we obtain q1

∂ 2 Λ ∂q1 ∂2 Λ ∂Λ + q1 = , j = 1, 2,..., s , 2 ∂q ∂ ∂ ∂ q q qj  ∂q1 j j 1

q1

∂2 Λ ∂q1 ∂2 Λ ∂Λ + q1 = , k = 2, 3,..., s . 2 ∂q ′ ∂qk′ ∂q1 ∂qk′ ∂q1 k

Comparing with the previous results, we may write ∂L ′ 1 ∂Λ ∂L ′ 1 ∂Λ = , j = 1, 2,..., s , = , k = 2, 3,..., s , ∂q j ∂qk′ q1 ∂q j q1 ∂qk′

wherefrom, using the above obtained relations, we get ∂L ′ 1 ∂L ∂L ′ ∂L = , j = 1, 2,..., s , = , k = 2, 3,..., s . ∂q j ∂qk′ ∂qk q1 ∂q j

Lagrange’s equations (18.2.38) lead to d ⎛ ∂L ′ ⎞ ∂L ′ − q1 = 0, k = 2, 3,..., s , ⎜ ⎟ ′ dt ⎝ ∂qk ⎠ ∂qk

or to d ⎛ ∂L ′ ⎞ ∂L ′ − = 0, k = 2, 3,..., s , dq1 ⎜⎝ ∂qk′ ⎟⎠ ∂qk

(18.2.77)

so that the functions qk (q1 ), k = 2, 3,..., s are determined by a system of s − 1 equations of second order of Lagrange type (the equations of Jacobi, who has put in evidence this method of calculation) in a (s − 1) -dimensional space. Thus, the number of degrees of freedom of the initial mechanical system has been reduced by a unity.

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76

From the expression of the function L ′ of Lagrangian type, one observes that – in general – the new independent variable q1 intervenes explicitly, so that it is no more possible to obtain a first integral of the type of Jacobi’s first integral. By means of the notations thus introduced, we notice that q1 = t − t0 =

q1

∫q

0 1

T dq1 , U +h

(U + h ) T , so that

(18.2.77')

completing thus the information given by the equations (18.2.77) about the dependence with respect to time.

18.2.3.6 Hidden Co-ordinates. Ignorable Co-ordinates We call, after Helmholtz, hidden co-ordinate qk that co-ordinate which does not intervene explicitly in the expression of the kinetic energy T , hence for which ∂T ∂qk = 0 ; because we can have h ≤ s hidden co-ordinates, it is convenient to denote them q1 , q2 ,..., qh . If the co-ordinate qk does not intervene explicitly in the expression of the kinetic potential L = T + U , corresponding to a natural system which admits a simple or generalized quasi-potential, hence if ∂L ∂qk = 0 , then this co-ordinate is called an ignorable co-ordinate (after W. Thomson) or a kinostenic coordinate (after J. J. Thomson); this denomination is used also in the case of a nonnatural mechanical system which admits a Lagrangian, with the condition (18.2.34'''). We can have r ≤ s ignorable co-ordinates q1 , q2 ,..., qr . Sometimes only the denomination of hidden co-ordinates is used; but we will use both the denominations of hidden and ignorable co-ordinates, to can make distinction in various situations. The mechanical systems in which appear ignorable co-ordinates are called – by some authors – gyroscopic systems (because such situations appear often in case of the gyroscope). If ∂U ∂qk = 0 , where U is a simple or generalized quasi-potential, then an eventually corresponding ignorable co-ordinate is also a hidden one. We notice that we can have ∂ (T + U ) ∂qk = 0 , without having necessarily ∂T ∂qk = 0 (Pars, L., 1965). In the case of h hidden co-ordinates q1 , q2 ,..., qh , the first h equations of Lagrange for holonomic constraints (18.2.29) read d ⎛ ∂T ⎞ = Qk , k = 1, 2,..., h . dt ⎜⎝ ∂qk ⎟⎠

(18.2.78)

If we have Qk = 0 too, then there result h first integrals ∂T = ck , k = 1, 2,..., h , ∂qk

(18.2.78')

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77

each first integral corresponding to a hidden co-ordinate. Assuming the existence of r ignorable co-ordinates q1 , q2 ,..., qr , Lagrange’s equations (18.2.38) take the form d ⎛ ∂L ⎞ = 0, k = 1, 2,..., r , dt ⎜⎝ ∂qk ⎟⎠

(18.2.79)

wherefrom the first integrals ∂L = C k , k = 1, 2,..., r . ∂qk

(18.2.79')

By convention, we denote pj =

∂L , j = 1, 2,..., s . ∂q j

(18.2.80)

In the case of a simple quasi-potential (or in the absence of a quasi-potential), it results pj =

∂T , j = 1, 2,..., s . ∂q j

(18.2.80')

If, in particular, the mechanical system is free or none holonomic constraint has been eliminated, the space of Lagrange being just the space E 3n , then – starting from the kinetic energy corresponding to the n particles – we can write ∂T ∂ ⎛1 n ⎞ mi vi2 ⎟ = m j v j . = ∂v j ∂v j ⎜⎝ 2 i∑ ⎠ =1

We notice that this represents just the magnitude of the momentum of the particle corresponding to the velocity v j . Starting from this observation, we will call the magnitudes p j , j = 1, 2,..., s , components of the generalized momentum; the first integrals (18.2.78') will be written in the form pk = ck , k = 1, 2,..., h ,

(18.2.78'')

while the first integrals (18.2.79') read pk = C k , k = 1, 2,..., r .

(18.2.79'')

If U is a generalized quasi-potential of the form (18.2.22), then it results ∂T = p j − U j , j = 1, 2,..., s , ∂q j

(18.2.81)

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78

and in case of hidden co-ordinates we obtain d ( p − U k ) = Qk . dt k

If we have Qk = 0 too, then we can write the first integrals pk − U k = ck , k = 1, 2,..., h .

(18.2.81')

We notice also that the notion of generalized momentum has a larger sense than the classical one, corresponding to the case in which the generalized co-ordinate is a length. Let us consider, e.g., a particle for which the kinetic energy is expressed in cylindrical coordinates ( T = (m 2)(r 2 + r 2 θ2 + z 2 ) ); in this case, pθ = ∂L ∂θ = ∂T ∂θ = mr 2 θ = m ( r × r )z . Hence, to a generalized co-ordinate which is an angle corresponds a

generalized momentum which is a component of a moment of momentum. If the relation (18.2.80') takes place, then the relation (18.2.78) reads pk =

dpk = Qk , k = 1, 2,..., h . dt

(18.2.82)

Hence, in case of a hidden co-ordinate and in the absence of a quasi-potential, we can write an universal theorem of mechanics. If the generalized co-ordinate is a length, then the generalized force is of the nature of a force (so that the product Qk δqk be a work), obtaining thus a theorem of momentum corresponding to the respective linear coordinate; but if the generalized co-ordinate is an angle, then the generalized force – from the same reasons – is of the nature of a moment, resulting a theorem of moment of momentum corresponding to the respective angular co-ordinate. As a matter of fact, we can state Theorem 18.2.6 (theorem of generalized momentum). In case of several hidden coordinates and in the absence of a quasi-potential, in the free motion of the representative point in the space of configurations, a theorem of generalized momentum of the form (18.2.82) is verified. In the case in which Qk = 0, k = 1, 2,..., h , we can write a conservation theorem of generalized momentum, which is – in fact – a first integral of the form (18.2.78''). As well, in case of generalized forces of the form (18.2.24) and of several ignorable co-ordinates, Lagrange’s equations lead to the relations pk =

dpk = Qk , k = 1, 2,..., r , dt

(18.2.82')

where Qk are non-potential generalized forces; hence, we can state

Theorem 18.2.6' (theorem of generalized momentum; second form). In case of several ignorable co-ordinates, in the free motion of the representative point in the space of configurations, a theorem of generalized momentum of the form (18.2.82') is verified.

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79

If Qk = 0, k = 1, 2,..., r , then we get a conservation theorem of generalized momentum, hence a first integral of the form (18.2.79'').

18.2.3.7 The Routh–Helmholtz Theorem We can state Theorem 18.2.7 (Routh–Helmholtz). If a discrete mechanical system S , with s degrees of freedom, has r ignorable co-ordinates, then the determination of the free motion of the representative point in the space of configurations is reduced to the integration of a system of differential equations of Lagrange type, corresponding to a mechanical system with s − r degrees of freedom and to the calculation of r quadratures. Indeed, let us perform the transformation of Routh, E.J. 1892, 1898]

R =

r

∂L

∑ ∂qk

qk − L .

k =1

(18.2.83)

The first integrals (18.2.79'), where L is of the form (18.2.34'), can be seen as a system of r linear equations in the generalized velocities qk , k = 1, 2,..., r ; the condition (18.2.34''') being fulfilled, we determine the functions qk = qk (qr + 1 , qr + 2 ,..., qs , qr + 1 , qr + 2 ,..., qs ;C 1 ,C 2 ,...,C r ; t ), k = 1, 2,..., r ,

where the generalized co-ordinates qr + 1 , qr + 2 ,..., qs are called palpable co-ordinates (unlike the ignorable co-ordinates). Replacing in (18.2.83), we get

R = R (qr +1 , qr + 2 ,..., qs , qr +1 , qr + 2 ,..., qs ;C 1 ,C 2 ,...,C r ; t ) . In this case, applying a variation δ to the function R of Routh, we obtain δR =

∂R δq j + j = r + 1 ∂q j s

∑

∂R δq j + j = r + 1 ∂q j s

∑

∂R

r

∑ ∂C k

k =1

δC k .

Noting that δL =

∂L δq j + j = r + 1 ∂q j s

∑

∂L qk = k = 1 ∂qk r

δ∑

r

∂L

∑ ∂qk

k =1

s

∂L

∑ ∂q j j =1

δqk +

δq j ,

r

∑ qk δC k

k =1

and taking into account the relation of definition (18.2.83), we get, as well, δR = −

∂L δq j − j = r + 1 ∂q j s

∑

∂L δq j + j = r + 1 ∂q j s

∑

r

∑ qk δC k .

k =1

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80

The variation δ being arbitrary, by comparison of the two relations, it results −

∂L ∂R ∂L ∂R = ,− = , j = r + 1, r + 2,..., s , ∂q j ∂q j ∂q j ∂q j qk =

∂R , k = 1, 2,..., r , ∂C k

so that, replacing in Lagrange’s equations (18.2.38), we get the equations

d ⎛ ∂R ⎞ ∂R − = 0, j = r + 1, r + 2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.84)

which are of Lagrange type, corresponding to a mechanical system with s − r degrees of freedom; these s − r equations determine the generalized co-ordinates q j = q j (t ) ,

j = r + 1, r + 2,..., s . Replacing these co-ordinates in Routh’s function, it results

R = R (C 1 ,C 2 ,...,C r ; t ) and one can obtain the other generalized co-ordinates in the form qk − qk0 =

t

∫t

0

∂R d t = 0, k = 1, 2,..., r , ∂C k

(18.2.85)

the theorem being thus justified. Starting from (18.2.83), Hertz introduced a generalized kinetic energy of the form

T = T − ⎛⎜ V + ⎝

r

⎞

∑ C k qk ⎟⎠ ,

k =1

the sum being considered as being a potential of forces resulting from a unknown cyclic motion; as a matter of fact, Hertz tried to define the potential energy of a mechanical system by the cyclic motion of several “ignorable masses”, which are added to the palpable (visible) masses of the mechanical system S , which justifies the denomination given to the corresponding generalized co-ordinates. Let be the classical model of atom, called “rotator”, which has a motion of rotation about a fixed axis in the space. Because the kinetic energy T = (m / 2) (r 2 + r 2 θ2 + z 2 ) is thus that ∂T / ∂θ = 0 , the theorem of areas is = ∂L / ∂θ mr 2 θ = const ; taking into account r = const , it results θ = ω0t + θ0 , ω0 , θ0 = const . We notice that the function R of Routh plays the rôle of a kinetic potential for the new problem which remains to be solved. But we mention that in this catastatic mechanical system appear always terms linear in the generalized velocities too (even in case of the catastatic mechanical systems for which the kinetic energy is a positive definite quadratic form and which is acted upon by generalized forces which admit a simple quasi-potential).

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81

18.2.3.8 Passing to an Equivalent System of Differential Equation of First Order In the absence of holonomic constraints, the space of configurations can be even the representative space E 3n . Indeed, the relation (18.2.1), written for a fixed t , leads to a metrics of the form ds 2 = g jk dq j dqk ,

(18.2.86)

where the coefficients of the Riemannian space R3n n

g jk =

∂r

∑ ∂qij

i =1

⋅

∂ri , j , k = 1, 2,..., s , ∂qk

(18.2.86')

can be obtained from the coefficients (18.2.15') taking masses equal to unity and maintaining the property det[ g jk ] ≠ 0 ; if the mechanical system is a free one, then s = 3n , hence an Euclidean space E 3 n , so that – by a linear transformation – we can obtain the normal form with constant coefficients

ds 2 =

∑ ⎡⎣⎢ ( dx1( i ) ) n

i =1

2

(

+ dx 2( i )

) + ( dx 3(i ) ) 2

2

⎤, ⎦⎥

(18.2.86'')

the square of the element of arc being expressed in the form of a sum of squares. If the time t is not fixed, ds 2 is an arbitrary polynomial of second degree in dq j (corresponding to the analogous form of the kinetic energy); but the character of the kinetic energy for a mechanical motion leads, by a convenient transformation, to an element of arc of the form (18.2.86'') too. But if there exist certain constraints, the space R3 n becomes a Riemannian space (of positive curvature), because – reducing the number of the configurative dimensions – the Euclidean character of the space is destroyed. In general, the state of a discrete mechanical system S is determined by its position and its motion in the neighbourhood of this position, hence by the set q j = q j (t ) , u j = q j = u j (t ), j = 1,2,..., s . We can thus conceive a representative space with 2s

dimensions, in which the representative point of co-ordinates q j , u j , j = 1, 2,..., s , does represent, at any moment, “the state” of the mechanical system S ; this space is called the state space or the phase space. In the case in which s = 3n , this space is 6n -dimensional and has not the character of a metric manifold; it is only an affine space and one cannot choose a common unity of measure for both types of co-ordinates, but it has a particular importance in the analytical representation of the motion. We have seen in Sect. 18.2.2.4 that one can replace Lagrange’s system of s equations of second order (18.2.45') by a system of 2s differential equations of first order (18.2.48), written in the normal form, the unknown functions being q j = q j (t ) and u j = u j (t ), j = 1, 2,..., s . This system of equations can be written also in the form

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82

d q1 dq du1 du2 dqs dus dt = 2 = ... = = = = ... = = . 1 u1 u2 us ϕ1 ϕ2 ϕs

(18.2.87)

If we succeed to find 2s first integrals of the form fl (q1 , q2 ,..., qs , u1 , u2 ,..., us ; t ) = C l , l = 1,2,..., 2s , where C l are arbitrary constants, then we can determine the

unknown functions q j = q j (t ;C 1 ,C 2 ,...,C 2 s ) , u j = u j (t ;C 1 ,C 2 ,...,C 2 s ) , obviously, the relations dq j dt = u j , j = 1, 2,..., s , taking place too. The problem of motion of the mechanical system S is thus entirely solved. Let us suppose that the mechanical system S is catastatic, so that the time t does not appear explicitly in the expression of the kinetic energy; in this case,, the equations of motion take the form (18.2.47') (we assume to have only holonomic constraints, which – in this case – are scleronomic), where Christoffel’s symbols of second kind do not depend explicitly on time. As well, we suppose that neither the generalized forces Q j∗ do not depend explicitly on time ( ∂Q j∗ ∂t = 0, j − 1,2,.., s ). In this case, nor in the system of equations (18.2.87) the time does not intervene explicitly in the first 2s ratios; neglecting the ratio dt 1 , we will try to integrate the system of equations

dq1 dq du1 d u2 dqs dus = 2 = ... = = = = ... = . ϕ1 ϕ2 ϕs u1 u2 us

(18.2.88)

If we succeed now to find the first integrals fl (q1 , q2 ,..., qs , u1 , u2 ,..., us ) = C l ,

l = 1,2,..., 2s − 1 , then we can determinate 2s − 1 unknown functions with respect to one the co-ordinates, e.g. q1 , in the form qk = qk (q1 ;C 1 ,C 2 ,...,C 2 s −1 ), k = 2, 3,..., s , u j = u j (q1 ;C 1 ,C 2 ,...,C 2 s −1 ), j = 1, 2,..., s .

Associating the equation dq1 u1 = dt 1 , we obtain t +τ =

dq

∫ u1 (q1 ;C1 ,C 2 ,...,Cs −1 ) ,

(18.2.88')

completing thus the integration constants by τ , the integration of the system of 2s linear equations (18.2.88) is thus reduced to the integration of a system of 2s − 1 linear equations and to a quadrature. Assuming that one can write a Jacobi first integral of the form (18.2.63) too, we can express then one of the generalized velocities, for instance q1 , in the form q1 = u1 (q1 , q2 ,..., qs , u2 , u3 ,..., us ; h ) , so that – instead of the system (18.2.88) – we

have to integrate a system of 2s − 2 linear differential equations

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83

d q1 dq du2 du 3 d qs dus = 2 = ... = = = = ... = , u1 u2 us ϕ2 ϕ3 ϕs

(18.2.89)

with ϕk = ϕk (q1 , q2 ,..., qs , u1 , u2 ,..., us ), k = 2, 3,..., s . Thus, the complete system of integrals of the equations of motion (18.2.88) is formed by 2(s − 1) integrals of the system (18.2.89), by Jacobi’s first integral (18.2.63) and by the integral (18.2.88'). We put thus in evidence the trajectory of the representative point in the state space. Such a study has been made by Hamilton in the phase space and we will present it systematically in the next chapter.

18.3 Other Problems Concerning Lagrange’s Equations There are also other problems concerning Lagrange’s equations which present a particular interest, as the forms taken by these equations in some special cases or different systems of equations of motion equivalent to them. Besides these results, we will deal with some significant applications of the analytical methods introduced by Lagrange.

18.3.1 New Forms of Lagrange’s Equations In the following, we consider some new forms of Lagrange’s equations, as the central equation of Lagrange, the equations of Nielsen, the equations of Tsenov and the Mangeron–Deleanu equations. As well, we will put in evidence the form taken by these equations in case of continuous mechanical system, in case of a non-inertial frame of reference and in case of the phenomenon of collision. 18.3.1.1 Lagrange’s Central Equation Let be a particle Pi of position vector ri and of mass mi , which belongs to a discrete mechanical system S . We can write the obvious relation

(

d dri ⋅ δri dt dt

)

=

( ) ⎥⎦⎤ + ddrt ⋅ ⎡⎢⎣ ddt ( δr ) − δ ( ddrt ) ⎤⎥⎦ .

d2 ri ⎡ 1 dri ⋅ δri + δ ⎢ 2 dt ⎣ 2 dt

2

i

i

i

Taking into account the relations (18.2.1), we have T (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) =

( ),

dri 1 n mi 2 i∑ dt =1

2

wherefrom, using the relations (18.2.28) too, we obtain ∂T δq = ∂q j j

n

∑ mi

i =1

( )

dri ∂ dri ⋅ δq j = dt ∂q j dt

n

∑ mi

i =1

dri ∂ri ⋅ δq = dt ∂q j j

n

∑ mi

i =1

The principle of virtual work, written in the form (18.2.27), leads to

dri ⋅ δri . dt

MECHANICAL SYSTEMS, CLASSICAL MODELS

84 n

∑ mi

i =1

d2 ri ⋅ δri = Q j δq j . dt 2

In this case, multiplying the above identity by mi and summing for all the indices i , it results d ⎛ ∂T ⎞ δq = Q j δq j + δΤ + dt ⎝⎜ ∂q j j ⎠⎟

( )

dri ⎡ d dri ⎤ ⋅ ( δri ) − δ . dt ⎢⎣ dt dt ⎥⎦

n

∑ mi

i =1

Observing that δΤ =

∂T ∂T δq j + δq , ∂q j ∂q j j

we can write this relation in the form ∂T ⎡ d ⎡ d ⎛ ∂T ⎞ ∂T ⎤ ⎤ ⎢ dt ⎜ ∂q ⎟ − ∂q − Q ⎥ δq j = ∂q ⎢⎣ δq j − dt ( δq j ) ⎥⎦ j j ⎣ ⎝ j ⎠ ⎦ n dr dri d − ∑ mi i ⋅ ⎡ δ − ( δri ) ⎤ . ⎢ ⎥⎦ dt ⎣ dt dt i =1

( )

(18.3.1)

We get thus, after Hamel, Lagrange’s central equation (Hamel, G., 1927). In case of holonomic constraints, the operator relations (18.1.77) take place for the position vectors in E 3 ; as well, the operator relations (18.2.13) take always place for the generalized co-ordinates in Λs . In these conditions, the equation (18.3.1) leads to the fourth form of the fundamental equation (the equation (18.2.27')). 18.3.1.2 Mechanical Systems with Separable Variables. Systems of Liouville Type There are many problems, e.g. in celestial mechanical or in atomic physics, which lead to a Lagrangian for which the kinetic energy and the simple potential of the generalized forces are of the form T =

1 s Vj (q j )q2j , U = 2∑ j =1

s

∑U j (q j ) , j =1

(18.3.2)

where U j ,Vj , j = 1, 2,..., s , are functions of only one variable of class C 1 . In this case, Lagrange’s equations (18.2.29) read Vj (q j )qj +

dU j 1 dV j 2 q = , j = 1, 2,..., s . 2 dq j j dq j

Multiplying by q j and integrating with respect to time, we get

(18.3.3)

Lagrangian Mechanics

85

1 V (q )q 2 − U j (q j ) = C j , j = 1,2,..., s , 2 j j j

(18.3.3')

where C j , j = 1, 2,..., s , are integration constants. Summing with respect to the index j . one obtain the first integral of mechanical energy; but we have thus first integrals corresponding to s particular mechanical energies (eventually, to the mechanical energies of s particles). By s quadratures of the form t + τj =

Vj (q j ) 1 dq j , j = 1, 2,..., s , 2 ∫ U j (q j ) + C j

(18.3.4)

we introduce the integration constants τ j , j = 1, 2,..., s , too. Thus, one obtains the relations which link the time to each generalized co-ordinate, the problem being thus entirely solved. The respective mechanical systems are mechanical systems with separable variables. A problem similar is that of the discrete mechanical systems, called systems of Liouville type, for which we have s

T =

s 1 s uk (qk ) ∑Vj (q j ) q 2j , U = ∑ 2 k =1 k =1

∑U j (q j ) j =1 s

∑ uk (qk )

(18.3.5)

k =1

and where U j ,Vj and uk , j , k = 1, 2,..., s , are functions of only one variable. Obviously, the case considered above is a particular case of a system of Liouville type for which uk = 1 s = const . By the substitution Vj (q j ) =

dWj (q j ) W j = , j = 1, 2,..., s , dq j q j

we can write T =

u s 2 1 s Wj , U = ∑U j (Wj ), u = ∑ 2 k =1 u j =1

s

∑ uk (Wk ) .

k =1

(18.3.6)

Lagrange’s equations relative to the new generalized co-ordinate Wj take the form d ( uW k dt

1 ∂u

s

) − 2 ∂W ∑W j2

=

k j =1

∂U , k = 1, 2,..., s . ∂Wk

Multiplying by 2uW k , we get d 2 2 ∂u ( u Wk ) − uW k ∂W dt k

s

∑W j2 j =1

= 2uW k

∂U . ∂Wk

MECHANICAL SYSTEMS, CLASSICAL MODELS

86

On the other hand, the first integral of mechanical energy is u s 2 Wj − U = h , h = const . 2∑ j =1

We are thus led to d 2 2 ( u Wk dt = 2W k

∂u ∂U + 2uW k ∂Wk ∂Wk

) = 2(h + U )Wk

∂ d [ ( h + U ) u ] = 2 [ huk (qk ) + U k (qk ) ] , k = 1, 2,..., s , ∂Wk dt

wherefrom one obtains the system of differential equations dW1 dW2 = = ... hu1 (q1 ) + U 1 (q1 ) + α1 hu2 (q2 ) + U 2 (q2 ) + α2 dWs dt = = 2 . u hus (qs ) + U s (qs ) + αs

(18.3.7)

Noting that u 2W k2 = 2(huk + U k ) + 2 αk , k = 1,2,..., s , summing with respect to all indices and taking into account the integral of mechanical energy, we find the condition s

∑ αk

k =1

= 0,

(18.3.7')

which must be verified by the constants αk , k = 1, 2,..., s , thus introduced. The equations (18.3.7) are with separate variables, the problem being reduced to quadratures. 18.3.1.3 Case of Continuous Mechanical Systems Let be a continuous mechanical system with a finite number of degrees of freedom, which occupies a domain V in E 3 and is acted upon by a volumic force F ( r ) (hence, by the force F dV on the element of volume); the torsor of these forces (resultant force and resultant moment) at the pole O is given by R =

∫V F (r )dV , MO

=

∫V r × F dV .

(18.3.8)

The kinetic energy is expressed in the form

T =

( ) dV ,

1 dr μ( r ) 2 ∫V dt

2

(18.3.9)

where μ ( r ) is the density, so that in the formulae (18.2.15), (18.2.15'), (18.2.15''), and (18.2.15''') we will have

Lagrangian Mechanics

87 g jk =

gj =

∂r

∫V μ(r ) ∂q j

∂r

∫V μ(r ) ∂q j

⋅

∂r dV , ∂qk

⋅ r dV , j , k = 1, 2,..., s , g 0 =

1 μ( r )r 2 dV . 2 ∫V

(18.3.9')

Analogously, the virtual work of the volumic forces will be given by δW =

∫V F ⋅ δr dV

= δq j

∂r

∫V F ⋅ ∂q j dV

= Q j δq j ,

so that the generalized forces will be Qj =

∂r

∫V F ⋅ ∂q j dV .

(18.3.10)

The principle of virtual work is expressed in the form ∂r ⎡ ⎤ ⎢ ∫V μ ( r )r ⋅ ∂q − Q j ⎥ δq j = 0 . j ⎣ ⎦

(18.3.11)

By a demonstration analogous to that in Sect. 18.2.2.1, where the sign plus is replaced by the sign integral, we find again the fourth form of the fundamental equation (18.2.27'). The results obtained are thus justified for rigid solids too.

18.3.1.4 Electromechanical Analogy We have seen in Sect. 8.2.2.10 that one can establish an interesting analogy between an R.L.C. circuit (an ohmic resistance R , a loading inductance L and a condenser of capacity C , connected in series with a generator having an electromotive force E (t ) ) and a particle P (of mass m → L , of damping coefficient k ′ → R and of coefficient of elasticity k → 1/C , acted upon by a given force F (t ) → E (t ) , where we have put in evidence the correspondence between various quantities), in a vibrating motion. Thus, we can introduce quantities analogue to those of mechanical nature in the form (we use well known notations)

1 ∂ ⎛ Rq 2 ⎞ 1 2 T = Wq 2 , Q = − ⎜ = − Rq , U = − q , 2 ∂q ⎝ 2 ⎟⎠ 2C

(18.3.12)

where (1/ 2)Rq 2 corresponds to Rayleigh’s function, q = q (t ) is the charge, while i = i (t ) = q (t ) is the intensity of the current. Lagrange’s equation d ⎛ ∂T ⎞ ∂T ∂U = +Q + F ⎜ ⎟− dt ⎝ ∂q ⎠ ∂q ∂q

leads to the equation (8.2.58).

MECHANICAL SYSTEMS, CLASSICAL MODELS

88

The above considerations can be applied to complex R.L.C. circuits, corresponding to discrete mechanical systems with a finite number of degrees of freedom; using the mentioned electromechanical energy one obtains a system of differential equations which allows the determination of the electric charges q (t ) in each circuit.

18.3.1.5 Nielsen’s Equations We have established Lagrange’s equations by means of the d’Alembert–Lagrange principle (the principle of virtual work); analogously, Jourdain’s principle (18.1.57'), written in the form n

∑ mi ai

i =1

n

⋅ δvi = ∑ Fi ⋅ δvi ,

(18.3.13)

i =1

leads to an equivalent system of equations of motion. Starting from the relation (18.2.14) and taking into account (18.2.4'), we may write

∂ai ∂ 2 ri ∂r ∂v qk + 2 i = 2 i , i = 1, 2,..., n , j = 1, 2,..., s , =2 ∂q j ∂q j ∂qk ∂q j ∂q j

(18.3.14)

On the other hand, we notice that Jourdain’s principle can be written for a given position of the mechanical system, at a fixed moment, varying the velocities; thus, it results δvi =

∂vi ∂r δq j = i δq j , ∂q j ∂q j

(18.3.15)

where we took into account (18.2.28). In this case, Jourdain’s principle takes the form ⎡

n

∑ ⎢⎣ −mi ai i =1

⋅

∂vi ∂r ⎤ + Fi ⋅ i ⎥ δq j = 0 . ∂q j ∂q j ⎦

(18.3.13')

Using the expression of the kinetic energy, we get n

dT = dt

∑ mi vi

i =1

⋅ ai ,

∂T = ∂q j

n

∑ mi vi

i =1

⋅

∂vi , ∂q j

so that

( ) = ∑m v

∂ dT ∂q j dt

n

i =1

i i

⋅

∂ai + ∂q j

n

∑ mi ai

i =1

⋅

n ∂vi ∂v = 2 ∑ mi vi ⋅ i + qj ∂q j ∂ i =1

n

∑ mi ai

i =1

⋅

∂vi , ∂q j

where we have used the relation (18.3.14). Taking into account the expression (18.2.18) of the generalized forces, the form (18.3.13') of Jourdain’s principle leads to

( )

∂ dT ⎡ ∂T ⎤ ⎢ 2 ∂q − ∂q dt + Q j ⎥ δq j = 0 . j j ⎣ ⎦

(18.3.16)

Lagrangian Mechanics

89

Noting that the variations δq j of the generalized velocities are independent and arbitrary, in case of holonomic constraints, we can write the equations obtained in 1935 by J. Nielsen in the form (Dolapčev, Bl., 1966)

( )

∂ dT ∂T −2 = Q j , j = 1,2,..., s . ∂q j dt ∂q j

(18.3.17)

These equations are governing the motion of the representative point P in the space Λs . In case of the non-holonomic constraints (18.2.10), one must use the method of Lagrange’s multipliers. Taking into account the relation (we use the operator relation (18.2.13))

d ⎛ ∂vi ⎞ d ⎛ ∂ri ⎞ ∂vi = = , i = 1, 2,..., n , j = 1,2,..., s , ⎜ ⎟ dt ⎝ ∂q j ⎠ dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.3.18)

we can write the derivative ∂v d ⎛ ∂T ⎞ d n mi vi ⋅ i = = ⎜ ⎟ dt ⎝ ∂q j ⎠ dt i∑ q j ∂ =1

n

∑ mi ai

i =1

⋅

∂vi + ∂q j

n

∑ mi vi

i =1

⋅

d ⎛ ∂vi ⎞ dt ⎜⎝ ∂q j ⎟⎠

in the form

( )

d ⎛ ∂T ⎞ ∂ dT ∂T , j = 1, 2,..., s . = − dt ⎜⎝ ∂q j ⎟⎠ ∂q j dt ∂q j

We see thus that Nielsen’s equations are equivalent to Lagrange’s equations. We mention also that, in some cases, Nielsen’s equations allow a more simple calculation that Lagrange’s ones. We consider also Gauss’s principle (the principle of the least action) (18.1.60') or (18.1.66), written in the form n

∑ mi ai

i =1

⋅ δai =

n

∑ Fi

i =1

⋅ δai

(18.3.19)

for a given position and velocity of the mechanical system. at a fixed moment, varying only the accelerations; in this case, the relation (18.2.14) allows to write ∂ai ∂r = i , ∂qj ∂q j

(18.3.20)

∂ai ∂r δq = i δqj . ∂qj j ∂q j

(18.3.15')

so that δai =

MECHANICAL SYSTEMS, CLASSICAL MODELS

90 The relation (18.3.19) takes the form n

⎡

∑ ⎢⎣ Fi

i =1

⋅

∂ri ∂r ⎤ − mi ai ⋅ i ⎥ δqj = 0 ∂q j ∂q j ⎦

(18.3.19')

and, taking into account the above results and (18.2.28), we obtain

( )

∂ dT ⎡ ∂T ⎤ ⎢ 2 ∂q − ∂q dt + Q j ⎥ δqj = 0 . j j ⎣ ⎦

(18.3.16')

Noting that, in case of holonomic constraints, the variations δqj of the generalized accelerations are independent and arbitrary, we find again the equations (18.2.17) of Nielsen. The principle of Gauss in finite form (18.2.60) can be written also in the form n

∑ mi ai

i =1

⋅ Δai − Q Δqj = 0 ,

(18.3.19'')

where we have put in evidence the finite variations of the accelerations ai and of the generalized accelerations qj ; this is the fifth form of the fundamental equation.

18.3.1.6 Tsenov’s Equations Starting from Gauss’s principle, written in the form (18.3.19) or in the form (18.3.19'), we can establish also other equations, equivalent to Lagrange’s equations. To do this, starting from the expression of the derivative dT / dt , let us calculate d2T = dt 2

n

n

i =1

i =1

∑ mi ai2 + ∑ mi vi

⋅ a(2) , i (2)

where we have introduced the accelerations of second order ai too). Using the equations (18.2.14), it results ) a(2 = i

(see Sect. 18.1.2.5

∂ri ∂2 ri ∂r d ⎛ ∂ri ⎞ + 2 qj + q qj qk + 2 i qj + ...  j ⎜ ⎟ ∂q j ∂q j ∂qk ∂q j dt ⎝ ∂q j ⎠ 2 ∂r ∂r ⎞ ⎛ ∂ ri = i qj + 3 ⎜ qk + i ⎟ qj + ..., ∂q j ∂q j ⎠ ⎝ ∂q j ∂qk

wherefrom ) a(2 = i

∂ri ∂v qj + 3 i qj + ... , ∂q j ∂q j

(18.3.21)

Lagrangian Mechanics

91

so that ) ∂a(2 ∂r i = i , ∂qj ∂q j

(18.3.20')

) ∂a(2 ∂v i =3 i . ∂qj ∂q j

(18.3.14')

as well as

Taking into account these results, we obtain n ∂a ∂ ⎛ d2T ⎞ 2 mi ai ⋅ i + = ∑ ⎜ ⎟ qj ∂qj ⎝ dt 2 ⎠ ∂ i =1

n

∑ mi vi

n n ∂ai ∂r ∂v = 2 ∑ mi ai ⋅ i + 3 ∑ mi vi ⋅ i , q qj ∂qj ∂ ∂ j i =1 i =1 (2)

⋅

i =1

so that the relation (18.2.19') leads to

⎧ 1 ⎡ ∂T

n

∑ ⎨⎩ 2 ⎣⎢ 3 ∂q j

i =1

−

∂ ⎛ d2T ⎞ ⎤ ⎫ + Q j ⎬ δqj = 0 . ∂qj ⎜⎝ dt 2 ⎟⎠ ⎦⎥ ⎭

(18.3.22)

Because, in case of holonomic constraints, the variations δqj of the generalized accelerations are independent, we can write the equations of I. Tsenov in the form ∂ ⎛ d2T ⎞ ∂T −3 = 2Q j , j = 1, 2,..., s . ⎜ ⎟ 2 ∂qj ⎝ dt ⎠ ∂q j

(18.3.23)

Taking into account the previous relations, we find easily that

( )

1 ⎡ ∂ ⎛ d2T ⎞ ∂T ⎤ ∂ dT ∂T −3 = −2 , j = 1,2,..., s . 2 ⎢⎣ ∂qj ⎜⎝ dt 2 ⎟⎠ ∂q j ⎥⎦ ∂q j dt ∂q j

(18.3.24)

Hence, the equations obtained in 1962 by Tsenov are equivalent to Nielsen’s equations (hence, to Lagrange’s equations too), in the case of ideal holonomic constraints (Dolapčev, Bl., 1966).

18.3.1.7 The Mangeron–Deleanu Equations Starting from the expression of the derivative d2T / dt 2 , we observe that n d3T ) 3 mi ai ⋅ a(2 = + ∑ i dt 3 i =1 n d4T (3) = 4 ∑ mi ai ⋅ ai + 4 dt i =1

n

∑ mi vi

i =1

n

∑ mi vi

i =1

⋅ a(i 3 ) ,

⋅ ai

(4)

+ ...,

MECHANICAL SYSTEMS, CLASSICAL MODELS

92

where, in the last expression, we have mentioned only terms which contain generalized accelerations till qj (and not generalized accelerations of higher order); by complete induction, one can show that n dmT ( m − 1) + m = m ∑ mi ai ⋅ ai dt i =1

n

∑ mi vi

i =1

⋅ a(i m ) + ... ,

(18.3.25)

where the terms which contain the generalized accelerations q (j m ) have been put in evidence. On the other hand, starting from (18.3.21) and using the operator relation (18.2.13), we obtain (3)

ai

=

∂ri  ∂v q j + 4 i qj + ... ∂q j ∂q j

and, in general, by complete induction, =

(m )

ai

∂ri ( m + 1) ∂v ( m ) + (m + 1) i q j + ..., i = 1, 2,..., n , q ∂q j j ∂q j

(18.3.26)

wherefrom ∂a(i m ) ( m + 1)

∂q j

=

(m ) ∂ri ∂ai ∂v , ( m ) = (m + 1) i , i = 1, 2,..., n , j = 1, 2,..., s . ∂q j ∂q ∂q j j

(18.3.26')

We can thus calculate ∂ ∂q j

(m )

( m − 1) m n ⎛ d T ⎞ = m m a ⋅ ∂ai ⎜ m ⎟ ∑ i i ∂q ( m ) + ⎝ dt ⎠ i =1 j

n

= m ∑ mi ai ⋅ i =1

n

∑ mi vi

i =1

⋅

∂a(i m ) ∂q j

(m )

+ ...

n ∂ri ∂v + (m + 1) ∑ mi vi ⋅ i + ..., j = 1, 2,..., s . qj ∂q j ∂ i =1

(18.3.25')

The relation ∂T 1 ⎡ ∂ ⎛ dmT ⎞ ⎢ ⎜ ⎟ − (m + 1) ∂q j m ⎢ ∂q ( m ) ⎝ dt m ⎠ ⎣ j

( )

⎤ ∂ dT ∂T ⎥ = −2 . ∂ ∂ q  d t qj j ⎥⎦

(18.3.24')

which generalizes the relation (18.3.24) is thus easily stated. Thus, we find the D. I. Mangeron–D. Deleanu equations (Lagrange’s generalized equations), obtained in 1962 in the form ∂ ∂q (j m )

m ⎛ d T ⎞ − (m + 1) ∂T = mQ , j = 1, 2,..., s , m ∈ ` . ⎜ m ⎟ j ∂q j ⎝ dt ⎠

(18.3.27)

Lagrangian Mechanics

93

These equations are equivalent to Lagrange’s equations, to Nielsen’s equations and to Tsenov’s equations, which result as particular cases for m = 1 and m = 2 , respectively. Let us consider a differential principle of the form n

∑ mi ai

( m − 1)

⋅ δai

i =1

=

n

∑ Fi

i =1

( m − 1)

⋅ δai

,m ∈`,

(18.3.28)

where we have put in evidence accelerations of higher order; assuming that this principle takes place for a given positions, a given velocity and given accelerations till the (m − 2) -th order inclusive, at a fixed moment, varying only the accelerations of the (m − 1) -th order, we can write δa(i m ) =

∂a(i m −1) (m ) ∂q j

δq (j m ) =

∂ri δq ( m ) , i = 1, 2,..., n , m ∈ ` . ∂q j j

(18.3.26'')

The differential principle (18.3.28) can be written in the form n

⎛

∑ ⎜⎝ Fi

i =1

⋅

∂ri ∂r ⎞ − mi ai ⋅ i ⎟ δqi( m ) = 0, m ∈ ` . ∂q j ∂q j ⎠

(18.3.28')

Starting from this principle we can obtain the Mangeron–Deleanu equations. Obviously, in this case, the ideal constraints are defined by the relation n

∑ Ri

i =1

⋅ δqi( m −1) = 0, m ∈ ` .

(18.3.29)

All the equations thus established can be used for holonomic mechanical systems, immaterial from the type of ideal constraints which are considered, which represents an interesting generalization from the mechanical point of view. In case of non-holonomic constraints, these equations are completed with constraint generalized forces, after Lagrange’s method; one uses the equations which correspond to different types of ideal constraints.

18.3.1.8 Case of a Non-inertial Frame of Reference If we report the discrete mechanical system S to a non-inertial frame of reference R (movable with respect to an inertial frame R ′ , considered fixed), then the

generalized co-ordinates q j , j = 1, 2,..., s , will specify its position with respect to this frame; as a matter of fact, the position of the system S is thus specified even with respect to the frame R ′ , because the motion of the frame R with respect to the frame R ′ is known. One uses Lagrange’s equations (18.2.29) in the form

d ⎛ ∂T ′ ⎞ ∂T ′ − = Q j , j = 1,2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.3.30)

MECHANICAL SYSTEMS, CLASSICAL MODELS

94

where the kinetic energy with respect to the frame R ′ is given by T′ =

1 n mi ( vO′ + vi + ω × ri )2 , 2 i∑ =1

vi being the velocities relative to the frame

(18.3.30')

R , while Q j = Q j (ql , ql ; t ) are the

generalized forces; because ri = ri (ql ; t ) , one obtains T ′ = T ′(ql , ql ; t ) , the problem being then reduced to that corresponding to an inertial frame. Another method of calculation is based on the theory of relative motion (see Sect. 10.2.1); one introduces thus the generalized forces Q (j t ) = Q (j t ) (ql , ql ; t ) and (C )

Qj

= Qj

(C )

(ql , ql ; t ), j = 1, 2,..., s , corresponding to the transportation generalized

forces and to the Coriolis generalized forces, respectively. Lagrange’s equations become

d ⎛ ∂T ⎞ ∂T (t ) (C ) − = Q j′ , Q j′ = Q j + Q j + Q j , j = 1, 2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.3.31)

where T = T (ql , ql ; t ) is the kinetic energy given by (calculated in the frame R )

T =

1 n mi vi2 . 2 i∑ =1

(18.3.31')

Gilbert elaborated in 1883 a mixed method, which is based – partially – on the theory of relative motion; one introduces thus a frame R with the origin at the pole O and with the axes parallel to the axes of the frame R ′ (which is not rotating about this frame). Hence, Lagrange’s equations read

∂U d ⎛ ∂T ⎞ ∂T − = Qj + , j = 1,2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j ∂q j

(18.3.32)

where U = U (ql , ql ; t ) is a potential given by

U =

n

∑ mi ri

i =1

⋅ aO′ = − M ρ ⋅ aO′ ,

(18.3.32')

corresponding to the centrifugal force. The kinetic energy T = T (ql , ql ; t ) with respect to the frame R is given by (see Sect. 11.2.2.2) T =

1 n 1 mi ( vi + ω × ri )2 = T + I Δ ω 2 + ω ⋅ KO , 2 i∑ 2 =1

(18.3.32'')

Lagrangian Mechanics

95

where KO and T are the moment of momentum with respect to the pole O and the kinetic energy, respectively, in the frame R , while I Δ is the moment of inertia of the mechanical system S with respect to the instantaneous axis of rotation Δ ; thus, the kinetic energy T with respect to the generalized co-ordinates and the generalized velocities is easily obtained. Using this method, Gilbert studies the influence of the rotation of the Earth on the motion of a heavy rigid solid of revolution, fixed at a fixed point O (different from C ), the principal axis of inertia Ox 3 being in rotation in a vertical plane linked to the Earth; the apparatus by which Gilbert verified experimentally the results theoretically obtained is called barogyroscope.

18.3.1.9 The Phenomenon of Collision In case of a phenomenon of collision, we start from the theorem of momentum, written in the form (13.1.8) for a particle Pi of the discrete mechanical system S subjected to constraints; a scalar product by ∂ri / ∂q j leads to mi vi′′ ⋅

∂ri ∂r ∂r ∂r − mi vi′ ⋅ i = Pi ⋅ i + PRi ⋅ i , ∂q j ∂q j ∂q j ∂q j

the position of the system S remaining the same before and after collision (we denoted by “second” the corresponding quantities after collision). Taking into account the relations (18.2.28) and summing for all the particles of the system S , we get the jump relations ⎛ ∂T ⎞ ⎛ ∂T Δ⎜ ≡⎜ ⎟ ⎝ ∂q j ⎠0 ⎝ ∂q j

⎞′′ ⎛ ∂T ⎟ − ⎜ ∂q ⎠ ⎝ j

⎞′ ⎟ = ⎠

Pj + PRj , j = 1, 2,..., s ,

(18.3.33)

where we have introduced the given generalized percussions and the constraint generalized percussions, respectively, in the form

Pj =

n

∑ Pi

i =1

⋅

∂ri ,P = ∂q j Rj

n

∑ PRi

i =1

⋅

∂ri , j = 1, 2,.., s , ∂q j

(18.3.33')

the index zero corresponding to an arbitrary moment of discontinuity t0 ∈ [ t ′, t ′′ ] , | t ′′ − t ′ |< ε, ε > 0 arbitrary.

Starting from the equations (18.2.32), integrating on the interval [t ′, t ′′ ] and passing to the limit in the sense of the theory of distributions, we can write

lim

∫

t ′′

t ′′ − t ′ → 0 + 0 t ′

t ′′ ∂T ⎛ ∂T ⎞ d⎜ dt = ⎟ − t ′′ −tlim ∫ ′ ∂q t ′ → + 0 0  q ∂ j ⎝ j ⎠

lim

∫

t ′′

t ′′ − t ′ → 0 + 0 t ′

(Q j + Rj )dt ;

noting that ∂T / ∂q j is a finite quantity and neglecting the given and constraint nonpercussive forces with respect to the corresponding percussive ones, we find again the jump relations (18.3.33), with

MECHANICAL SYSTEMS, CLASSICAL MODELS

96 t ′′

Q j dt t ′′ − t ′ → 0 + 0 ∫t ′ lim

=

n

∂r

∑ ∂qij

i =1

⋅

=

t ′′

Fi t ′′ − t ′ → 0 + 0 ∫t ′

t ′′

lim

n

Fi dt = ∑ Pi t ′′ −t ′ → 0 + 0 ∫t ′ i =1

lim

⋅

⋅

∂ri dt ∂q j

∂ri = ∂q j

Pj ,

and similar relations for the constraint generalized percussions. Introducing the generalized momenta (18.2.80), considering Lagrange’s equations in the form (18.2.36) and observing that ∂U / ∂q j is a finite quantity, one can write the jump relations (18.3.33) in the form

( Δp j )0

= p j′′ − p j′ = Pj + PRj , j = 1, 2,..., s ,

(18.3.33'')

We state thus Theorem 18.3.1 (theorem of generalized momentum). The jump of the generalized momentum of a discrete mechanical system subjected to constraints, corresponding to a generalized co-ordinate at a moment of discontinuity, is equal to the sum of the given and constraint generalized percussions, which correspond to the same generalized co-ordinate and which act upon the same system at that moment. This theorem corresponds to the Theorem 13.1.2 stated in the space E 3 . Let us consider, in general, the case of a mechanical system S subjected to holonomic constraints, the position of which at the moment t ′ is definite by the generalized co-ordinates q1′ , q2′ ,..., qs′ (after eliminating the holonomic constraints); at the moment t0 ∈ [ t ′, t ′′ ] , h new holonomic constraints are suddenly introduced, so that the motion of the system S is suddenly perturbed. while at the moment t ′′ the generalized velocities have finite variations, passing from q j′ to q j′′, j = 1, 2,..., s , the position of the considered system remaining practically the same. The preceding constraints of the system S are permanent, while the new introduced constraints can become permanent or can disappear after the interval of percussion. The new introduced constraints are expressed by relations of the form ϕl (q1 , q2 ,..., qs ) = 0, l = 1, 2,..., h . Making a change of variable by which one replaces the co-ordinates ql by the co-ordinates rl = ϕl , the new constraints imposed to the system S will be expressed in the form rl = 0, l = s − h + 1, s − h + 2,..., s ; we assume to choose the generalized co-ordinates just in this form, having qs − h + 1 = 0, qs − h + 2 = 0,..., qs = 0 . If the new constraints introduced are temporary, then these co-ordinates will no more be equal to zero after the moment t ′′ ; otherwise, they remain equal to zero. The arbitrary virtual generalized displacements δq1 , δq2 ,..., δqs − h take place, while the virtual generalized displacements δqs − h + 1 , δqs − h + 2 ,..., δqs must vanish. We remain with Lagrange’s equations

d ⎛ ∂T ⎞ ∂T − = Q j , j = 1, 2,..., s − h , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.3.34)

Lagrangian Mechanics

97

for the interval of percussion, the constraints generalized forces new introduced disappearing from the calculation, because the corresponding virtual work vanishes Rs − h + 1 δqs − h + 1 + Rs − h + 2 δqs − h + 2 + ... + Rs δqs = 0 .

(18.3.34')

Noting that the variations of the generalized velocities are due only to the constraint generalized forces, the intensity of which has increased very much but which have disappeared from the calculation in the considered system of generalized co-ordinates, it results that ∂T / ∂q j and Q j are finite quantities; by integration on the interval of percussion and by passing to the limit, we find the jump relations ⎛ ∂T ⎞ ⎛ ∂T Δ⎜ ≡⎜ ⎟ ⎝ ∂q j ⎠0 ⎝ ∂q j

⎞′′ ⎛ ∂T ⎟ − ⎜ ∂q ⎠ ⎝ j

⎞′ ⎟ = 0, j = 1, 2,..., s − h , ⎠

(18.3.34'')

which are linear and homogenous with respect to the differences q j′′ − q j′ , j = 1, 2,..., s . One can thus state that the generalized momenta corresponding to the generalized co-ordinates, which do not vanish at the moment of collision, have the same values before and after this phenomenon (have not a jump). The generalized co-ordinates q1 , q2 ,..., qs have such values, in the equations (18.3.34''), that qs − h + 1 , qs − h + 2 ,..., qs do vanish at the moment of collision. In exchange, the generalized velocities qs − h + 1 , qs − h + 2 ,..., qs must not necessarily vanish, neither before, nor after collision; they equate to zero after collision if the new introduced constraints are permanent. In this last case, the relations (18.2.34'') determine entirely the generalized velocities q1′′, q2′′,..., qs′′− h , hence the state of generalized velocities after collision. Beside this particular case, we have at our disposal only s − h equations for s unknown generalized velocities, so that we must make supplementary hypotheses for the mathematical modelling of the mechanical phenomenon.

Fig. 18.8 A circular disc rolling without sliding on an axis in a vertical plane

As Beghin and Rousseau have shown in 1903, in case of a non-holonomic mechanical system, the constraint generalized forces are finite, so that they disappear from calculation, as well as the given generalized forces; hence, in this case, the jump relations (18.3.34'') can be applied too. Let be, e.g., a homogenous circular disc, of mass M and radius R , which moves in the vertical plane Ox1x 2 ; the disc strikes the Ox1 -axis at the moment t0 and continues to roll

MECHANICAL SYSTEMS, CLASSICAL MODELS

98

without sliding on this axis. Before the collision, the position of the disc depends on three parameters: the co-ordinates ρ1 and ρ2 of the centre C of the disc and the angle of rotation θ , taken counterclockwise (Fig. 18.8). As an effect of the collision, two new constraints are introduced: the contact with the Ox1 -axis (hence ρ2 = R ) and the rolling of the disc on this axis ( ρ1 = Rθ ). Choosing conveniently the origin O , one introduces the generalized co-ordinates q1 = ρ1 , q2 = ρ2 − R , q 3 = ρ1 − Rθ , so that the constraint relations are expressed in the form q2 = 0, q 3 = 0 . Denoting by IO = MiO2 the moment of inertia of the disc with respect to the pole O , where iO is the corresponding radius of gyration, we can write the kinetic energy in the form T =

1 M ( iO2 θ2 + ρ12 + ρ22 2

1

( ) ( q

⎡ iO ⎣ R

) = 2M ⎢

2

1

⎤ − q3 )2 + q12 + q22 ⎥ . ⎦

We have at our disposal only a relation of the form (18.3.34''), corresponding to the generalized co-ordinate q1 , which does not vanish at the collision moment, that is

( ) [q ′′ − q ′ − ( q ′′ − q ′ ) ] + q ′′ − q ′ = 0 . iO R

2

1

1

3

3

1

But q 3 = 0 after collision, hence we have q3′′ = 0 too. Returning to the old variables, we can write

( iR ) ( ρ ′′ − Rθ′ ) + ρ ′′ − ρ ′ = 0 , O

2

1

1

1

the velocity of the centre of the disc in a rolling motion on the Ox1 -axis being given by ρ1′′ =

R ( Rρ1′ + iO2 θ ′ ) R2 + iO2

.

(18.3.35)

If Rρ1′ + iO2 θ ′ = 0 , then the disc stops.

18.3.2 Applications In what follows we present several applications, interesting by themselves, to illustrate the results previously obtained: the motion of a particle in various cases of loading, the problem of two particles and the problem of two centres; as well, we consider some cases of motion of the rigid solid: the plane motion of a bar, the double pendulum, the sympathetic pendulums etc.

Lagrangian Mechanics

99

18.3.2.1 Motion of a Particle Using the expression T = (m / 2)( x12 + x22 + x 32 ) of the kinetic energy, in case of a single particle P , Lagrange’s equations (18.2.29) take the form ( q j = x j , j = 1, 2, 3 ) d ( mx j dt

∂r

) = F ⋅ ∂x

j

+F⋅

∂x k ik = F ⋅ ik δkj = F ⋅ i j = Fj = Q j , k = 1, 2, 3 , ∂x j

so that we find again the equations of motion (6.1.22'). In the case of cylindrical co-ordinates q1 = r , q2 = θ , q 3 = z , the kinetic energy (see the formula (5.1.13'')) has the form T = (m / 2)(r 2 + r 2 θ2 + z 2 ) , resulting Lagrange’s equations ( r = r cos θi1 + r cos θi2 + z i 3 ) d ∂r ( mr ) − mr θ2 = F ⋅ = F ⋅ (cos θi1 + sin θi2 ) = F ⋅ ir = Fr = Qr , dt ∂r d ∂r ( mr 2 θ ) = F ⋅ ∂θ = F ⋅ [ r ( − sin θi1 + cos θi2 ) ] = F ⋅ ( r iθ ) = rFθ = Qθ , dt d ∂r ( mz ) = F ⋅ = F ⋅ i 3 = Fz = Qz , dt ∂z

being thus led to the equations of motion (6.1.26), (6.1.26'). In particular, we obtain the plane motion in polar co-ordinates. As well, in the case of spherical co-ordinates q1 = r , q2 = θ , q 3 = ϕ (see the formula (5.1.12'')), we can write T = (m / 2)(r 2 + r 2 θ2 + r 2 sin2 θϕ 2 ) , while Lagrange’s equations read ( r = r sin θ cos ϕi1 + r sin θ sin ϕi2 + r cos θi 3 ) d ∂r ( mr ) − mr ( θ2 + sin2 θϕ 2 ) = F ⋅ dt ∂r = F ⋅ (sin θ cos ϕi1 + sin θ sin ϕi2 + cos θi 3 ) = F ⋅ ir = Fr = Qr , d ∂r mr 2 θ ) − mr 2 sin θ cos θϕ 2 = F ⋅ ( dt ∂θ = F ⋅ [ r ( cos θ cos ϕi1 + cos θ sin ϕi2 − sin θ i 3 ) ] = F ⋅ ( r i θ ) = rFθ = Qθ , d ∂r ( mr 2 sin2 θϕ ) = F ⋅ ∂ϕ dt = F ⋅ [ r ( sin θ sin ϕi1 + sin θ cos ϕi2 ) ] = F ⋅ ( r iϕ ) = rFϕ = Qϕ ,

wherefrom we get the equations of motion (6.1.25), (6.1.25'). If the support of the force F pierces the Ox 3 -axis, then Qϕ = 0 , so that we can write the first integral

mr 2 sin2 θϕ = const ,

(18.3.36)

according to which the projection of the particle P on the plane Ox1x 2 moves after the law of areas.

MECHANICAL SYSTEMS, CLASSICAL MODELS

100

Analogously, we find again the equations (6.1.24''') in arbitrary curvilinear coordinates. Let be, e.g., the quadric Γ x12 x 22 x 32 + + −1 = 0 α1 α2 α3

(18.3.37)

and the quadrics homofocal to this one x12 x 22 x 32 + + −1 = 0. α1 − λ α2 − λ α3 − λ

(18.3.37')

If, to fix the ideas, α1 > α2 > α3 , then the equation (18.3.37') represents an ellipsoid for λ < α3 , an one-sheet hyperboloid if α3 < λ < α2 or a two-sheet hyperboloid if α2 < λ < α1 . Through each point P (x1 , x 2 , x 3 ) pass three homofocal quadrics Γ i , i = 1, 2, 3 . Supposing that the co-ordinates x1 , x 2 , x 3 are given, the equation (18.3.37') of third degree in λ has three real roots q 3 < α3 < q2 < α2 < q1 < α1 , as one can easily verify; the root q1 leads to a two-sheet hyperboloid, the root q2 leads to an one-sheet hyperboloid, while the root q 3 correspond to an ellipsoid. These three quadrics are orthogonal two by two. We obtain thus the elliptical co-ordinates (introduced by Jacobi) q1 , q2 , q 3 of the point P . Obviously, we have x12 x 22 x 32 ( λ − q1 ) ( λ − q2 ) ( λ − q 3 ) , + + −1 = − − − f (λ ) α1 λ α2 λ α3 λ f (λ ) = ( α1 − λ ) ( α2 − λ ) ( α2 − λ ) ;

multiplying by αi = λ and making then λ = αi , i = 1, 2, 3 , we can express the Cartesian co-ordinates as functions of the elliptic co-ordinates in the form x i2 =

( αi − q1 ) ( αi − q2 ) ( αi − q 3 ) , i ≠ j ≠ k ≠ i , i , j , k = 1, 2, 3 . ( αj − αi ) ( αk − αi )

(18.3.38)

Calculating the logarithmic derivatives, we obtain the element of arc ds 2 =

1 ( A dq 2 + A2 dq22 + A3 dq 32 ) , 4 1 1

(18.3.39)

where Ai = =

( qi

x12

( qi − a1 )

2

+

x 22

( qi − a 2 )

2

+

x 32

( q 3 − a1 )2

− q j ) ( qi − qk ) ( ! ) , i ≠ j ≠ k ≠ i , i , j , k = 1, 2, 3. f (qi )

(18.3.39')

Lagrangian Mechanics

101

If the intersections of the surfaces q j = const and qk = const taken two by two are the curves C i , i ≠ j ≠ ≠ k ≠ i , i , j , k = 1, 2, 3 (Fig. 18.9), then the element of arc on these curves is given by dsi =

1 Ai dqi ( ! ) , i = 1, 2, 3 , 2

(18.3.40)

so that ds 2 = dsi dsi .

(18.3.39'')

Fig. 18.9 Elliptic co-ordinates in E 3

The kinetic energy is calculated in the form T =

( )

m ds 2 dt

2

=

m ( A q2 + A2q22 + A3q32 ) , 8 1 1

(18.3.41)

allowing thus to obtain Lagrange’s equations. We decompose the given force F along the tangents to the curves C 1 ,C 2 ,C 3 , hence let be F = F1 + F2 + F3 ; the corresponding virtual work will be F ⋅ δr = Qi δqi = Fi δsi . Taking into account (18.3.40), it results Qi =

1 F 2 i

Ai ( ! ) , i = 1, 2, 3 .

(18.3.42)

In case of a plane motion, we can use elliptic co-ordinates in the plane Ox1x 2 , the corresponding formulae being obtained by particularization (we make x 3 = 0, q 3 = 0 ) from the above ones. The homofocal conics will be an ellipse and a hyperbola (Fig. 18.10). If the given forces are conservative or quasi-conservative, then we can use the results in Sect. 18.1.1.3 concerning natural systems. Lagrange’s equations can be written also in the vector form

MECHANICAL SYSTEMS, CLASSICAL MODELS

102

d ⎛ ∂T ′ ⎞ ∂T ′ = F, ⎜ ⎟− dt ⎝ ∂v ⎠ ∂r

(18.3.43)

with respect to a non-inertial frame of reference R (the generalized co-ordinates are taken with respect to this frame); in this case, the kinetic energy is given by T ′ = (m / 2)( vO′ + v + ω + r )2 . We notice that vO′2 is a known function of time and can be expressed as the total derivative with respect to time of another function; hence, this term can be omitted from calculation (Lagrange’s function – inclusive the kinetic energy – is determined excepting the total derivative with respect to time of an arbitrary function). On the other hand, we notice that m vO′ ⋅ ( v + ω × r ) = m vO′ ⋅

dr ⎡ d ( vO′ ⋅ r ) ⎤ = m⎢ − aO′ ⋅ r ⎥ ; t dt d ⎣ ⎦

Fig. 18.10 Elliptic co-ordinates in E 2

the total derivative with respect to time can be, as well, neglected. Thus, we can use the conventional kinetic energy 1 1 T = m v2 + m v ⋅ ( ω × r ) − m aO′ ⋅ r . 2 2

(18.3.44)

We can calculate dT = m ( v + ω × r ) ⋅ dv − m [ aO′ + ω × v + ω × ( ω × r ) ] ⋅ dr ,

wherefrom ∂T ∂T = m ( v + ω × r ), = − m [ aO′ + ω × v + ω × ( ω × r ) ] . ∂v ∂r

(18.3.44')

The equation (18.3.43) leads thus to the equation of motion m

dv = F − m [ aO′ + ω × r + ω × ( ω × r ) ] − 2m ω × v , dt

(18.3.45)

Lagrangian Mechanics

103

written with respect to the non-inertial frame of reference R ; the transportation force and the Coriolis force are thus put in evidence (see Sect. 10.2.1.1 too)

18.3.2.2 Case of a Particular Conservative Force We consider the motion of a particle P acted upon by a conservative force which derives from the simple potential

U ( r , θ , ϕ ) = f (r ) +

g (θ ) h (ϕ ) + 2 , r2 r sin2 θ

(18.3.46)

expressed in spherical co-ordinates. Noting that Qr = ∂U / ∂r , Qθ = ∂U / ∂θ and Qϕ = ∂U / ∂ϕ , we obtain Lagrange’s equations in the form h (ϕ ) ⎤ ⎡ ⎢⎣ g ( θ ) + r 2 sin2 θ ⎥⎦ , d 2 ⎡ sin 2 θ ⎤ 2 ( r 2 θ ) − r 2 sin 2 θϕ 2 = , g ′( θ ) − h (ϕ ) 2 dt mr 2 ⎣⎢ r sin 4 θ ⎦⎥ h ′(ϕ ) d 2 ( r sin2 θϕ ) = 2 2 , dt mr sin θ

r − r ( θ2 + sin2 θϕ 2

1

) = m f ′(r ) −

2 mr 3

(18.3.47)

where we have denoted by “prime” the derivative of the functions f , g and h with respect to the corresponding variables. The third equation (18.3.47) can be written also in the form mr 2 sin2 θϕ ( r 2 sin2 θϕ ) =

m d ( r 4 sin 4 θϕ 2 2

) = h ′(ϕ )ϕ dt

= h ′(ϕ )dϕ = dh (ϕ ),

so that we obtain the first integral r 4 sin 4 θϕ 2 =

2 [ h (ϕ ) + C 1 ] , m

(18.3.48)

where C 1 is an integration constant; as well, the second equation (18.3.47) reads 2C 1 sin 2 θ d 1 sin 2 θ ⎡ 4 2 2 ⎡ ( r 2 θ ) − , g ′( θ ) ⎤ = 2 r sin 4 θϕ 2 − h (ϕ ) ⎤ = m ⎦⎥ mr 2 sin 4 θ ⎣⎢ dt ⎦⎥ r sin 4 θ ⎣⎢ mr 2

where we took into account the first integral previously found. Multiplying by r 2 θ , one obtains 2r 2 θ d ( r 2 θ ) = d ( r 4 θ2 =

) = m ⎡⎢⎣ g ′( θ )θ dt + 2C 1 2

(

)

2 ⎡ 1 ⎤, g (θ ) − C1 d m ⎢⎣ sin2 θ ⎦⎥

sin θ cos θ  ⎤ θ dt ⎥⎦ sin 4 θ

MECHANICAL SYSTEMS, CLASSICAL MODELS

104 resulting the first integral r 4 θ2 =

2 m

⎡ g (θ ) + C1 + C 2 ⎢⎣ sin2 θ

⎤, ⎥⎦

(18.3.48')

where C 2 is a new integration constant. Eliminating the terms r θ2 and r sin2 θϕ 2 between the first equation (18.3.47) and the first integrals (18.3.48), (18.3.48'), we obtain r − 2

C2 1 = f ′(r ) . 3 m mr

Multiplying by rdt and integrating, we get the third first integral r 2 + 2

C2 1 = [ 2 f (r ) + C 3 ] , m mr 2

(18.3.48'')

where C 3 is a new integration constant. As a matter of fact, one can obtain this result starting from the first integral of the mechanical energy and taking into account the first integrals previously obtained. The first integral (18.3.48'') contains only one space variable, so that C dr 1 ⎡ 2 f (r ) − 2 22 + C 3 ⎤ , = ± f1 (r ), f1 (r ) = m ⎣⎢ dt ⎦⎥ r

(18.3.49)

wherefrom, by a quadrature, one gets t as a functions of r and then r = r (t ) , introducing the fourth integral constant C 4 . Analogously, the first integral (18.3.48') leads to

r

2

dr dθ 2 = ± , g1 ( θ ) = m f1 (r ) g1 ( θ )

⎡ g (θ ) − C1 + C ⎤ . 2 ⎥ ⎢⎣ ⎦ sin2 θ

(18.3.49')

By two quadratures, we obtain θ = θ (r ) and then θ = θ (t ) , a new integration constant C 5 being introduced. Finally, the first integral (18.3.48) allows to write dϕ dθ 2 = ± 2 , h1 (ϕ ) = [ h (ϕ ) + C 1 ] , m h1 (ϕ ) sin θ g1 ( θ )

(18.3.49'')

where we have used the previous result; hence, one obtains ϕ = ϕ ( θ ) and then one gets ϕ = ϕ (t ) , appearing the integration constant C 6 . The integration constants C k , k = 1, 2,..., 6 , are then determined by means of the initial conditions. The case of loading of a particle P , considered above, is useful in many particular cases of great interest.

Lagrangian Mechanics

105

18.3.2.3 Problem of Two Particles Let be two particles P1 and P2 of masses m1 and m2 , m1 ≥ m2 , respectively; the mass centre O is on the segment of a line P1P2 , so that m1r1 = m2 r2 , r1 = OP 1 , r2 = OP 2 , r1 ≤ r2 (Fig. 18.11). In spherical co-ordinates, we have P1 (r1 , θ , ϕ ) , P2 (r2 , π − θ , π + ϕ ) , and the kinetic energy can be expressed in the form

Fig. 18.11 Problem of two particles

T =

m1 2 m r1 + r12 θ2 + r12 sin2 θϕ 2 ) + 2 ( r22 + r22 θ2 + r22 sin2 θϕ 2 ) . ( 2 2

We introduce the notations m1r12 + m2 r22 = mr 2 , m1r12 + m2 r22 = mr 2 , with

r1 + r2 = r ,

(18.3.50)

where r = P1P2 ; we get r1 =

m2 m1 r , r2 = r, m1 + m2 m1 + m2

so that m1r12 + m2 r22 =

m1m2 2 m1m2 2 r , m1r12 + m2 r22 = r . m1 + m2 m1 + m2

Comparing with the notations which have been introduced, we must have 1 1 1 = + . m m1 m2

(18.3.50')

Thus, the motion of relation about the mass centre O is specified by the generalized coordinates q1 = r , q2 = θ , q 3 = ϕ , corresponding to three degrees of freedom. The kinetic energy will be thus given by

MECHANICAL SYSTEMS, CLASSICAL MODELS

106 T =

1 m ( r 2 + r 2 θ2 + r 2 sin2 θϕ 2 ) . 2

(18.3.51)

Assuming that the two particles are acted upon only by forces of Newtonian attraction, the potential function being U = f

m1m2 , r

(18.3.51')

we obtain the case considered in the preceding section, with f (r ) = fm1m2 / r and g ( θ ) = h (ϕ ) = 0 ; the problem is thus reduced to quadratures.

18.3.2.4 Problem of Two Centres Let be a particle subjected to the attraction of two fixed centres C 1 and C 2 , in an inverse proportion to the squares of the distances rk = PC k , k = 1, 2 , to these centres; the potential of the corresponding forces is U =

k1 k2 + , r1 r2

(18.3.52)

where k1 , k2 > 0 are positive constants. We can study this problem in spherical coordinates too, by means of the results obtained in Sect. 18.3.2.2. But we can study the problem also with the aid of the elliptical co-ordinates introduced in Sect. 18.3.2.1. If, in this order of ideas, we denote q1 = λ =

1 1 ( r1 + r2 ) , q2 = μ = ( r1 − r2 ) , 2 2

(18.3.53)

then the curves λ = const represent ellipses, the semi-major axes of which are equal to λ , and the curves μ = const are hyperbolae the semi-major axes of which are equal to μ ; these curves are homofocal (they have common foci C 1 and C 2 ) and orthogonal one to the other. Observing that min(r1 + r2 ) = 2c , max r1 − r2 = 2c , 2c = C 1C 2 , it results the relation of condition λ ≥ c ≥ μ ; in this case, the equations of the families of ellipses and hyperbolae will be

x12 x2 x2 x2 + 2 2 2 = 1, 12 + 2 2 2 = 1 . 2 c −μ λ λ −c μ

(18.3.54)

By logarithmic differentiation, we can calculate the kinetic energy in the form

T =

m 2 ( x + x22 2 1

)=

μ 2 m 2 ⎛ λ 2 + λ − μ2 ) ⎜ 2 ( 2 2 c 2 − μ2 ⎝λ −c

Taking into account (18.3.53), we obtain the potential

⎞ ⎟. ⎠

(18.3.55)

Lagrangian Mechanics

U =

107

1 [ ( k1 + k2 ) λ − ( k1 − k2 ) μ ] , λ 2 − μ2

(18.3.55')

observing thus that we have a system of Liouville type (see Sect. 18.3.1.2). Because u1 (q1 ) = λ2 , u2 (q2 ) = − μ2 , U 1 (q1 ) = (k1 + k2 )λ , U 2 (q2 ) = (k1 − k2 ) μ, m m , V2 (q2 ) = − 2 , 2 λ −c μ − c2 dμ dλ dW1 = m , dW2 = m , 2 2 2 λ −c c − μ2 u = λ2 − μ2 , α1 = α, α2 = − α, V1 (q1 ) =

2

the system of equations (18.3.7) leads to

∫

dμ = ψ( μ)

2 β , m 1

dλ − μ2 ϕ (λ ) ∫

dμ = ψ(μ )

2 (t + β2 ), m

dλ − ϕ (λ )

∫ ∫λ

2

(18.3.56)

with the notations ϕ (λ ) = (λ 2 − c 2 )[ hλ 2 + (k1 + k2 )λ + α ], ψ ( μ ) = ( μ2 − c 2 )[ h μ2 + (k1 − k2 )μ + α ],

(18.3.56')

where h is the constant of mechanical energy, while α, β1 and β2 are three other integration constants. We denote by λ1 , λ2 and μ1 , μ2 pairs of two simple roots, differing from ±c , of the equations ϕ (λ ) = 0 and ψ ( μ ) = 0 , respectively, and let be the integrals ω11 =

λ2

∫λ

1

ω21 =

λ2

∫λ

1

λ

2

dλ , ω12 = ϕ (λ ) dλ , ω22 = ϕ (λ )

dμ , ψ(μ) μ2 2 dμ ∫μ1 μ ψ ( μ ) . μ2

∫μ

1

(18.3.57)

We introduce also the quantities ν1 and ν2 by relation ω11 ν1 + ω12 ν2 = πβ1 , ω21ν1 + ω22 ν2 = π (t + β2 ) .

(18.3.58)

The relations (18.3.56) define thus λ and μ as double periodical (elliptic) functions of β1 and t + β2 or (with the relations (18.3.58)) of ν1 and ν2 . If a relation of the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

108

n1 ω11 + n2 ω12 = 0, n1 , n2 ∈ ` , takes place, hence if ω11 and ω12

are commensurable

quantities, then the motion is strictly periodic with the period 2T = 2(n1 ω21 + n2 ω22 ) . If the roots λ1 , λ2 and μ1 , μ2 , respectively, are double, then we have so called motions of libration (called periodical conditioned motions too), important in the classical quantum theory.

18.3.2.5 Motion of an Electrized Particle in an Electromagnetic Field Let be an electrized particle P of electric charge q (eventually, an electron), which is in motion with the velocity v in an electromagnetic field { E, B } , where E is the intensity of the electric field and B is the magnetic induction (see Sect. 18.1.1.4). Lorentz’s generalized force (18.1.36) derives from the generalized quasi-potential (18.2.37'). We obtain thus Lagrange’s kinetic potential 1 2

L = m v2 + q ( A ⋅ v − A0 ) ,

(18.3.59)

where A = A ( x1 , x 2 , x 3 ; t ) is the vector magnetic quasi-potential, while the scalar one is A0 = A0 (x1 , x 2 , x 3 ; t ) . If one observes that dL / dvi = mvi + qAi , i = 1, 2, 3 , then Lagrange’s equations (18.3.38) read d ( mvi + qAi ) = q ( v j Aj ,i − A0,i dt

) = 0, i

= 1, 2, 3 .

We assume that the velocities do not depend on the point but only on the time. Calculating the total derivative dAi / dt = ( ∂Ai / ∂x j )v j + Ai , Lagrange’s equations become m

dvi = q ⎡⎣ ( Aj ,i − Ai , j ) v j − Ai − A0,i ⎤⎦ , i = 1, 2, 3 . dt

(18.3.60)

The relations (18.1.36') and (18.1.36'') can be scalarly written in the form Bi =∈ijk Ak , j , Ei = −( Ai + A0,i ), i = 1, 2, 3 .

It is seen that the equations (18.3.60) represent, in fact, the projections on the three axes of co-ordinates of Newton’s equation of motion m

dv = q(E + v × B) , dt

(18.3.60')

where we have Lorentz’s generalized force (18.1.36); for this, one takes into account the above relations and the relation ( v × B )i =∈ijk v j Bk =∈ijk v j ∈klm Am ,l

= ( δil δjm − δim δjl ) v j Am ,l = ( Aj ,i − Ai , j ) v j .

Lagrangian Mechanics

109

To take into account the relativistic aspect of the problem, one must introduce the mass m = m0 1 − (v / c )2 , where m0 is the mass at rest, while c is the velocity of propagation of light in vacuum.

18.3.2.6 Motion of the Rigid Solid A free rigid solid S has six degrees of freedom and we can use, e.g., the ′ , i = 1, 2, 3 , q 4 = ψ , q 5 = θ , q 6 = ϕ , where xOi ′ generalized co-ordinates qi = xOi are the co-ordinates of a point O of it with respect to an inertial frame of reference R ′ , while ψ , θ and ϕ are Euler’s angles, which specify the rotation about the point O . If a certain point P0 of the solid is compelled to slide (without friction) on a given surface, then it will have only two degrees of freedom, appearing thus a holonomic constraint relation; analogously, if the point P0 is compelled to stay on a given curve, then there result two holonomic constraint relations. Especially, if to the point P0 is imposed a given motion, then the rigid solid is subjected to three holonomic constraints; in particular, the point P0 (taken – in this case – as a pole O of a non-inertial frame R ) can be fixed. In each of these cases, one searches convenient generalized coordinates, by eliminating the respective constraints; thus, in case of a rigid solid with a fixed point, we can choose as generalized co-ordinates Euler’s angles, the kinetic energy being of the form (15.1.13), where I 1 , I 2 , I 3 are the principal moments of inertia with respect to the fixed point. We notice that ψ is a hidden co-ordinate, so that d ⎛ ∂T ′ ⎞ = Qψ . dt ⎜⎝ ∂ψ ⎟⎠

(18.3.61)

For instance, in case of the motion of a heavy homogenous rigid solid of revolution of mass M , which slides frictionless on a fixed horizontal plane (see Sect. 17.1.2.4), the kinetic energy reads T′ =

1 {M ( ρ1′2 + ρ 2′2 ) + [ Mf ′2 ( θ ) + J ] θ2 + J ψ 2 sin2 θ + I 3 (ϕ + ψ cos θ )}, 2 (18.3.62)

where ρ1′ , ρ2′ and ρ3′ = f ( θ ) are the co-ordinates of the mass centre C with respect to the inertial frame of reference R ′ and where J = I 1 = I 2 ; the potential of its weight is U = − Mgf ( θ ) .

(18.3.62')

We can use thus the generalized co-ordinates q1 = ρ1′ , q2 = ρ2′ , q 3 = ϕ, q 4 = ψ, q 5 = θ . The first four equations of Lagrange are written in the form ( ρ1′ , ρ2′ , ψ and ϕ are hidden and ignorable co-ordinates) d ⎛ ∂T ′ ⎞ d ⎛ ∂T ′ ⎞ d ∂T ′ ⎞ d ⎛ ∂T ′ ⎞ = 0, = 0, ⎛⎜ = 0, ⎜ = 0, ⎟ ⎜ ⎟ ⎜ ⎟ dt ⎝ ∂ρ1′ ⎠ dt ⎝ ∂ρ 2′ ⎠ dt ⎝ ∂ϕ ⎠ dt ⎝ ∂ψ ⎟⎠

MECHANICAL SYSTEMS, CLASSICAL MODELS

110 being thus led to the first integrals

ρ1′ = νC′01 , ρ 2′ = νC′02 , ϕ + ψ cos θ = ω30 ,

J ψ sin2 θ + I 3 (ϕ + ψ cos θ ) cos θ = KC′ 3 ,

(18.3.63)

where we have used the notations in Sect. 17.1.2.4. Instead of the fifth equation of Lagrange, we can use the first integral of mechanical energy, the problem being thus entirely formulated. An important case is that of the rigid solid for which three non-collinear points have a given motion in a fixed (or mobile) plane; the problem is thus reduced to the motion of a mechanical system in a plane (or of a plane solid rigid), which has only three degrees of freedom, corresponding to the translation of a point (two degrees of freedom) and to the rotation about this point (one degrees of freedom). In the case of a system of rigid solids, one must keep in mind the connections between them. E.g., a hinge between two rigid solids involves the introduction of three holonomic constraints (we can imagine a free rigid solid, the second one having a point with a given motion).

18.3.2.7 Plane Motion of a Rigid Straight Bar Let be a rigid straight bar AB which moves frictionless in the plane O ′x1′x 2′ , being acted upon by forces in direct proportion to the mass and to the distance to the O ′x1′ -axis (we have chosen a system of particular forces, to fix the ideas). The position of the bar is specified by the co-ordinates ρ1′ and ρ2′ of the mass centre C and by the angle θ made with the fixed axis O ′x1′ (Fig. 18.12); we can thus choose the generalized co-ordinates q1 = ρ1′ , q2 = ρ2′ , q 3 = θ .

Fig. 18.12 Plane motion of a straight rigid bar

Using Koenig’s theorem, we can write the kinetic energy in the form T′ =

1 ⎡ M ( ρ1′2 + ρ 2′2 ) + I θ2 ⎦⎤ , 2⎣

(18.3.64)

where M is the mass of the bar and I is its moment of inertia with respect to the centre of mass C . Let us consider a bar element dx along the Cx -axis, which is attracted by the O ′x1′ -axis by the force − k μ ( ρ2′ + x sin θ )dx , k = const , where μ is the linear

Lagrangian Mechanics

111

density of the bar; this force −(k μ / 2)( ρ2′ + sin θ )2 dx . Noting that B

∫ A μ dx

= M,

B

∫A μx dx

derives

= 0,

B

∫A μx

2

from

the

simple

potential

dx = I ,

we get U = −

2 k B k ′ ( sin ) ρ x θ μ dx = − ( M ρ2′2 + I sin2 θ ) . + 2 2 ∫A 2

(18.3.64')

One obtains, easily, k Q1 = 0, Q2 = − kM ρ2′ , Q3 = − I sin 2 θ . 2

(18.3.64'')

We are thus led to Lagrange’s equations

ρ1′ = 0, ρ2′ = − k ρ2′ , 2 θ = − k sin 2 θ .

(18.3.65)

The mass centre C describes a sinusoid along the O ′x1′ -axis, its projection on this axis having a uniform motion; the bar oscillates about the O ′x1′ -axis with a frequency equal (in case of small oscillations with sin θ ≅ θ , hence sin 2 θ ≅ 2 θ ) to the frequency of the oscillations of the projection of the centre C on the O ′x 2′ -axis.

18.3.2.8 Double Pendulum We return to the double pendulum, considered in Sect. 17.1.1.2. This mechanical system has two degrees of freedom and we can choose as generalized co-ordinates the angles made by O1C 1 and O2C 2 , respectively, with the descendent vertical ( q1 = θ1 , q2 = θ2 ); we have denoted by O1 the suspension pole of the solid S1 with the mass centre C 1 and by O2 the hinge between this solid and the solid S2 , with the mass centre C 2 (see Fig. 17.1, a too). The kinetic energy of the mechanical system S ≡ {S1 , S2 } is given by T′ =

{

}

2 2 1 2 1 1 I θ + M ⎡ ρ ′(2 ) ⎤ + ⎡⎣ ρ 2′(2 ) ⎤⎦ + I 2 θ22 , 2 1 1 2 2 ⎣ 1 ⎦ 2

(18.3.66)

where I 1 is the moment of inertia of the rigid solid S1 with respect to the O ′x1′ -axis, while I 2 is the moment of inertia of the rigid solid S2 , of mass M 2 , with respect to (2 ) (2 ) the axis which passes through the centre C 2 of co-ordinates ρ1′ , ρ2′ and is parallel

to the axis O1′x 3′ . We notice that ρ1′

(2 )

we obtain thus

= l cos θ1 + l2 cos θ2 , ρ2′

(2 )

= l sin θ1 + l2 sin θ2 ;

MECHANICAL SYSTEMS, CLASSICAL MODELS

112 T′ =

1 1 ( I + M 2l 2 ) θ12 + 2 ( I 2 + M 2l22 ) θ22 + M 2ll2 θ1θ2 cos( θ2 − θ1 ) . 2 1

(18.3.66')

As well, the potential of the weight forces is U = M1gl1 cos θ1 + M 2 g ( l cos θ1 + l2 cos θ2 ) .

(18.3.67)

Lagrange’s equations read

( I1

+ M 2l 2 ) θ1 + M 2ll2 ⎡⎣ cos ( θ2 − θ1 ) θ2 − sin ( θ2 − θ1 ) θ22 ⎤⎦ = − ( M 1l1 + M 2l ) g sin θ1 ,

( I2

+ M 2l22 ) θ2 + M 2ll2 ⎡⎣ cos ( θ2 − θ1 ) θ1 + sin ( θ2 − θ1 ) θ12 ⎤⎦

(18.3.68)

= − M 2l2 g sin θ2 .

We find thus again (17.1.4), which have been studied and integrated by Bradistilov and by Anca Zlătescu. The small oscillations of the double pendulum have been considered in Sect. 17.1.1.2.

18.3.2.9 Sympathetic Pendulums The sympathetic pendulum has been considered directly with the aid of Newton’s equations in Sect. 17.1.1.3. We take again the problem in case of the two physical pendulums of masses M 1 and M 2 , the gravity forces being applied at the mass centres C 1 and C 2 , respectively; we denote O1C 1 = l1 , O2C 2 = l2 , where O1 and O2 are the

poles through which pass the axes of suspension. The two pendulums are connected by an elastic spring Q1Q2 , so that the point Q1 is on the straight line O1C 1 , while the point Q2 is on the straight line O2C 2 , with O1Q1 = O2Q2 = a (Fig. 18.13). The spring Q1Q2 is characterized by the elastic constant k , so that the magnitude of the force

which arises in the spring is given by k | O1O2 − Q1Q2 | , where k = const (unlike the case considered in Sect. 17.1.1.3, where the elastic constant depends on the mass of the physical pendulum upon which acts the spring, the position of the points Q1 and Q2 playing no one rôle). The position of this mechanical system is specified by the generalized co-ordinates q1 = θ1 and q2 = θ2 , these ones being the angles made by O1C 1 and O2C 2 , respectively, with the vertical line. The kinetic energy is given by T =

1 ( I θ2 + I 2 θ22 ) , 2 1 1

(18.3.69)

where I 1 and I 2 are the moments of inertia of each pendulum with respect to the corresponding axis of suspension. The potential U is expressed in the form U = g (M 1l1 cos θ1 + M 2l2 cos θ2 ) −

k (O O − Q1Q2 2 1 2

)2 .

Lagrangian Mechanics

113

But Q1Q22 = [O1O2 + a ( sin θ2 − sin θ1 ) ] + a 2 (cos θ2 − cos θ1 )2 2

= O1O22 + 2aO1O2 ( sin θ2 − sin θ1 ) + 2a 2 [ 1 − cos ( θ2 − θ1 ) ] .

In the case of small oscillations, we have sin θ1 ≅ θ1 , sin θ2 ≅ θ2 , cos θ1 ≅ 1 − θ12 / 2 , cos θ2 ≅ 1 − θ22 / 2 , so that

Q1Q22 + 2aO1O2 ( θ2 − θ1 ) + a 2 ( θ2 − θ1 )2 = [O1O2 + a ( θ2 − θ1 ) ] . 2

Hence, in case of small oscillations, it results U = g ( M1l1 + M 2l2 ) − −

1 ( M1gl1 + ka 2 ) θ12 2

1 ( M 2 gl2 + ka 2 ) θ22 + ka 2 θ1θ2 . 2

(18.3.69')

Fig. 18.13 Sympathetic pendulums

Lagrange’s equations read ( L = T + U ) k ∂ ( O O − Q1Q2 )2 = 0, 2 ∂θ1 1 2 k ∂ 2 I 2 θ2 + M 2 gl2 sin θ2 + O1O2 − Q1Q2 ) = 0 . ( 2 ∂θ2 I 1 θ1 + M 1gl1 sin θ1 +

(18.3.70)

In case of small oscillations, we get I 1 θ1 + M 1gl1 θ1 + ka 2 ( θ1 − θ2 ) = 0, I 2 θ2 + M 2 gl2 θ2 + ka 2 ( θ1 − θ2 ) = 0 .

(18.3.70')

In the particular case of two identical physical pendulums ( M 1 = M 2 = M , l1 = l2 = l , I 1 = I 2 = I ), we may write

MECHANICAL SYSTEMS, CLASSICAL MODELS

114

∂2 ∂2 2 θ θ ω θ θ 0, + + + = ( ) ( ) ( θ1 − θ2 ) + ω 2 ( θ1 − θ2 ) = 0 , 0 1 2 1 2 ∂t 2 ∂t 2

where we have introduced the pulsations ( l ′ is the length of the synchronous mechanical pendulum) ω02 =

1 Mgl g = , ω 2 = ( Mgl + 2ka 2 2 I l′

) = ω02

+

2ka 2 . I

(18.3.71)

Hence, we obtain 1 [ ( θ 0 + θ20 ) cos ω0t + ( θ10 − θ20 ) cos ωt ] 2 1 1 1 0 1 + ⎡⎢ θ1 + θ20 ) sin ω0t + ( θ10 − θ20 ) sin ωt ⎤⎥ . ( 2 ⎣ ω0 ω ⎦ 1 θ2 (t ) = [ ( θ10 + θ20 ) cos ω0t − ( θ10 − θ20 ) cos ωt ] 2 1 1 0 1 − ⎡⎢ ( θ + θ20 ) sin ω0t − ω ( θ10 − θ20 ) sin ωt ⎤⎥ , 2 ⎣ ω0 1 ⎦ θ1 (t ) =

(18.3.72)

where θk0 = θk (0) , θk0 = θk (0), k = 1, 2 , corresponding to the initial moment t = 0 . If θ10 = θ0 , θ20 = 0, θ10 = θ20 = 0 , then we get

(ω θ0 ( cos ω0t + cos ωt ) = θ0 cos 0 2 (ω θ θ2 (t ) = 0 ( cos ω0t − cos ωt ) = θ0 sin 0 2

θ1 (t ) =

− ω )t (ω cos 0 2 − ω )t (ω sin 0 2

+ ω )t , 2 + ω )t , 2

(18.3.72')

These results correspond to those in Sect. 17.1.1.3, excepting to the different definition of the elastic constant k .

Chapter 19 Hamiltonian Mechanics The motion of the representative point P ∈ Λs is governed by a system of s differential equations of second order (Lagrange’s equations) in Lagrangian mechanics. In 1834, W. R. Hamilton had the idea to use a representative space with 2s dimensions, the motion of a representative point in this space being specified by a system of 2s linear differential equations (Hamilton’s equations); The new formulation (in the frame of Hamiltonian mechanics) is equivalent to the Lagrangian formulation for discrete mechanical systems with holonomic, ideal constraints, being put a restrictive condition: these systems must be natural (as a matter of fact, the considered systems must admit a Lagrangian). These equations have remarkable analytical properties, leading to a rigorous and elegant mathematical formulation of the quantic model of mechanics. Especially, the Hamilton– Jacobi partial differential equation allows to pass from the matric quantum mechanics to the undulatory mechanics. In this order of ideas, after a study of Hamilton’s equations, we will consider in detail the Hamilton–Jacobi method (Arnold, V.I., 1976; Dobronravov, V.V., 1976; Goldstein, H., 1956; Lurie, A.I., 2002; Routh, E.J., 1892, 1898; Ter Haar, D., 1964).

19.1 Hamilton’s Equations After establishing some results with a general character, including Hamilton’s equations, we make a study of several expressions which play a special rôle: Lagrange’s brackets and Poisson’s brackets; these results will be illustrated by some applications to particular mechanical systems.

19.1.1 General Results To pass from the space of configurations to the phase space, one introduces the canonical co-ordinates and one obtains Hamilton’s canonical equations; after putting in evidence some properties of the latter ones, one considers Routh’s equations, as well as other equivalent equations. 19.1.1.1 Canonical Co-ordinates. Phase Space. Associate Expressions To study a natural discrete mechanical system S of n particles, subjected to holonomic ideal constraints, we consider Lagrange’s equations (18.2.38) (where we have introduced the kinetic potential L = T + U ), which describe the motion of the representative point P (q1 , q2 ,..., qs ) in the space of considerations Λs . To pass from this P.P. Teodorescu, Mechanical Systems, Classical Models, © Springer Science+Business Media B.V. 2009

115

MECHANICAL SYSTEMS, CLASSICAL MODELS

116

system of equations of second order to a system of equations of first order, one has an infinity of possibilities to proceed (e.g., as in Sect. 18.2.3.8). Thus, in Sect. 18.2.3.6 there have been introduced the generalized momenta by means of relations (18.2.80). Taking into account (18.2.81) and (18.2.15), (18.2.15'), and (18.2.15''), we can write p j = g jk qk + g j + U j , j = 1, 2,..., s .

(19.1.1)

Introducing the normalized algebraic complement g jk of the element g jk of the determinant g = det[ g jk ] ≠ 0 , which verify the relations (18.2.44), we obtain, easily, q j = g jk pk − g jk (gk + U k ), j = 1, 2,..., s .

(19.1.1')

Let Λ2′s be the space of the points P (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) ; let us introduce also the representative space Γ 2s of the representative points P (q1 , q2 ,..., qs , p1 , p2 ,..., ps ) . We notice that the formulae (19.1.1), (19.1.1') establish a one-to-one correspondence between the two spaces. A state of the considered discrete system S is represented, as we have shown in the preceding chapter, by a representative point in the space Λs , hence by a point ′ , and this one is represented, in a one-to-one mode, by a representative in the space Λ2s point in the space Γ 2s . The representative space Γ 2s is called the phase space (or the Gibbs’s space). The co-ordinates q1 , q2 ,..., qs , p1 , p2 ,..., ps of the corresponding representative point P (the set of generalized co-ordinates and the set of generalized momenta) are called canonical co-ordinates (Hamiltonian co-ordinates); the generalized momenta p j are coordinates conjugate to the generalized co-ordinates q j . We replace thus the study of the motion of the mechanical system S in E 3 by the study of the motion of the representative point P in the space Γ 2s . Having to do only with holonomic constraints, which are eliminated by passing to generalized co-ordinates, the representative point P ∈ Γ 2s is a free point. To study the motion of the point P , we will express the quantities of energetical nature used in the space Λs by means of the canonical co-ordinates in the space Γ 2s ; by passing  to canonical co-ordinates, a function F (q j , q j ; t ) becomes a new function F (q j , p j ; t ) ,  called the associate expression of the function F , so that F (q j , q j ; t ) = F (q j , p j ; t ) . The quantities (18.2.15'), (18.2.15'') read 1 1 1 g jk q j qk = g jk p j pk − g jk (g j + U j ) pk + g jk (g j + U j )( gk + U k ), 2 2 2 g j q j = g jk g j pk − g jk g j ( gk + U k ),

where we took into account the relations (18.2.44). The kinetic energy will take the form

Hamiltonian Mechanics

117  T = T2 + T1 + T0 ,

(19.1.2)

where we have put in evidence a quadratic form, a linear form and a constant with  respect to the generalized momenta, given by (we notice that Tk ≠ Tk , k = 0,1, 2 ) T2 =

1 jk 1 g p j pk , T1 = − g jkU j pk , T0 = − g jk (g j gk − U jU k ) + g 0 . 2 2

(19.1.2')

We get thus the associate expression of the kinetic energy by means of the canonical coordinates. If the generalized forces derive from a simple quasi-potential, then we have U j = 0, j = 1, 2,..., s , so that T1 = 0 ; as well, if the mechanical system is scleronomic, then it results g j = 0, j = 1,2,..., s , and g 0 = 0 , so that T0 = (1/ 2)g jkU jU k . If both conditions are fulfilled (the mechanical system is scleronomic and the generalized forces  derive from a simple quasi-potential), then we obtain T1 = T0 = 0 , so that T = T2 . In this case, the relations (19.1.1) read p j = g jk qk ,

(19.1.1'')

the generalized momenta vanishing together with the generalized velocities. This property does no more take place if the mentioned conditions are not fulfilled. After a partial differentiation of (18.2.15) with respect to q j , we multiply by q j and sum for all the values of the index j ; it results q j

∂T2 ∂T1 ∂T0 ∂T = q j + q j + q j = 2T2 + T1 , ∂q j ∂q j ∂q j ∂q j

where we used Euler’s theorem concerning the homogeneous functions. Adding T1 + 2T0 to both members of this relation and using the relation (18.2.81) and the notations (18.2.15''), (18.2.15'''), we may write T =

1 1 q p + (g − U j )q j + g 0 , 2 j j 2 j

(19.1.3)

obtaining thus a new remarkable expression for the kinetic energy. By an analogous calculation, the associate expression of the generalized quasipotential (18.2.22) will be of the form

 U = U 1 + U 0 , U 1 = g jk pk , U 0 = − g jkU j (gk + U k ) + U 0 ,

(19.1.4)

hence a sum of a linear form and a constant with respect to the generalized momenta. In  case of a simple quasi-potential (U j = 0 ), we have U = U 0 = U 0 . Observing that T1 + U 1 = 0 , the associate expression of the kinetic potential (18.2.34) is given by

MECHANICAL SYSTEMS, CLASSICAL MODELS

118

 1 L = L2 + L0 , L2 = T2 , L0 = − g jk (g j + U j )(gk + U k ) + g 0 + U 0 , 2

(19.1.5)

hence it will be a sum a quadratic form and a constant with respect to the generalized momenta. In case of a scleronomic mechanical system and of forces which derive from a simple quasi-potential we have L0 = U 0 , while L2 is a positive definite form in the generalized momenta. We notice that one can make the transformation (18.2.80) in case of a non-natural mechanical system too, if the condition (18.2.34''') is fulfilled. Indeed, in this case the Jacobian of the functions ∂L ∂q j is the Hessian of the kinetic potential; thus, the theorem of implicit functions allows to calculate q j = q j (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ,

j = 1,2,..., s . Obviously, these relations are linear with respect to the generalized momenta, so that the most important results previously obtained remain valid. 19.1.1.2 Donkin’s Theorem. Legendre’s Transformation Let be a function X = X (x1 , x 2 ,..., xs ) of class C 2 , for which the Hessian is non-zero

⎡ ∂2 X ⎤ det ⎢ ⎥ ≠ 0. ⎣ ∂x j ∂x k ⎦

(19.1.6)

Let us consider a transformation of variables generated by the function X in the form yj =

∂X , j = 1, 2,..., s . ∂x j

(19.1.7)

We can state Theorem 19.1.1 (Donkin). A transformation (19.1.7) for which the condition (19.1.6) is fulfilled being given, there exists a transformation, inverse to this one, generated by a function Y = Y (y1 , y2 ,..., ys ) of class C 1 , by means of the relations xj =

∂Y , j = 1, 2,..., s . ∂y j

(19.1.7')

The two functions are linked by the relation (we assume that the variables x i are expressed with the aid of the variables y j ; as well, the function Y too) X + Y = x j yj .

(19.1.8)

If X = X ( x1 , x 2 ,..., xs ; α1 , α2 ,..., αm ) , where αk , k = 1, 2,..., m , are given parameters, then we have Y = Y (y1 , y2 ,..., ys ; α1 , α2 ,..., αm ) too, so that

Hamiltonian Mechanics

119 ∂Y ∂X =− , k = 1,2,..., m . ∂αk ∂αk

(19.1.9)

Indeed, the Hessian of the function X coincides with the Jacobian of the functions in the second member of the relations (19.1.7); on the basis of the condition (19.1.6), the theorem of implicit functions shows that, using the mentioned relations, we can obtain x j = x j (y1 , y2 ,..., ys ), j = 1, 2,..., s . Calculating the function Y from the relation (19.1.8) and expressing it by means of the variables y j , j = 1, 2,..., s , we obtain ∂x ∂Y ∂ ∂X ∂x k . = ( x y − X ) = k yk + x j − ∂y j ∂y j k k ∂y j ∂x k ∂y j

Taking into account the relations (19.1.7), we get the relations (19.2.7'), which are thus justified. If in the function X intervene also the parameters αj , j = 1, 2,..., m , then these ones intervene in the direct transformation (19.1.7) too; hence, we have to do with x j = x j (y1 , y2 ,..., ys ; α1 , α2 ,..., αm ) , j = 1, 2,..., s , as well as with Y = Y (y1 , y2 ,..., ys ; α1 , α2 ,..., αm ) . We can calculate ∂x ∂Y ∂ ∂X ∂x k ∂X ∂X , = − = − ( x y − X ) = k yk − ∂α j ∂α j k k ∂α j ∂x k ∂α j ∂α j ∂α j

where we took into account (19.1.7). The theorem stated by Donkin in 1854 is thus completely proved. The passing from the variables x j to the variables y j , considered above, is known as Legendre’s transformation. For instance, in the mathematical modelling of thermodynamics, where X = H is the enthalpy, −Y = G is Gibbs’s function of state, s = 1 , x1 = S is the entropy, while y1 = T is the absolute temperature, takes place the relation H − G = ST ,

(19.1.10)

the product ST being the bound energy; the relations ∂H ∂G = T, = −S ∂S ∂T

(19.1.10')

take place. 19.1.1.3 Canonical Equations We will use Donkin’s theorem and Legendre’s transformation to establish the equations of motion of the representative point P ∈ Γ 2s . We take thus X = L , x j = q j , y j = p j , α j = q j , j = 1, 2,..., s , as well as αm = t , m = s + 1 ; because of

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the condition (18.2.34'''), the condition (19.1.6) is fulfilled. We introduce the function Y = H , called Hamilton’s function (Hamiltonian), in the form

H = p j q j − L .

(19.1.11)

  If H = p j q j − L , then it results H = H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) . We may write pj =

∂L ∂H , , q = ∂q j j ∂p j

∂L ∂H ∂L ∂H , =− , = ∂q j ∂q j ∂t ∂t

(19.1.12) (19.1.12')

for j = 1,2,..., s . If in Lagrange’s equations (18.2.38) we take into account the first relations (19.1.12), then we can replace these relations by the equivalent system pj =

∂L ∂L . , p j = − ∂q j ∂q j

(19.1.13)

Using the first relations (19.1.12') and associating the last relations (19.1.12), there result the equations of motion (given by Hamilton in 1834) of the representative point P in the space Γ 2s in the normal form q j =

∂H ∂H , p = − , j = 1, 2,..., s . ∂p j j ∂q j

(19.1.14)

These equations are the canonical equations of Hamiltonian mechanics (Hamilton’s equations) (Hamilton, W.R., 1890). Using the relation of definition (19.1.11), we get  ∂H ∂L ⎞ ∂qk ⎛ ∂L ⎞ ⎛ = − p j + ⎜ pk − ⎟ ∂q + ⎜ p j − ∂q ⎟ , ∂p j ∂ q  ⎝ j j ⎠ k ⎠ ⎝  ∂ q  ∂H ∂ L ⎛ ⎞ k = q j + ⎜ pk − . ∂p j ∂qk ⎟⎠ ∂p j ⎝

If the system (19.1.13) is verified, then one obtains Hamilton’s equations. On the other hand, if the canonical equations are verified, then the second relations obtained above lead to the first equations (19.1.13); indeed, one obtains a homogeneous system of  linear algebraic equations for which the determinant of the coefficients det[ ∂qk / ∂p j ]  is the inverse of the determinant det[ ∂p j / ∂qk ] , which is thus the Hessian (18,2,34'''), hence being non-zero (if L = T + U , then this determinant is equal to 1/ g ). Then, the first above relations lead to the last equations (19.1.13). We can thus state that Hamilton’s equations are equivalent to the equations (19.1.13), which – at their turn – are equivalent to Lagrange’s equations.

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We can calculate the first variation of the Lagrangian in the form δL =

∂L ∂L δq + δq = p j δq j + p j δq j , ∂q j j ∂q j j

(19.1.15)

which represents the sixth form of the basic equation; taking into account the operator relation (18.1.77), this relation takes the remarkable form δL =

d ( p δq ) , dt j j

(19.1.15')

in case of holonomic constraints. We can also write δ ( p j q j − L

) = q j δp j

− p j δq j .

For Hamilton’s function, we obtain, analogically, δΗ =

∂H ∂H δq + δp . ∂q j j ∂p j j

Using the relation of definition of Hamilton’s function and comparing these results, we find again the canonical equations (19.1.14). Hamilton’s equations form a system of 2s differential equations of first order in the unknown functions q j = q j (t ), p j = p j (t ), j = 1, 2,..., s . Taking into account the one-to-one link between the generalized velocities and the generalized momenta, we can put the initial conditions in the form

q j (t0 ) = q j0 , p j (t0 ) = p j0 .

(19.1.14')

Hence, we must know the position P0 of the representative point P at the initial moment. The number of the unknowns and of the equations became double, but their order diminished by a unity, which represents a great advantage from the point of view of computation; we remark also some properties of symmetry of the system of equations (19.1.14). We can write Hamilton’s equations in the matric form R = ZHR

(19.1.14'')

too, where R = [ q1 , q2 ,..., qs , p1 , p2 ,..., ps ]T , ∂H ∂H ∂H ∂H ∂H ∂H ⎤ T , ,..., , , ,..., HR = ⎡⎢ ∂qs ∂p1 ∂p2 ∂ps ⎥⎦ ⎣ ∂q1 ∂q 2

are column matrices, respectively, and Z is the matrix

(19.1.14''')

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122

⎡ 0 I⎤ Z = ⎢ ⎥. ⎣ −I 0 ⎦

(19.1.14IV)

The matrices 0 and I are the null and the unit matrices, respectively, with s rows and s columns. If, in Donkin’s theorem, we take X = T , x j = q j , y j = p j , j = 1,2,..., s , so that the generalized momenta are given by p j = ∂T / ∂q j , j = 1,2,..., s , while the function Y = Φ is specified by Φ = p j q j − T ,

(19.1.16)

then we obtain Hamilton’s equations in the form q j =

  ∂Φ ∂Φ , p = − + Q j + Rj , j = 1,2,..., s , ∂p j j ∂q j

(19.1.16')

corresponding to a non-natural and non-holonomic mechanical system, hence to Lagrange’s equations (18.2.32). If Q j , j = 1, 2,..., s , is the non-conservative part of the generalized force, then Hamilton’s equations read q j =

 ∂H ∂H , p j = − + Q j + Rj , j = 1,2,..., s . ∂p j ∂q j

(19.1.16'')

In case of holonomic constraints ( Rj , j = 1,2,..., s ) and of quasi-conservative forces which derive from a simple potential ( Q j = 0, j = 1, 2,..., s ) we find again the canonical equations; in case of a generalized potential appears a supplementary term due to the modality in which the generalized momenta have bean introduced. The advantage to introduce these quantities by the relations (18.2.80) is thus put in evidence; as a matter of fact, we can thus consider the case of an arbitrary Lagrangian too. 19.1.1.4 General Theorems. First Integrals For the initial value problem (19.1.14), (19.1.14') one can state Theorem 19.1.2 (of existence and uniqueness; Cauchy–Lipschitz). If Hamilton’s function H is of class C 1 with respect to the space variables and of class C 0 with respect to time on the ( 2s + 1 )-dimensional interval D , specified by q j0 − Q j0 ≤ q j ≤ q j0 + Q j0 , p j0 − Pj0 ≤ p j ≤ p j0 + Pj0 , t0 − T0 ≤ t ≤ t0 + T0 , Q j0 , Pj0 ,T0 = const, j = 1,2,..., s ,

and defined in the space Cartesian product of the phase space (of canonical co-ordinates q1 , q2 ,..., qs , p1 , p2 ,..., ps ) by the time space (of co-ordinate t ), and if Lipschitz’s conditions

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123

∂ ∂ H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) − H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ∂q j ∂q j μj s ⎡ μ 1 ⎤ ≤ ⎢⎣ τ λk qk − qk + λk pk − pk ⎥⎦ , T k∑ =1 ∂ ∂ H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) − H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ∂p j ∂p j ≤

1 νjT

∑ ⎡⎢⎣ λk s

k =1

qk − qk +

τ ⎤ p − pk ⎥ , μλk k ⎦

hold for j = 1,2,..., s , where T > 0 is a time constant independent on q j , p j and t , the time constant τ and the constant μ of the nature of a mass are equal to unity, while λk , μj , ν j are constants equal to unity (dimensionless if the corresponding generalized co-ordinate is a length and having the dimension of a length if the corresponding generalized co-ordinate is non-dimensional), then there exists a unique solution q j = q j (t ), p j = p j (t ) of the system (19.1.14), which satisfies the initial conditions (19.1.14') and is definite an the interval t0 − T ≤ t ≤ t0 + T , where q j0 τ p j0 ⎡ ,T T ≤ min ⎢T0 , λ j , V μλj V ⎣

⎤ ∂H ⎡ τ ∂H ⎤ ⎥ , V = max ⎢ μλ ∂q , λj ∂p in D ⎥ , j j ⎣ j ⎦ ⎦

without summation with respect to the index j . The continuity of the derivatives of first order of the function H with respect to the canonical co-ordinates on the interval D ensures the existence of the solution, according to Peano’s theorem. Lipschitz’s conditions must be also fulfilled for the uniqueness of the solution; but the latter conditions can be replaced (as in the case of other theorems of uniqueness) by other ones less restrictive, according to which the partial derivative of second order of the function H with respect to the canonical coordinates must exist and must be bounded in absolute value on the interval D . As a matter of fact, the conditions in the Theorem 19.1.2 are sufficient conditions of existence and uniqueness, which are not necessary too. As in the case of other analogous theorems (e.g., the Theorem 18.2.1) one can obtain a prolongation of the solution for t ∈ [t1 , t2 ] , corresponding to a certain interval of time in which the considered phenomena takes place, or even for t ∈ [t0 , ∞ ) or t ∈ ( ∞, t1 ] or t ∈ ( −∞, ∞ ) .

We can state also theorems on the continuous or analytical dependence of the solution on a parameter, analogue to the Theorems 6.1.3 and 6.1.4; as well, we state Theorem 19.1.3 (on the differentiability of the solution). If, in the neighbourhood of a point P (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ∈ D , the functions ∂H / ∂q j , ∂H / ∂p j , j = 1,2,..., s , H = H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) , are of class C k , then the solution

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124

q j (t ) and p j (t ) of the system (19.1.14) which satisfy the initial conditions (19.1.14')

are of class C k +1 in a neighbourhood of the point P . We call first integral of the system of canonical equations (19.1.14) a function f = f (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) of class C 1 , which is reduced to a constant along the integral curves (trajectories in the phase space) q j = q j (t ; c1 , c2 ,..., c2 s ), p j = p j (t ; c1 , c2 ,..., c2 s ), j = 1, 2,..., s ,

(19.1.17)

of this system ( ck , k = 1, 2,..., 2s , are, obviously, integration constant); hence df =

∂f ∂f dq j + dp j + f dt = 0 ∂q j ∂p j

(19.1.18)

or df ∂f ∂f q + p + f = 0 = ∂q j j ∂p j j dt

(19.1.18')

along the integral curves. The canonical equations (19.1.14) can be written in the form d q1 dq dp1 dp2 dqs dps dt = 2 = ... = = = = ... = = ∂H ∂H ∂H ∂H ∂H ∂H 1 − − − ∂p1 ∂p2 ∂ps ∂q1 ∂q2 ∂qs

(19.1.19)

too, for integration being necessary 2s independent first integrals. If we determine l ≤ 2s first integrals fk (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) = ck , ck = const, k = 1, 2,..., l ,

(19.1.20)

the matrix ∂ ( f1 , f2 ,..., fl ) ⎡ ⎤ M = ⎢ ⎣ ∂ (q1 , q2 ,..., qs , p1 , p2 ,..., ps ) ⎥⎦

being of rank l , then all these integrals are functionally independent (independent first integrals) and we can express l unknown functions of the system (19.1.19) with respect to the other ones; replacing in this system of equations, the problem is reduced to the integration of a system of equations with 2s − l unknowns (hence, with a smaller number of unknowns). If l = 2s , then all the first integrals are independent, so that the system (19.1.20) determines all the unknown functions. We notice that, if l > 2s , then the first integrals (19.1.20) are no more independent; hence, one can set up at the most 2s independent first integrals.

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125

Solving the system (19.1.20) for l = 2s , we obtain (the matrix M is a square one of order 2s , for which det M ≠ 0 ) the solution (19.1.17), hence a general integral of the system of equations (19.1.14). One puts thus in evidence 2s integration constants. Imposing the initial conditions (19.1.14'), it results q j (t0 ; c1 , c2 ,..., c2 s ) = q j0 , p j (t0 ; c1 , c2 ,..., c2 s ) = p j0 , j = 1,2,..., s ; the conditions (19.1.14') being independent,

we may write ⎡ ∂ (q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ) ⎤ det ⎢ ⎥ ≠0 ∂ (c1 , c2 ,..., c2 s ) ⎣ ⎦

and we get ck = ck (t0 ;q10 , q 20 ,..., qs0 , p10 , p20 ,..., ps0 ) , k = 1, 2,..., 2s , using the theorem of implicit functions. Thus, it results, finally (for j = 1,2,..., s ), q j = q j (t ; t0 , q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ), p j = p j (t ; t0 , q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ).

(19.1.21)

Hence, in the frame of the conditions of the theorem of existence and uniqueness, the canonical equations and the corresponding initial conditions determine, univocally, the motion of the discrete mechanical system S , subjected to holonomic ideal constraints, in a finite interval of time; by prolongation, the statement can be valid for any t . The deterministic aspect of mechanics is thus put in evidence in the Hamiltonian presentation too. If f1 = c1 and f2 = c2 are two distinct first integrals, then f ( f1 , f2 ) = 0 is a first integral too; any other first integral is expressed, in general, as a function of the 2s distinct first integrals of Hamilton’s equations. 19.1.1.5 Hamilton’s Function Starting from the relation of definition (19.1.11), passing to canonical co-ordinates, taking into account (18.2.80), (18.2.34') and using Euler’s theorem concerning homogeneous functions, we can write Hamilton’s function in the form

 ∂L      H = p j q j − L = q − L = 2L2 + L1 − L = L2 − L0 . ∂q j j If we take into account also the relations (18.2.34''), it results H =

   E , E = T2 − T0 + V ,

(19.1.22)



where E is the generalized mechanical energy, given by (18.2.68) and expressed in canonical co-ordinates (V = −U in case of a simple quasi-potential or V = −U 0 in case of a generalized quasi-potential). In case of a scleronomic mechanical system

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126

    ( T2 = 0, T0 = 0 ) we have E = T2 + V = T + V = E , obtaining the mechanical energy of the considered mechanical system. Observing, on the basis of the formulae (18.2.15') and (19.1.1'), that  2T2 − g jk g jl [ pl − (gl + U l )]g km [ pm − (gm + U m )] = g jk [ p j − ( g j + U j )][ pk − ( gk + U k )],

we can represent Hamilton’s functions in the form H = H 2 + H1 + H 0 ,

(19.1.23)

where H 2 = T2 = H0 =

1 jk g p j pk , H 1 = − g jk (g j + U j ) pk , 2

1 jk g (g j + U j )( gk + U k ) − g 0 + V , 2

(19.1.23')

hence a sum of a quadratic form, of a linear form and of a constant with respect to the generalized momenta. In case of scleronomic constraints we have g 0 = 0 , g j = 0, j = 1,2,..., s , so that H 1 = − g jkU j pk , H 0 =

1 jk g U jU k + V , 2

(19.1.23'')

and in case of generalized forces which derive from a simple potential it result U j = 0 , j = 1,2,..., s , so that H 1 = − g jk g j pk , H 0 =

1 jk g g j gk − g 0 + V ; 2

(19.1.23''')

if both conditions are fulfilled, then H 1 = 0, H 0 = V . If g jk = δ jk , then we have g jk = δ jk , where δ jk is Kronecker’s tensor; we can write T2 =

1 1 1 1 δ jk q j qk = q j q j , H 2 = T2 = δ jk p j pk = p j p j , 2 2 2 2

(19.1.24)

hence in the form of a sum of squares. In this case, the generalized co-ordinates are orthogonal (they are called normal co-ordinates), the generalized momenta having the same property. We can calculate dH ∂H ∂H = q j + p + H . dt ∂q j ∂p j j

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127

Taking into account the equations (19.1.14), we can state that the relation dH ∂H = = H dt ∂t

(19.1.25)

takes place along the integral curves of Hamilton’s system of equations. If Hamilton’s  function does not depend explicitly on time ( H = 0 , hence E = 0 ), then the generalized mechanical energy H (q1 , q 2 ,..., qs , p1 , p2 ,..., ps ) =

E (q1 , q 2 ,..., qs , p1 , p2 ,..., ps ) = h , h = const, (19.1.26)

is a first integral of the canonical system (19.1.14). This first integral corresponds to Jacobi’s first integral or to a first integral of Jacobi type of Lagrange’s system of equations, which can be written if L = 0 (see Sect. 18.2.3.4); as a matter of fact, H = 0 leads to L = 0 and reciprocally, on the basis of the last relation (19.1.12'). The mechanical system is, in this case, a generalized conservative system (as in the case in which L = 0 ). If the mechanical system is scleronomic, then we find again the first integral of the mechanical energy expressed in canonical co-ordinates   ( H = T + V = E = h ), as a particular case; we have to do with a simple or  generalized potential (U = 0 ) in this case, so that ∂ (T2 − T0 )/ ∂t + ∂V / ∂t = 0 . We notice that to have H = 0 it is not necessary that the mechanical system be scleronomic. Indeed, from (19.1.23') we see that it is necessary and sufficient to have g jk = 0, g j + U j = 0, g0 = V ;

(19.1.27)

the first relations above lead also to g jk = 0 for j , k = 1, 2,..., s , Taking into account (18.2.15'), (18.2.15''), we can state that these relations can take place only and only if the derivative ∂ri / ∂q j , i , j = 1, 2,..., s , do not depend explicitly on time, i.e. if ri = ri′ + ri′′ , ri′ = ri′ (q1 , q2 ,..., qs ) , ri′′ = ri′′(t ), i = 1, 2,..., s . This statement remains

valid for Jacobi’s first integral or for the first integrals of Jacobi type too; in particular, for ri′′ = 0, i = 1, 2,..., s , when the mechanical system is scleronomic, then one obtains the integral of mechanical energy. 19.1.1.6 Reduction of the Number of First Integrals As we have seen above, the function H is a first integral of the canonical system of equations if H = 0 ; hence, it is necessary to determine other 2s − 1 first integrals. If we write Hamilton’s equations in the form (19.1.19), then we notice that – in this case – we can neglect the last ratio in this system; indeed, to integrate the remaining system of equations, there are sufficient 2s − 1 first integrals, which do not depend explicitly on time. Thus we can determine 2s − 1 canonical co-ordinates as functions of one of them. To fix the ideas, we assume that ∂H / ∂p1 ≠ 0 ; we find thus the generalized co-

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128

ordinates q j = q j (q1 ; c1 , c2 ,..., c2 s −1 ), j = 2, 3,..., s , and the generalized momenta pi = pi (q1 ; c1 , c2 ,..., c2 s −1 ), i = 1, 2,..., s . In this case, it results the function of

Hamilton H = H (q1 ; c1 , c2 ,..., c2 s −1 ) , where we have put in evidence the integration constants. From the first and the last ratio (19.1.19) we get −1

⎛ ∂H ⎞ t = ∫⎜ ⎟ dq1 + c2 s = F (q1 ; c1 , c2 ,..., c2 s ) , ⎝ ∂p1 ⎠

(19.1.28)

the function F containing he last integration constant c2 s too. Observing that the derivative ∂F / ∂q1 = ( ∂H / ∂p1 )−1 ≠ 0 , the theorem of implicit functions allows to calculate q1 = q1 (t ; c1 , c2 ,..., c2 s ) . Introducing then in the above results, we obtain the general integrals (19.1.17). The system of equations (19.1.19) is of the form (15.1.24), so that one can use the theory of Jacobi’s multiplier (see Sect. 15.1.1.4), as well as the theory of the last multiplier (see Sect. 15.1.1.5). In this case, x j = q j , xs + j = p j , X j = ∂H / ∂p j , Xs + j = −∂H / ∂q j , j = 1, 2,..., s , and x 2 s + 1 = 1 , n = 2s + 1 . We notice that the

divergence of the vector which has as components the denominators in (19.1.19) vanishes ∂ ∂q1

∂ ⎛ ∂H ⎞ ∂ ⎛ ∂H ⎞ ⎛ ∂H ⎞ ⎜ ∂p ⎟ + ∂q ⎜ ∂p ⎟ + ... + ∂q ⎜ ∂p ⎟ s ⎝ s ⎠ ⎝ 1⎠ 2 ⎝ 2 ⎠ ∂ ⎛ ∂H ⎞ ∂ ⎛ ∂H ⎞ ∂ ⎛ ∂H ⎞ ∂ + − + − + ... + ⎜− ⎟ + ( I ) = 0, ∂p1 ⎜⎝ ∂q1 ⎟⎠ ∂p2 ⎜⎝ ∂q2 ⎟⎠ ∂ps ⎝ ∂qs ⎠ ∂t

hence the condition (15.1.30) is fulfilled; a constant is thus a multiplier of the sequence of equations (19.1.19), while – according to the Theorem 15.1.4'' – any non-constant multiplier is a first integral of this system of differential equations. Hence, there are sufficient 2s − 1 first integrals to determine the general solution of the respective system. If H = 0 , then there remain sufficient 2s − 2 first integrals for a complete study of the canonical equations. For instance, in case of the problem of two particles – at a first view – there are necessary 2 ⋅ 6 = 12 first integrals; according to what has been shown above, in case of scleronomic mechanical systems and of conservative forces there are sufficient 2 ⋅ (6 − 1) = 10 first integrals. If the two particles are subjected to no one constraint, being acted upon only by forces of Newtonian attraction, we are just in the mentioned case; the conservation theorem of momentum leads to six first integrals, the conservation theorem of the moment of momentum allows to write three first integrals and the conservation theorem of mechanical energy given also a first integral, so that the integration of the corresponding system of canonical equations is reduced to quadratures. Let be a scleronomic conservative Hamiltonian system with two degrees of freedom. It is possible, in this case, to write the integral of the mechanical energy, i.e. H = (q1 , q2 , p1 , p2 ) = h , h = const ; one obtains the differential system

Hamiltonian Mechanics

129 d q1 dq 2 dp1 dp2 dt = = = = . ∂H ∂H ∂H ∂H 1 − − ∂p1 ∂p2 ∂q1 ∂q2

Let us suppose that another first integral f (q1 , q2 ; p1 , p2 ) = c , c = const , which does not contain explicitly the time, is known. The two first integrals being independent, we can calculate p j = p j (q1 , q2 ; h , c ), j = 1, 2 ; replacing in Hamilton’s equations, this system is reduced to dq1 dq = 2 = dt , ∂H ∂H ∂p1 ∂p2

with the unknown functions qk = qk (t ), k = 1, 2 , for which the last multiplier M = det[ ∂ ( p1 , p2 )/ ∂ (h , c )] satisfies the equation (see Sect. 15.1.1.5 too)

∂M ⎞ ⎛ ∂M ⎞ ∂ ⎛⎜ M ⎟ ∂ ⎜ M ∂p ⎟ p ∂ ⎝ ⎝ 1 ⎠ 2 ⎠ + = 0. ∂q1 ∂q2

Hence, ⎛ ∂H

∂H

⎞

∫ M ⎜⎝ ∂p2 dq1 − ∂p1 dq2 ⎟⎠ = const is a first integral of the last differential system. We notice that H and f depend on h and c by means of the generalized momenta p1 and p2 . Because dH / dh = 1 and dH / dc = 0 , it results

∂H ∂p1 ∂H ∂p2 ∂H ∂p1 ∂H ∂p2 + = 1, + = 0. ∂p1 ∂h ∂p2 ∂h ∂p1 ∂c ∂p2 ∂c One obtains analogous results from df / dh = 0 , df / dc = 1 . The determinant M being non-zero, we can write ∂p ∂H = 2,M ∂p1 ∂c ∂p ∂f M = 2,M ∂p1 ∂h

M

∂p ∂H = − 1, ∂p2 ∂c ∂p ∂f = − 1. ∂p2 ∂h

The last first integral becomes ∂χ = const, χ(q1 , q2 ; h , c ) = ∫ ( p1 dq1 + p2 dq2 ) . ∂c

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130

Then, taking into account M det[ ∂ (H , f )/ ∂ ( p1 , p2 )] = 1 , the reduced system of differential equations allows to write ∂f ∂f dt = M ⎛⎜ dq 2 − dq1 ⎞⎟ , p p ∂ ∂ ⎝ 1 ⎠ 2

wherefrom t =

∂χ + const . ∂h

The integration of the differential system is thus reduced to a quadrature, which determines the function χ . 19.1.1.7 Routh’s Equations It is convenient, in some cases, to use as parameters which describe the motion of the mechanical system S a part of the Hamiltonian co-ordinates and a part of the Lagrangian ones; one obtains thus a “mixed method” to approach the problem. We assume that the motion is specified by Hamilton’s variables (denoted by Greek indices) q α , pα , α = 1, 2,..., r and Lagrange’s variables (denoted by Roman indices) q j , q j , j = r + 1, r + 2,..., s , the set of which form Routh’s variables. We effect a partial Legendre transformation of the form pα =

∂L , α = 1, 2,..., r , ∂qα

(19.1.29)

which allows to determine qα = qα (q β , pβ , q j , q j ; t ) = qα (q1 , q2 ,..., qs , p1 , p2 ,..., pr , qr + 1 , qr + 2 ,..., qs ; t ), α = 1, 2,..., r .

Applying a variation δ to Lagrange’s function L = L (q α , qα , q j , q j ; t ) , we obtain δL =

r

∂L

∑ ⎛⎜⎝ ∂qα δqα

α =1

+

∂L δqα ⎞⎟ + ∂qα ⎠

∂L ⎛ ∂L ⎜ ∂q δq j + ∂q δq j j j j =r +1 ⎝ s

∑

⎞ ⎟. ⎠

Taking into account (19.1.29), we notice that r

δ ∑ pα qα = α =1

r

∂L

∑ ∂qα δqα

α =1

+

r

∑ qα δpα .

α =1

wherefrom r ∂L ⎛ r ⎞ δ ⎜ ∑ pα qα − L ⎟ = − ∑ ⎛⎜ δq α − qα δpα ⎞⎟ − q ∂ α ⎝ ⎠ ⎝ α =1 ⎠ α =1

∂L ⎛ ∂L ⎞ ⎜ ∂q δq j + ∂q δq j ⎟ . j j ⎠ j =r +1 ⎝ s

∑

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131

We introduce Routh’s function

R =



 ∑ pαqα − L r

,

(19.1.30)

α =1

where “the cap” indicates the expression of the generalized velocities qα , α = 1, 2,...,r , by means of Routh’s variables, so that R = R (q α , pα , q j , q j ; t ) ; it results δR =

r

∂R

∑ ⎝⎛⎜ ∂qα

α =1

δq α +

∂R δpα ⎞⎟ + ∂pα ⎠

∂R ⎛ ∂R ⎞ ⎜ ∂q δq j + ∂q δq j ⎟ . j j ⎠ j =r +1 ⎝ s

∑

Identifying the above relations, where we take into account the notation (19.1.30), we can write ∂L ∂R ∂R , qα = , α = 1, 2,..., r , = ∂q α ∂q α ∂pα ∂L ∂R ∂L ∂R − = = ,− , j = r + 1, r + 2,..., s . ∂q j ∂q j ∂q j ∂q j −

Lagrange’s equations (18.2.38) led thus Routh, in 1876, to the equations qα =

∂R ∂R , p α = − , α = 1, 2,..., r , ∂pα ∂q α

d ⎛ ∂R ⎞ ∂R − = 0, j = r + 1, r + 2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(19.1.31) (19.1.31')

called Routh’s equations (see Sect. 18.2.3.7 too); we remark that a part of these equations are of Hamilton type (Routh’s function plays the rôle of Hamilton’s function) and another part of Lagrange type (Routh’s function plays the rôle of Lagrange’s kinetic potential), which justifies the name of “mixed method” given above. In the case in which the generalized co-ordinates q α , α = 1,2,..., r , are ignorable we find again the Routh–Helmholtz theorem (see Sect. 18.2.3.7 too). If the holonomic constraints to which is subjected the mechanical system S are also scleronomic, then the kinetic energy is expressed in the form T =

1 s s ∑ g jk (qr +1 , qr + 2 ,..., qs )qj qk . 2∑ j = 1 k =1

(19.1.31'')

We also assume that the generalized forces are quasi-conservative, deriving from a simple quasi-potential U = U (q1 , q2 ,..., qs ; t ) . In this case, pα =

∂Τ = ∂qα

r

∑ gαβ qβ

β =1

+

s

∑

j =r +1

g α j q j , α = 1, 2,..., r .

(19.1.32)

MECHANICAL SYSTEMS, CLASSICAL MODELS

132

Because [ g αβ ] is a square matrix of rank r and det[ g αβ ] ≠ 0 , corresponding to the condition (18.2.34'''), we get r

∑ g αβ pβ

qα =

β =1

−

s

∑

j =r +1

hα j q j , α = 1, 2,..., r ,

(19.1.32')

where g αβ is the normalized algebraic complement of the element g αβ in det[ g αβ ] , while r

∑ g αβ g β j ,

hα j =

β =1

α = 1, 2,..., r , j = r + 1, r + 2,..., s .

(19.1.32'')

Obviously, the coefficients g αβ and hα j are functions only on the palpable coordinates q j , j = r + 1, r + 2,..., s . Observing that g jk =

 r ∂2T ∂ ⎛ ∂T ∂T ∂qα = + ∑ ⎜ ∂q j ∂qk ∂q j ⎝ ∂qk α =1 ∂qα ∂qk ∂ ⎛ ∂T ⎞ ∂ r qα − hαk pα , = ⎜ ⎟ ∂q j ⎝ ∂qk ⎠ ∂q j α∑ =1

⎞ ⎟ qα ⎠

there results r

∑ gα j hαk

g jk = g jk −

α =1

= g jk −

r

r

∑ ∑ g αβ gα j g βk , j , k

α =1 β =1

= r + 1, r + 2,..., s . (19.1.33)

Analogically,  ∂2T ∂ = = ∂pα ∂pβ ∂pα

g αβ

=

r

∑ g γβ δγα

γ =1

g jα ∂ = ∂q j

r

∑ g βα pβ

β =1

∂T ∂qγ ∂ = ∂ p ∂ p α β γ =1 r

∑ ∂qγ

r

∑ g γβ pγ

γ =1

= g αβ , α, β = 1,2,..., r ,

 ∂2T ∂ = = ∂q j ∂pα ∂q j

r

∂T ∂qβ ∂pα

∑ ∂qβ

β =1

= 0, α = 1, 2,..., r , j = r + 1, r + 2,..., s ,

so that we can write s  1 r r 1 s T = ∑ ∑ g αβ pα pβ + g q q . ∑ ∑ 2 α =1 β =1 2 j = r + 1 k = r + 1 jk j k

We denote

(19.1.34)

Hamiltonian Mechanics

133 T∗ =

s 1 s g q q . ∑ ∑ 2 j = r + 1 k = r + 1 jk j k

(19.1.35)

Let us introduce Routh’s potential U∗ = U −

1 r r αβ ∑ g pα p β 2 α∑ =1 β =1

(19.1.36)

and the generalized potential Π = U∗ +

s

r

∑ ∑ hα j pαqj .

j =r + 1 α =1

(19.1.36')

Taking into account the relation of definition (19.1.30) and the generalized velocities (19.1.32'), we obtain Routh’s function in the form

R = −(T ∗ + Π ) .

(19.1.37)

From (19.1.34), (19.1.35), and (19.1.36), it results T −U = T∗ −U∗.

(19.1.38)

Lagrange’s equations (19.1.31') can be thus considered as being the equations of motion of a reduced system, which admits −R (given by (19.2.37)) as kinetic potential and which has the same mechanical energy as the initial system. If, in particular, the kinetic energy of the initial mechanical system does not contain products of the generalized velocities qα of the ignorable co-ordinates, hence if g α j = 0, α = 1, 2,..., r , j = r + 1, r + 2,..., s , then the kinetic energy T is obtained as a sum of two quadratic forms: a form in the generalized velocities qα and a form in the generalized velocities q j ; such a mechanical system is a gyroscopic uncoupled one. In this case, hα j = 0, α = 1,2,..., r , j = r + 1, r + 2,..., s , while Π = U ∗ , as well, T∗ =

s 1 s g q q . ∑ ∑ 2 j = r + 1 k = r + 1 jk j k

(19.1.35')

As it can be easily seen, if the initial system is conservative (U = 0 ), then the reduced system has the same property (U ∗ = 0 ). If one has ∂U / ∂q α = 0 too, then it results ∂L / ∂q α = 0, α = 1, 2,..., r , so that the ignorable co-ordinates are hidden ones too. The motions in which vary only the hidden co-ordinates are called hidden motions; such type of motion occurs, e.g., in case of mechanical systems which contain gyroscopes. H. Hertz dealt in 1894 with these problems, considering the potential energy of a conservative system as being the kinetic energy of some hidden motions.

MECHANICAL SYSTEMS, CLASSICAL MODELS

134 19.1.1.8 Cyclic Co-ordinates

A generalized co-ordinate q α which does not intervene explicitly in Hamilton’s function ( ∂H / ∂q α = 0 ) is called, after Helmholtz, cyclic co-ordinate; let us admit thus that the first r co-ordinates are cyclic ( α = 1,2,...,r ). Taking into account the first relation (19.1.12'), we can state that an ignorable co-ordinate (for which ∂L / ∂q α = 0 ) is a cyclic one too. We are thus led to r first integrals pα = bα , bα = const , α = 1,2,...,r ; taking into account the results in the preceding subsection, we can state that the cyclic co-ordinates do not intervene explicitly neither in Routh’s function, so that the equations (19.1.31) lead to the same results as the Routh–Helmholtz theorem (see Sect. 18.2.3.7). The Lagrange type equations (19.1.31) remain to be integrated. If we deal with the system of canonical equations, after the determination of the mentioned r first integrals, we remain with the system of equations (19.1.14) for j = r + 1, r + 2,..., s (a system of 2(s − r ) partial differential equations of first order with 2(s − r ) unknown functions). Hamilton’s function will be of the form H = H (q j , p j , bα ; t ) ≡ H (qr + 1 , qr + 2 ,..., qs , pr + 1 , pr + 2 ,..., ps , b1 , b2 ,..., br ; t ) ;

thus, by integration of the reduced canonical system, we get q j = q j (t ; ak , bk , bα ) ≡ q j (t ; ar + 1 , ar + 2 ,..., as , b1 , b2 ,..., bs ) , p j = p j (t ; ak , bk , bα ) ≡ p j (t ; ar + 1 , ar + 2 ,..., as , b1 , b2 ,..., bs ), ak , bk = const, j , k = r + 1, r + 2,..., s .

Replacing in Hamilton’s function, we obtain H = H (t ; ak , a j , bk , bα ) ≡ H (t ; ar + 1 , ar + 2 ,..., as , b1 , b2 ,..., bs ) .

Introducing in the first r equations of the first group of equation (19.1.14), we find also the cyclic co-ordinates qα =

∂H

∫ ∂bα

dt + aα , α = 1, 2,..., r .

(19.1.39)

Hence, q α = q α (t ; aα , bα , bj ) ≡ qα (t ; a1 , a2 ,..., as , b1 , b2 ,..., bs ) , only the cyclic coordinates depending on all the integration constant. If, in particular, all the generalized co-ordinates are cyclic ( r = s ), then all the generalized momenta are constant, while H = H (t ;b1 , b2 ,..., bs ) ; the cyclic co-ordinates are obtained by s quadratures, intervening the other s integration constants. If the mechanical system is also scleronomic, then it results ∂H / ∂bj = ω j , ω j = const , j = 1, 2,..., s , the generalized co-ordinates being of the form q j = ω j t + a j , j = 1, 2,..., s ,

(19.1.39')

Hamiltonian Mechanics

135

which justifies the denomination of cyclic co-ordinates. The problem of integration of Hamilton’s equations (19.1.14) is thus reduced to the determination of a transformation of generalized co-ordinates to maintain the form of the canonical equations, the new generalized co-ordinates being cyclic. As we will see, the intervention of the one or more cyclic co-ordinates may lead to a simplification of many methods of calculation. 19.1.1.9 The Pfaff Form of Hamilton. The Bilinear Covariant. Whittaker’s Equations Let be the differential form ω =

n

∑ Xi (x1 , x 2 ,..., xn )dxi ,

(19.1.40)

i =1

where the functions Xi , i = 1, 2,..., n , of class C 1 are definite on a certain domain in \n ; we assume that ω is not a total differential. An expression of this form is called Pfaffian (a Pfaff form). Let us suppose that x j = x j (u1 , u2 ), j = 1, 2,..., n , depending on two parameters. Let be the differential quantities dx j =

∂x j ∂x j du1 , δx j = du . ∂u1 ∂u2 2

Noting that d ( δx j

)=

∂2 x j d u d u , δ ( dx j ∂u1 ∂u2 2 1

)=

∂2 x j du d u ∂u2 ∂u1 1 2

and assuming that x j are functions of class C 2 in the two parameters, we have d( δx j ) = δ(dx j ) , according to Schwartz’s theorem. If we denote

ωd =

n

n

i =1

i =1

∑ Xi dxi , ωδ = ∑ Xi δxi

(19.1.41)

and take into account the permutation relation of the differential operators, then we can write ω ′ = ω ( d, δ ) = δωd − dωδ =

n

∑ ω j δx j , j =1

⎛ ∂Xi ∂X j ⎞ ωj = ∑ ⎜ − dx , j = 1, 2,..., n , ∂x i ⎟⎠ i i = 1 ⎝ ∂x j n

(19.1.41')

obtaining thus the bilinear covariant of the form ω . Let us make the change of variables

x j = x j (y1 , y2 ,..., yn ), x j ∈ C 1 , j = 1, 2,..., n ,

(19.1.42)

MECHANICAL SYSTEMS, CLASSICAL MODELS

136

so that det[ ∂x j / ∂yk ] ≠ 0 on a domain in \n . Denoting ∂x

n

∑ Xi ∂yki , k

Yk =

i =1

= 1, 2,..., n ,

(19.1.42')

we obtain

ω = ω′ =

n

∑ ωk δyk , ωk

=

k =1

n

∑Yk dyk ,

k =1

∂Y

∑ ⎛⎜⎝ ∂ykl n

l =1

−

(19.1.43)

∂Yk ⎞ dy , k = 1, 2,..., n . ∂yl ⎟⎠ l

(19.1.43')

The Pfaff form ω and its bilinear covariant ω ′ maintain their form. The necessary and sufficient conditions to have ω ′ ≡ 0 for any δx j , considered arbitrary, are ω j = 0, j = 1, 2,..., n , being thus led to the differential system associate to the Pfaff form n

⎛ ∂X

∑ ⎜⎝ ∂x ji

i =1

−

∂X j ⎞ dx = 0, j = 1, 2,..., n . ∂x i ⎟⎠ i

(19.1.44)

The transform of this system, by the change of variables (19.1.42), will – obviously – be ∂Y

∑ ⎛⎜⎝ ∂ykl n

l =1

−

∂Yk ⎞ dy = 0, k = 1, 2,..., n . ∂yl ⎟⎠ l

(19.1.44')

As it can be easily seen, if ω is a total differential, then ω ′ = 0 ; consequently, we can state that the Pfaff forms ω and

c ω + dF = c ω +

n

∂F

∑ ∂xi

i =1

dx i ,

where c = const , and F = F ( x1 , x 2 ,..., x n ) is a function of class C 2 , have the same bilinear covariant. By definition, the Pfaff form of Hamilton is ω = p j dq j − H dt ,

(19.1.45)

where we have introduced the canonical variables q j , p j and the time t (hence, n = 2s + 1 ); the bilinear covariant of the form is

ω ′ = δωd − dωδ = δp j dq j − δHdt − ( dp j δq j − dH δt )

(

)

∂H ∂H ∂H ⎛ ⎞ ⎛ ⎞ dt ⎟ δq j + ⎜ dq j − dt ⎟ δp j + dH − dt δt . = − ⎜ dp j + ∂ ∂ ∂t q p j j ⎝ ⎠ ⎝ ⎠

Hamiltonian Mechanics

137

The differential system of Hamilton associated to the Pfaff form is formed by the canonical equations (19.1.14) and the differential outcome (19.1.25) of them. By introducing the notion of exterior differential in the sense of Elie Cartan, the Pfaff form of Hamilton can lead to a new differential principle of mechanics; choosing conveniently the Hamiltonian, one can build up new mathematical models (e.g., the invariantive model of O. Onicescu). In case of a generalized conservative system ( H = 0 ) one obtains the first integral (19.1.26) of the generalized mechanical energy. Assuming that ∂H / ∂p1 ≠ 0 (at least one the components of the generalized momentum must be non-zero, otherwise the motion of the mechanical system could not take place), the theorem of implicit functions leads to p1 = − K (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; h ) . The corresponding Pfaff form will be ω =

s

∑ p j dq j

j =2

− Kdql − hdt ,

(19.1.46)

where K plays the rôle of Hamilton’s function and q1 the rôle of time; indeed, according to a previous notice, the form ω and the form ω + hdt ( hdt is an exact differential) have the same bilinear covariant. In this case, the associate differential system will be q j′ =

∂K ∂K dK ∂K , , p′ = − , j = 1, 2,..., s , = ∂p j j ∂q j d q1 ∂q1

(19.1.47)

where q j′ = dq j / dq1 , p j′ = dp j / dq1 , j = 1,2, 3,..., s ; these 2(s − 1) equations of Hamilton type are known as Whittaker’s equations. One can thus determine the canonical co-ordinates q j = q j (q1 ; a2 , a 3 ,..., as , b2 , b3 ,..., bs , h ), p j = p j (q1 ; a2 , a 3 ,..., as , b2 , b3 ,..., bs , h ), j = 1, 2,..., s ;

(19.1.47')

replacing then in the function K , we obtain K = K (q1 ; a2 , a 3 ,..., as , b2 , b3 ,..., bs , h ) , hence the generalized momentum p1 too. Finally, Hamilton’s equation q1 = ∂H / ∂p1 allows to put in evidence the dependence on time of the canonical co-ordinates, introducing the last integration constant too. Besides, these considerations can be put in connection to those made in Sect. 19.1.1.6. Denoting

L =

s

∑ p j q j′

j =2

−K,

(19.1.48)

where L ′ = L ′(q1 , q2 ,..., qs , q2′ , q 3′ ,..., qs′ ) , and effecting a Legendre transformation, we can replace Whittaker’s equations (of Hamilton type) by Jacobi’s equations (of Lagrange type) (see Sect. 18.2.3.5).

MECHANICAL SYSTEMS, CLASSICAL MODELS

138

19.1.2 Lagrange’s Brackets. Poisson’s Brackets In what follows one introduces Lagrange’s and Poisson’s brackets, in connection to the system of canonical equations; these brackets allow to determine first integrals of the mentioned equations and put in evidence interesting properties of the integration constant. 19.1.2.1 Lagrange’s Brackets. Lagrange’s Theorem Let be the function x i = x i ( ξ1 , ξ2 ,..., ξn , η1 , η2 ,..., ηn ), yi = yi ( ξ1 , ξ2 ,..., ξn , η1 , η2 ,..., ηn ), i = 1, 2,..., n ,

(19.1.49)

of class C 1 . We call Lagrange’s brackets (denoted by square brackets) the sums ⎡ ∂ (x , y ) ⎤

∂x ∂y ⎞ ⎛ ∂x ∂yi − i i ⎟, ∂ ξ ∂ ξk ∂ξ j ⎠ k ⎦ k ⎣ i =1 i =1 ⎝ n n ∂x ∂y ⎞ ⎡ ∂ (x , y ) ⎤ ⎛ ∂x ∂y [ η j , ηk ] = ∑ det ⎢ ∂ ( η i , ηi ) ⎥ = ∑ ⎜ ∂ηi ∂η i − ∂η i ∂ηi ⎟, j j j ⎠ k ⎦ k k ⎣ i =1 i =1 ⎝ n n ∂x i ∂yi ⎞ ⎡ ∂ ( x i , yi ) ⎤ ⎛ ∂x i ∂yi [ ξj , ηk ] = ∑ det ⎢ ∂ ( ξ , η ) ⎥ = ∑ ⎜ ∂ξ ∂η − ∂η ∂ξ ⎟, k ⎦ k k j j j ⎠ ⎣ i =1 i =1 ⎝ n

n

[ ξj , ξk ] = ∑ det ⎢ ∂ ( ξ i , ξi ) ⎥ = ∑ ⎜ ∂ξ i j j

(19.1.49')

for j , k = 1, 2,..., n ; the denomination of brackets is given because these quantities are sums of brackets. Lagrange’s brackets are, obviously, anticommutative (skewsymmetric with respect to the indices j and k )

[ ξj , ξk ] = − [ ξk , ξj ], [ η j , ηk ] = − [ ηk , η j ], [ ξj , ηk ] = − [ ηk , ξj ] .

(19.1.49'')

In quantum mechanics it is used to denote Lagrange’s brackets in the form { ξ j , ξk } ,

{ η j , ηk } , { ξj , ηk } . Let be q j = q j (t ; a , b ) , p j = p j (t ; a , b ), j = 1,2,..., s , the integrals of the canonical system (19.2.14), where we have put in evidence the integration constants a and b ; the corresponding Lagrange’s bracket is written in the form (the two constants play the rôle of parameters) ∂q j ∂a [ a ,b ] = ∂q j ∂b

Let us calculate

∂p j ∂q j ∂p j ∂q j ∂p j ∂a . = − ∂p j ∂a ∂b ∂b ∂a ∂b

(19.1.50)

Hamiltonian Mechanics

139

∂q j ∂p j ∂q j ∂p j ∂q j ∂p j ∂q j ∂p j d [ a ,b ] = + − − , dt ∂a ∂b ∂a ∂b ∂b ∂a ∂b ∂a

where we assume that the two parameters are independent on t , so that (d / dt )( ∂ / ∂c ) = ( ∂ / ∂c )(d / dt ) , where c can be a or b ; taking into account Hamilton’s equations, we may write ∂q j ∂2 H ∂qk ∂2 H ∂pk = + , ∂c ∂qk ∂p j ∂c ∂pk ∂p j ∂c ∂p j ∂2 H ∂qk ∂2 H ∂pk = − − , ∂c ∂qk ∂q j ∂c ∂pk ∂q j ∂c

where, analogously, we make – successively – c equal to a or b . We get ∂q ∂q j ⎞ ∂p ∂p j ⎞ ∂2 H ⎛ ∂qk ∂q j ∂2 H ⎛ ∂pk ∂p j d [ a ,b ] = − k + − k ⎜ ⎟ dt ∂qk ∂q j ⎝ ∂a ∂b ∂b ∂a ⎠ ∂pk ∂p j ⎜⎝ ∂a ∂b ∂b ∂a ⎟⎠ ∂q ∂p j ⎞ ∂q ∂q j ⎞ ∂2 H ⎛ ∂qk ∂p j ∂2 H ⎛ ∂pk ∂q j . + − k + − k ⎜ ⎟ ⎜ ∂qk ∂p j ⎝ ∂a ∂b ∂b ∂a ⎠ ∂pk ∂q j ⎝ ∂a ∂b ∂b ∂a ⎟⎠

Observing that the mixed derivatives of Hamilton’s function do not depend on the order of differentiation (we assume that H ∈ C 2 ), it results that the first two sums vanish (the derivatives of the function H are symmetric with respect to the indices k and j , while the brackets are skew-symmetric with respect to the same indices); if in the last sums we replace the dummy indices k and j by j and k , respectively, then we remark that the last two sums vanish too. Hence, d[a , b ]/ dt = 0 , so that

[ a , b ] = const ;

(19.1.51)

we can state Theorem 19.1.4 (Lagrange). Lagrange’s bracket corresponding to two integration constants is constant along the trajectory of the representative point P in the phase space Γ 2s . 19.1.2.2 Poisson’s Bracket. Properties Let be the functions ξi = ξi (x1 , x 2 ,..., x n , y1 , y2 ,..., yn ), ηi = ηi (x1 , x 2 ,..., x n , y1 , y2 ,..., yn ), i = 1, 2,..., n ,

of class C 1 . We call Poisson’s brackets (denoted by usual brackets) the sums

(19.1.52)

MECHANICAL SYSTEMS, CLASSICAL MODELS

140 ⎡ ∂( ξ , ξ ) ⎤

⎛ ∂ξ j ∂ξk ∂ξ j ∂ξk ⎞ , − ∂yi ∂yi ∂x i ⎟⎠ i i ⎦ ⎣ i =1 i =1 n n ⎡ ∂ ( η j , ηk ) ⎤ ⎛ ∂η j ∂ηk ∂η j ∂ηk ⎞ , = ∑ det ⎢ = ∑⎜ − ⎥ ∂yi ∂x i ⎟⎠ ⎣ ∂ ( x i , yi ) ⎦ i =1 ⎝ ∂x i ∂yi i =1 n n ⎡ ∂ ( ξ j , ηk ) ⎤ ⎛ ∂ξ j ∂ηk ∂ξ j ∂ηk ⎞ = ∑⎜ − , = ∑ det ⎢ ⎥ ∂yi ∂x i ⎟⎠ ⎣ ∂ (x i , yi ) ⎦ i =1 ⎝ ∂x i ∂yi i =1 n

( ξj , ξk ) = ∑ det ⎢ ∂ (xj , yk ) ⎥ ( η j , ηk ) ( ξj , ηk )

=

n

∑ ⎜⎝ ∂xi

(19.1.52')

for j , k = 1, 2,..., n ; these brackets are anticommutative too (skew-symmetric with respect to the indices j and k )

( ξj , ξk ) = − ( ξk , ξj ) , ( η j , ηk ) = − ( ηk , η j ) , ( ξ j , ηk ) = − ( ηk , ξj ) .

(19.1.52'')

In quantum mechanics, Poisson’s brackets are usually denoted in the form [ ξ j , ξk ] , [ η j , ηk ] , [ ξ j , ηk ] .

Let us put in evidence the connection between Lagrange’s and Poisson ’s brackets in a unitary form; we will consider 2n functions f j = f j (x1 , x 2 ,..., x n , y1 , y2 ,..., yn ) , j = 1, 2,..., 2n , for which ∂ ( f1 , f2 ,..., f2 n ) ⎡ ⎤ ≠ 0; det ⎢ ⎣ ∂ (x1 , x 2 ,..., x n , y1 , y2 ,..., yn ) ⎥⎦

so that, in this case, we can calculate x i = x i ( f1 , f2 ,..., f2 n ) , yi = yi ( f1 , f2 ,..., f2 n ) , i = 1, 2,..., n . Observing that 2n

2 n ∂f ∂y ∂f ∂x k k = δ jk , ∑ i = 0, ∂fi i = 1 ∂x j ∂fi

∑ ∂xij

i =1 2n

2 n ∂f ∂y ∂f ∂x k k = 0, ∑ i = δ jk , ∂fi ∂ y ∂ f j i i =1

∑ ∂yij

i =1

∂fp ∂xl ∂fp ∂yl ⎞ + = δpq , ∂fq ∂yl ∂fq ⎟⎠

∑ ⎛⎜⎝ ∂xl n

l =1

for j , k = 1, 2,..., n and p , q = 1, 2,..., 2n , we obtain, easily, 2n

∑ [ fi , fj ] ( fi , fk ) = δ jk , j , k

i =1

In detail, we can write

= 1, 2,..., 2n .

(19.1.53)

Hamiltonian Mechanics

141

2n

∑ {[ ξi , ξj ] ( ξi , ξk ) + [ ηi , ξj ] ( ηi , ξk )} = δ jk

,

∑ {[ ξi , ξj ] ( ξi , ηk ) + [ ηi , ξj ] ( ηi , ηk )} = δ jk

,

i =1 n

i =1 n

∑ {[ ξi , η j ] ( ξi , ξk ) + [ ηi , η j ] ( ηi , ξk )} = δ jk ,

.

(19.1.53')

i =1 2n

∑ {[ ξi , η j ] ( ξi , ηk ) + [ ηi , η j ] ( ηi , ηk )} = δ jk

,

i =1

for j , k = 1, 2,..., n . In case of a Hamiltonian system, let be the functions

ϕ = ϕ (q j , p j ; t ) ,

ψ = ψ (q j , p j ; t ) of class C ; the corresponding Poisson’s bracket is 1

[ ϕ, ψ ] =

∂ϕ ∂q j

∂ψ ∂q j

∂ϕ ∂p j

∂ψ ∂p j

=

∂ϕ ∂ψ ∂ϕ ∂ψ − . ∂q j ∂p j ∂p j ∂q j

(19.1.54)

We mention the obvious properties (ϕ , ϕ ) = 0, (ϕ ,C ) = 0, (C ϕ, ψ ) = C (ϕ , ψ ), C = const, (ϕ , ψ ) = −( ψ , ϕ ), ( − ϕ , ψ ) = −(ϕ , ψ ) ,

(19.1.55)

as well as the properties ( f ,q j ) = −

∂f ∂f , ( f , pj ) = − , j = 1, 2,..., s , ∂p j ∂q j

(q j , qk ) = 0, ( p j , pk ) = 0, (q j , pk ) = −( p j , qk ) = δ jk , j , k = 1, 2,..., s .

(19.1.55') (19.1.55'')

Starting from the relation of definition (19.1.54), it results, as well, (ϕ1 + ϕ2 , ψ ) = (ϕ1 , ψ ) + (ϕ2 , ψ ), (ϕ1ϕ2 , ψ ) = ϕ1 (ϕ2 , ψ ) + ϕ2 (ϕ1 , ψ ),

(19.1.55''')

where we have used the formulae of differentiation of the sum and of the product of two functions. The formula of differentiation of a function of function leads to (ϕ , ψ ) = (ϕ , ψ ) =

m

∂ϕ

∑ (Pj , ψ ) ∂Pj

,

∂ϕ ∂ψ ⎛ ∂ϕ ∂ψ − ∂ ∂ P Pk ∂Q j k j =1 k =1 m

m

∑ ∑ ⎜⎝ ∂Qj

(19.1.56)

j =1

⎞ ⎟(Q j , Pk ) , ⎠

(19.1.56')

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142

where ϕ = ϕ (Q j , Pj ) , ψ = ψ (Q j , Pj ) , with Q j = Q j (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ), Pj = Pj (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ), j = 1, 2,..., m .

We can calculate ∂2 ϕ ∂ψ ∂ϕ ∂2 ψ ∂2 ϕ ∂ψ ∂ϕ ∂2 ψ ∂ ( ϕ, ψ ) = + − − ∂t ∂t ∂q j ∂p j ∂q j ∂t ∂p j ∂t ∂p j ∂q j ∂p j ∂t ∂q j ∂ϕ ∂ψ ∂ϕ ∂ψ ∂ϕ ∂ψ ∂ϕ ∂ψ = − + − , ∂q j ∂p j ∂p j ∂q j ∂q j ∂p j ∂p j ∂q j

where we assume the continuity of the derivatives of second order and where the point indicates the differentiation with respect to time; there results ∂ (ϕ , ψ ) = (ϕ , ψ ) + (ϕ, ψ ) . ∂t

(19.1.57)

Introducing the differential operator Dϕ =

∂ϕ ∂ ∂ϕ ∂ , − ∂q j ∂p j ∂p j ∂q j

(19.1.58)

where ϕ ∈ C 1 is a given function, we can express Poisson’s bracket in the form ( ϕ, ψ ) = Dϕ ψ .

(19.1.58')

Let be χ = χ (q j , p j ; t ) ; we assume that the functions ϕ , ψ and χ are of class C 2 . Taking into account the expression of Poisson’s brackets in the form (19.2.58'), we can write ( ϕ, ( ψ , χ ) ) − ( ψ , ( ϕ , χ ) ) = ( Dϕ Dψ − Dψ Dϕ ) χ = Dχ ∂ϕ ∂ψ ∂2 ∂ϕ ∂2 ϕ ∂ ∂ϕ ∂ψ ∂2 ⎡ ∂ϕ ∂2 ψ ∂ = ⎢ + − − ∂q j ∂qk ∂p j ∂pk ∂q j ∂p j ∂pk ∂qk ∂q j ∂pk ∂p j ∂qk ⎣ ∂q j ∂p j ∂qk ∂pk 2 2 2 ∂ϕ ∂ ψ ∂ ∂ϕ ∂ψ ∂ ∂ϕ ∂ ψ ∂ ∂ϕ ∂ψ ∂2 − − + + ∂p j ∂q j ∂qk ∂pk ∂p j ∂qk ∂q j ∂pk ∂p j ∂q j ∂pk ∂qk ∂p j ∂pk ∂q j ∂qk ∂ψ ∂ϕ ∂2 ∂ψ ∂2 ϕ ∂ ∂ψ ∂ ∂2 ⎛ ∂ψ ∂2 ϕ ∂ −⎜ + − − ∂q j ∂qk ∂p j ∂pk ∂q j ∂p j ∂pk ∂qk ∂q j ∂pk ∂p j ∂qk ⎝ ∂q j ∂p j ∂qk ∂pk ∂ψ ∂2 ϕ ∂ ∂ψ ∂ϕ ∂2 ∂ψ ∂2 ϕ ∂ ∂ψ ∂ϕ ∂2 ⎞ ⎤ χ. − − + + ∂p j ∂pk ∂q j ∂qk ⎟⎠ ⎦⎥ ∂p j ∂q j ∂qk ∂pk ∂p j ∂qk ∂q j ∂pk ∂p j ∂q j ∂pk ∂qk

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143

Observing that the operators of second order are of the form

∂ψ ∂ϕ ⎞ ∂2 ⎛ ∂ϕ ∂ψ ⎜ ∂q ∂q − ∂q ∂q ⎟ ∂p ∂p = 0 j j k ⎠ k ⎝ j k or of the form

−

∂ϕ ∂ψ ∂2 ∂ψ ∂ϕ ∂2 + = 0, ∂q j ∂pk ∂p j ∂qk ∂p j ∂qk ∂q j ∂pk

the commutator D will be also an operator of first order; we remain with ∂χ ⎛ ∂ϕ ∂2 ψ ∂ϕ ∂2 ψ ∂ψ ∂2 ϕ ∂ψ ∂2 ϕ ⎞ + − − ⎜ ∂qk ⎝ ∂q j ∂q j ∂pk ∂q j ∂p j ∂pk ∂q j ∂p j ∂pk ∂p j ∂q j ∂pk ⎟⎠ ∂χ ⎛ ∂ϕ ∂2 ψ ∂ϕ ∂2 ψ ∂ψ ∂2 ϕ ∂ψ ∂2 ϕ ⎞ − + − − ∂pk ⎜⎝ ∂p j ∂q j ∂qk ∂q j ∂p j ∂qk ∂q j ∂p j ∂qk ∂p j ∂q j ∂qk ⎟⎠ = Dχ ( ψ, ϕ ) = − Dχ (ϕ , ψ ) = − ( χ,(ϕ , ψ ) ) .

Finally, we get the Poisson–Jacobi identity ( ϕ , ( ψ , χ ) ) + ( ψ , ( χ , ϕ ) ) + ( χ, ( ϕ , ψ ) ) = 0 .

(19.1.59)

As a matter of fact, in the left member of this identity one observes that each term will have as factor a derivative of second order of one of the functions ϕ , ψ or χ . But all the derivatives of second order of the function χ vanish, as it has been shown; the expression in the above identity is obtained by circular permutations of the first Poisson bracket, so that the derivatives of second order of the functions ϕ and ψ will be equal to zero too, the identity being thus proved. If we take ϕ = q j or ϕ = p j and ψ = H , we obtain (corresponding to the formulae (19.1.55'))

(qj ,H ) =

∂H , ( pj , H ∂p j

)=

−

∂H , j = 1, 2,..., s . ∂q j

(19.1.60)

Hamilton’s equations (19.1.14) take the form q j = ( q j , H ) , p j = ( p j , H

)j

used in the quantic modelling of mechanics.

= 1, 2,..., s ,

(19.1.60')

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144

19.1.2.3 First Integrals of the Canonical System Let be f = f (q j , p j ; t ) a first integral of Hamilton’s canonical system, which is constant along the trajectory of the representative point P ∈ Γ 2s ; thus, we will have df ∂f ∂f = q + p + f = 0 . dt ∂q j j ∂p j j

Taking into account the equations (19.1.14), this necessary condition becomes

( f , H ) + f = 0 ,

(19.1.61)

where we use Poisson’s bracket. As well, starting from the partial differential equation of first order ∂f ∂H ∂f ∂H − + f = 0 , ∂q j ∂p j ∂p j ∂q j

we can write the associate characteristic system (the Lagrange–Charpit sequence) in the form (19.1.19), hence in the form of Hamilton’s equations. Hence, we can state Theorem 19.1.5 (Poisson). The function f = f (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ∈ C 1 is a first integral of the canonical system (19.1.14) only and only if the relation (19.1.61) (which is thus a necessary and sufficient condition) holds. If f = 0 , hence if the function f does not depend explicitly on time ( f = f (q j , p j ) ), then the condition (19.1.61) becomes (f ,H ) = 0 .

(19.1.61')

In the particular case f = H we find again the first integral of the generalized mechanical energy. Performing the partial differentiation of the relation (19.1.61) with respect to time and taking into account (19.1.57), we can write ( f , H ) + ( f , H ) + f = 0 ; if H = 0 , corresponding to a generalized conservative mechanical system; then we remain with the relation

( f , H ) +

∂f =0 ∂t

and we can state Theorem 19.1.6 (Poisson). If f = f (q j , p j ; t ) is a first integral for Hamilton’s equations of a generalized conservative mechanical system, then f , f,... are, as well, first integrals for this system. If we make χ = H in the Poisson–Jacobi identity, that one takes the form (ϕ ,( ψ, H )) + ( ψ,( H , ϕ )) + (H ,(ϕ, ψ )) = 0 ; assuming that

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145

(ϕ , H ) + ϕ = 0, ( ψ, H ) + ψ = 0

(19.1.62)

and taking into account the properties (19.1.55), (19.1.57), we can write ((ϕ , ψ ), H ) +

∂ (ϕ , ψ ) = 0 . ∂t

(19.1.62')

We state thus Theorem 19.1.7 (Jacobi–Poisson). If we suppose that the functions ϕ = ϕ (q j , p j ; t ), ψ = ψ (q j , p j ; t ) ∈ C 2 are first integrals of the canonical system (19.1.14), then Poisson’s bracket (ϕ , ψ ) is a first integral of the system. Starting from the first integrals H , ϕ , ψ , one can build up new first integrals of the form ϕ , ϕ,..., ψ , ψ,...,( H , ϕ ),( H , ψ ),(ϕ,(ϕ , ψ )),(ϕ ,(H , ψ )) etc., by means of the last two theorems. Thus, one can obtain, at a given moment, a constant or a first integral which is not independent on the previous first integrals; indeed, the canonical system admits at the most 2s distinct first integrals. If, by setting up a Poisson bracket of the first integrals ϕ and ψ , we get (ϕ , ψ ) = C , C = const

(19.1.63)

(ϕ , ψ ) = C (ϕ, ψ ), C = const ,

(19.1.63')

or

which is – practically – the same thing, then we say that the two first integrals are canonically conjugate. Moreover, if, in particular, (ϕ , ψ ) = 0 ,

then we say that the first integrals More general, let us consider α = 1, 2,...,r ; after S. Lie, the ( fα , fβ ) = fαβ ( f1 , f2 ,..., fr ), α, β

(19.1.64)

ϕ and ψ are in involution.

the functions fα = fα (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) , set of these functions forms a functional group if = 1, 2,..., r , hence if Poisson’s bracket of two

functions of this set is a function of the functions which belong to the set. These notions have been used by S. Lie in the study of Hamilton’s system of equations. 19.1.2.4 Theorem of Lie. Theorem of Liouville Let us suppose that r first integrals f j = f j (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) of the canonical system are known, so that

( f j , H ) + fj = 0, j = 1, 2,..., r . Assuming that these first integrals are in involution, we can write

(19.1.65)

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146

( f j , fk ) ≡ 0, j , k = 1, 2,..., r .

(19.1.65')

Thus, the system of functions { f j , j = 1, 2,..., r } will be a system in involution. Let us introduce the conjugate variables p0 and q 0 = t , so that ∂H / ∂p0 = 0 , ∂f j / ∂p0 = 0, j = 1, 2,..., r . Let be, in general, the functions ϕ and ψ of variables q 0 , q1 , q2 ,..., qs , p0 , p1 , p2 ,..., ps . We can set up Poisson’s bracket ∂ϕ ∂ψ ∂ϕ ∂ψ − + (ϕ , ψ ) , ∂t ∂p0 ∂p0 ∂t

(ϕ , ψ )0 =

(19.1.66)

where we put the index zero so to specify the introduction of the variables q 0 and p0 ; if ∂ϕ / ∂p0 = ∂ψ / ∂p0 = 0 , then we have (ϕ , ψ )0 = (ϕ, ψ ) . Denoting f0 = H + p0 and observing that fj + ( f j , H ) = ( f j , f0 )0 = 0 , we can express the relations (19.1.65), (19.1.65') in the unitary form ( f j , fk )0 = 0, j , k = 1, 2,..., r .

(19.1.67)

We assume that the equations f j = c j , c j = const, j = 1, 2,..., r , and H + p0 = 0 lead to pk = ϕk (q1 , q2 ,..., qs , pr +1 , pr + 2 ,..., ps , c1 , c2 ,..., c2 s ; t ), p0 = H

= − H = ϕ0 (q1 , q2 ,..., qs , pr + 1 , pr + 2 ,..., ps , c1 , c2 ,..., c2 s ; t ) ,

pk = ϕk

(19.1.68)

for k = 1, 2,..., r . If u is one of the variables q 0 , q1 , q2 ,..., qs , p0 , pr +1 , pr + 2 ,..., ps , then we can write ∂f j + ∂u

r

∂f j ∂ϕα = 0, j = 1, 2,..., r , ∂u

∑ ∂pα

α =0

or ∂f j = ∂u

r

∂f j ∂ ( pα − ϕα ) , j = 1, 2,..., r . ∂u

∑ ∂pα

α =0

We notice that these equations become identities if u is one of the variable p0 , p1 , p2 ,..., pr . We can thus write ∂f j ∂fk = ∂qi ∂pi

r

r

∂f j ∂f ∂ ( pα − ϕα ) ∂ ( pβ − ϕβ ) , i = 0,1, 2,..., s . ∂qi ∂pi

∑ ∑ ∂pα ∂pkβ

α =0 β =0

Taking the double of the skew-symmetric part with respect to the indices j and k , it results

( fj , fk )0

=

r

r

∂f j ∂f

∑ ∑ ∂pα ∂pkβ ( pα

α =0 β =0

− ϕα , pβ − ϕβ )0 , j , k = 0,1, 2,..., m .

(19.1.69)

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147

The conditions (19.1.67) read r

∂f j

∑ ∂pα z α( k )

∂f

r

∑ ∂pkβ ( pα

= 0, z α( k ) =

α =0

β =0

− ϕα , pβ − ϕβ ) ,

α, j , k = 0,1, 2,..., r .

(19.1.67')

Fixing k , one obtains a system of r + 1 linear algebraic equations with det[ ∂f j / ∂pα ] ≠ 0 , by the previous hypothesis; we get z ( k ) = 0 , hence also a system of r + 1 linear algebraic equations with the same non-zero determinant of the coefficients. Finally, ( pα − ϕα , pβ − ϕβ ) = 0 or ( p j − ϕ j , pk − ϕk )0 = 0, j , k = 0,1, 2,..., r .

(19.1.70)

In an expanded form, we can write ∂ϕ j ∂ϕk ⎞ ⎛ ∂ϕ j ∂ϕk ⎜ ∂q ∂p − ∂p ∂p ⎟ σ σ σ σ ⎠ σ =r +1 ⎝ ∂ϕ j ∂ϕ = − k , j , k = 0,1, 2,..., r . ∂qk ∂q j

( ϕj , ϕk )sσ =r +1

=

s

∑

(19.1.70')

These conditions show that the functions f1 , f2 ,..., fr are first integrals of the canonical system, being two by two in involution. Replacing p j by ϕ j , j = 1, 2,..., r , −H by ϕ0 and dt by dq 0 in the Pfaff form of Hamilton (19.1.45), we obtain the form ω =

r

∑ ϕ j dq j

+

j =0

s

∑

pσ dq σ ,

(19.1.71)

σ =r +1

the bilinear covariant of which is given by ∂ϕ j ⎞ δq ρ ⎟ dq j p ∂ ρ j =0 ⎝ k =0 ρ =r +1 ⎠ s s r s s ∂ϕ j ∂ϕ j ⎛ ⎞ ∂pσ dq k + ∑ dpρ ⎟ δq j + ∑ ∑ δpρ dq σ − ∑ ⎜ ∑ p q p ∂ ∂ ∂ ρ ρ k j =0 ⎝ k =0 σ =r +1 ρ =r +1 ρ =r +1 ⎠ ω ′ = δωd − dωδ =

− −

⎛

s

∂ϕ j

∂pσ dpρ δq σ = σ = r + 1 ρ = r + 1 ∂pρ s

s

∑ ∑

∂ϕk dq ρ − ρ = r + 1 ∂q ρ s

∑

r

∑ ⎜ ∑ ∂qk

∂ϕk ⎤ dpρ ⎥ δqk + p ∂ ρ ⎦ ρ =r +1 s

∑

δqk +

⎡⎛

r

s

∑

∂ϕ j

r

∑ ⎢ ⎜ ∑ ∂qk

k =0 ⎣ ⎝ j =0 s

⎛

r

∂ϕ j

−

∂ϕk ⎞ dq ∂q j ⎠⎟ j

∑ ⎜ ∑ ∂qσ dq j

−

∂pσ ⎞ dpρ ⎟ δq σ p ∂ ρ ⎠ ρ =r +1

σ =r +1 ⎝ j = 0 s ϕ ∂ ⎛ ∂pσ j ⎞ dq j + ∑ dq σ ⎟ δpρ . + ∑ ⎜∑ p p ∂ ∂ ρ ρ ⎠ ρ =r +1 ⎝ j = 0 σ =r +1 s

r

s

∑

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148

The associate differential system, obtained by equating to zero the brackets which multiply the differential variations δq σ and δpσ , is written in the form r ∂ϕ ∂ϕ j j dq j , dpσ = − ∑ dq j , σ = r + 1, r + 2,..., s . p q ∂ ∂ σ σ j =0 j =0 r

dq σ = − ∑

(19.1.72)

The equations obtained by annulling the square brackets which multiply the differential variations δqk are identically verified if one takes into account the conditions (19.1.70') and the equations (19.1.72). The functions ϕ j , j = 0,1, 2,..., r , being thus determined, the system (19.1.72) becomes a system of 2(s − r ) equations with 2(s − r ) unknown functions. Corresponding to Morera’s method of integration, we introduce an arbitrary parameter λ , so that q j = q j (λ ), j = 0,1, 2,..., r . Denoting r

K (qr + 1 , qr + 2 ,..., qs , pr + 1 , pr + 2 ,..., ps , c1 , c2 ,..., c2 s ; λ ) = − ∑ ϕ j j =0

dq j , dλ

(19.1.73)

the system of equation (19.1.71) reads q σ′ =

dq σ ∂K dpσ ∂K = , pσ′ = = − , σ = r + 1, r + 2,..., s . dλ ∂pσ dλ ∂q σ

(19.1.72')

As a matter of fact, starting from the Pfaff form ω =

s

∑

σ =r +1

pσ dq σ − Kdλ ,

(19.1.71')

one obtains the same result. We can thus state Theorem 19.1.8 (S. Lie). If the system of 2s canonical equations (19.1.14) admits r , r ≤ s , first integrals in involution, resolvable with respect to r , of the generalized momenta, then the integration of this system is reduced to the integration of a system of 2(s − r ) canonical equations in s − r pairs of conjugate variables and to quadratures. The generalized co-ordinates q1 , q2 ,..., qr are then determined, using the relation ϕ k = −

∂H , k = 1, 2,..., r . ∂qk

(19.1.74)

In the particular case r = s , we notice that ∂ϕ j / ∂pk = 0, j , k = 0,1, 2,..., s , and from (19.1.70') it results ∂ϕ j ∂ϕk = , j , k = 0,1, 2,..., s . ∂qk ∂q j

(19.1.75)

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149

Thus, we can state that the existence (necessary and sufficient conditions) of a function S = S (q 0 , q1 , q2 ,..., qs ; a1 , a2 ,..., as ) is ensured, so that ∂S ∂S = ϕj , = S = ϕ0 = − H ∂q j ∂q 0

p j = ϕj

, j = 1, 2,..., s .

(19.1.75')

Hence, dS =

s

∂S

∑ ∂q j

j =0

dq j =

s

∑ ϕ j dq j ,

j =0

(19.1.75'')

and we can state Theorem 19.1.9 (Liouville). If the system of 2s canonical equations (19.1.14) admits s first integrals in involution, resolvable with respect to the generalized momenta, then the integration of this system is reduced to quadratures. We notice that the function S satisfies the partial differential equation of first order ∂S ∂S ∂S ⎞ , ,..., S + H ⎛⎜ q 0 , q1 ,..., qs , = 0, ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.1.76)

called the Hamilton–Jacobi equation, and the condition

⎡ ∂2S ⎤ ⎡ ∂ϕ j ⎤ det ⎢ = det ⎢ ⎥ ⎥ ≠ 0, ⎣ ∂ak ⎦ ⎣ ∂q j ∂ak ⎦ assuming that the function ϕ j

(19.1.76')

are resolvable with respect to the constant

ak , j , k = 1, 2,..., s ; in this case, S is the complete integral of the Hamilton–Jacobi

equation (see Sect. 19.2.1.1 too). If H = 0, fj = 0, j = 1,2,..., s , hence if the Hamiltonian and the first integrals do not contain explicitly the time, then the functions ϕ j have the same property, so that S (q1 , q2 ,..., qs , a1 , a2 ,..., as ; t ) = S (q1 , q2 ,..., qs , a1 , a2 ,..., as ) + T (t ; a1 , a2 ,..., as ) .

From (19.1.75'), it results T = ϕ0 ; but (19.1.75) shows that ϕ0 = − h , h = const , obtaining S = − ht + S . Let be a generalized conservative mechanical system ( H = 0 ), the canonical system (19.1.14) of which admits s − 1 independent first integrals f j , j = 1, 2,..., s − 1 , in involution, which do not contain explicitly the time ( fj = 0 ). The s th first integral is H and the other first integrals must verify the conditions ( f j , H ) = 0, j = 1, 2,..., s − 1 ; hence, H is also in involution with these first integrals and we can apply Liouville’s theorem.

MECHANICAL SYSTEMS, CLASSICAL MODELS

150

If, in case of the motion of a heavy rigid solid about a fixed point, beside the first integral of the moment of momentum and the first integral of the kinetic energy we can set up a third integral, in involution with the first two ones, then the integration of the equations of motion is reduced to quadratures, according to Liouville’s theorem. 19.1.2.5 Other Formulations in the Spaces Λs and Γ 2s We have built up the Lagrangian formalism in the space Λs to study the motion of the representative point P , while in the space Γ 2s we have used the Hamiltonian formalism to determine the motion of the corresponding representative point; but these formalisms are not the only ones which can be used. For instance, starting from the relation (19.2.15'), we notice that one can write d δ ⎡ ( pj q j ⎣⎢ dt

)−L

⎤ = q δp + q δp j j j j ⎦⎥

too. Denoting ϕ =

d (p q dt j j

)−L

(19.1.77)

,

we obtain qj =

∂ϕ ∂ϕ , q = , j = 1, 2,..., s , ∂p j j ∂p j

(19.1.77')

hence, relations of the form (19.1.13); one obtain thus, easily, a system of differential equations of second order

d ⎛ ∂ϕ ⎞ ∂ϕ − = 0, j = 1,2,..., s , dt ⎜⎝ ∂p j ⎟⎠ ∂p j

(19.1.77'')

analogue to Lagrange’s equations (the place of the generalized co-ordinates is taken by the generalized momenta). But, to can set up the function ϕ , we must solve the system of equations (19.1.13) with respect to q j and q j , function of pk and pk , which is difficult – in general – this system being non-linear in the generalized co-ordinates. We can start also from the relation δ ( q j p j − L

) = q j δp j

− p j δq j .

We denote

ψ = q j p j − L

(19.1.78)

and obtain the system of equations qj =

∂ψj ∂ψj . , pj = − ∂p j ∂q j

(19.1.78')

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In this case too, the difficulty consists in expressing the generalized co-ordinates as functions of the derivatives with respect to time of the generalized momenta, using the second subsystem (19.1.13).

19.1.3 Applications We consider, firstly, the motion of a single particle; we pass then to a Hamiltonian treatment of a system of two or more particles, as well as to the case of a rigid solid or of a system of rigid solids. 19.1.3.1 Motion of a Particle In general, the generalized momentum is defined by the relation (18.2.80). In case of a generalized quasi-potential intervenes the vector part in the form (18.2.81); to simplify the presentation, we assume, in what follows, that the generalized forces derive from a simple quasi-potential, the generalized momenta being thus given by the relations (18.2.80'). We use the results obtained in Sect. 18.3.2.1. In case of a single particle P (x1 , x 2 , x 3 ) , the representative point in Γ 6 is expressed by P (q1 , q2 , q 3 , p1 , p2 , p3 ), q j = x j , j = 1, 2, 3 , so that the kinetic energy is given by

T = (m / 2)(x12 + x22 + x 32 ) , while the generalized momenta (equal to the usual components of the momentum) are p j = mx j , j = 1, 2, 3 .

(19.1.79)

Hamilton’s function becomes (we use the formula of definition (19.1.11) with (18.2.3.4)) 1 p p − U ( x1 , x 2 , x 3 ; t ) , 2m j j

H =

(19.1.79')

while the canonical equations (19.1.14) read x j =

1 ∂U p , p = , j = 1, 2, 3 . m j j ∂x j

(19.1.79'')

In cylindrical co-ordinates ( q1 = r , q2 = θ , q 3 = z ), we obtain

pr = mr , pθ = mr 2 θ , pz = mz .

(19.1.80)

Hamilton’s function being H =

(

)

1 1 pr2 + 2 pθ2 + pz2 − U (r , θ , z ; t ) , 2m r

There result the canonical equations

(19.1.80')

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152

1 1 1 pr , θ = pθ , z = pz , m m mr 2 ∂U ∂U 1 2 ∂U , p θ = , p z = . pr = p + 3 θ ∂ ∂ θ ∂z r mr r =

(19.1.80'')

If ∂U / ∂θ = ∂U / ∂z = 0 , then θ and z are cyclic co-ordinates. As well, in spherical co-ordinates ( q1 = r , q2 = θ , q 3 = ϕ ) we can write

pr = mr , pθ = mr 2 θ, pϕ = mr 2 sin2 θϕ ,

(19.1.81)

so that H =

(

)

1 1 1 pr2 + 2 pθ2 + 2 pϕ2 − U (r , θ , ϕ ; t ) . 2m r r sin2 θ

(19.1.81')

We get Hamilton’s equations 1 1 1 pr , θ = p , ϕ = pϕ , 2 θ 2 m mr mr sin2 θ ∂U 1 1 , pr = pθ2 + pϕ2 + ∂r sin2 θ mr 3 ∂U ∂U cot θ , pϕ = . p θ = pϕ2 + 2 2 ∂ϕ ∂ϕ mr sin θ

r =

(

)

(19.1.81'')

If ∂U / ∂ϕ = 0 , then ϕ is a cyclic co-ordinate. We notice that, in case of a simple potential of the form (18.3.46), we find a system of canonical equations equivalent to Lagrange’s equations (18.3.47). In case of a central force of Newtonian attraction we have U = k / r , k > 0 , so that ∂U / ∂r = − k / r 2 , ∂U / ∂θ = ∂U / ∂ϕ = 0 .

As a matter of fact, in case of a central force the trajectory is a plane curve; if the intensity of this force is function only on the distance to the fixed pole, Hamilton’s function will be given by (in polar co-ordinates, in the plane of the motion, the fixed pole being the origin of the co-ordinate axes) H =

(

)

m 2 1 1 ( r + r 2 θ2 ) − U (r ) = 2m pr2 + 2 pθ2 − U (r ) . 2 r

(19.1.82)

There result Hamilton’s equations in Γ 4 r =

1 1 1 2 ∂U pr , θ = p , pr = pθ + , p θ = 0 , 2 θ m ∂r mr mr 3

(19.1.82')

θ being a cyclic co-ordinate; we obtain pθ = mC , C = const , and then r 2 θ = c ,

hence the first integral of areas. Eliminating the generalized momentum pr and taking into account the first integral previously obtain, we can write

Hamiltonian Mechanics

153 r =

C2 1 ∂U + . 3 m ∂r r

(19.1.82'')

We find thus the equation of motion of the particle along the vector radius (see Sect. 8.1.1.1, formula (8.1.3)); the first integral H = h corresponds to the first integral (8.1.6'). In elliptical co-ordinates (see Sect. 18.3.2.1), the formula (18.3.41) leads to p1 =

m m m A q , p = A2q2 , p3 = A3q3 ; 4 1 1 2 4 4

Hamilton’s function will be given by H =

2 ⎛ p12 p22 p32 ⎞ + + −U , m ⎜⎝ A1 A2 A3 ⎟⎠

(19.1.83)

with U = U (q1 , q2 , q 3 ; t ) , allowing thus to write the canonical equations in the form q1 = p j =

4 p1 4 p2 4 p3 , q = , q = , m A1 2 m A2 3 m A3

p32 ∂A3 ⎞ 2 ⎛ p12 ∂A1 p22 ∂A2 + + ⎟ , j = 1, 2, 3 . m ⎜⎝ A12 ∂q j A22 ∂q j A32 ∂q j ⎠

(19.1.83')

The generalized momenta p j , given by (19.1.79), are the components of the momentum H = p ; the components of the moment of momentum KO = r × p will be KOi = ∈ijk x j pk = m ∈ijk x j xk , j = 1, 2, 3 .

(19.1.84)

We can thus form Poisson’s brackets ∂p j ∂pk ∂p j ∂pk , − ∂x i ∂pi ∂pi ∂x i ∂ (∈jlm xl pm ) ∂ (∈knr x n pr ) ∂ (∈jlm xl pr ) ∂ (∈knr x n pm ) (KOj , KOk ) = − ∂x i ∂pi ∂pi ∂x i ( p j , pk ) =

=∈jlm ∈kni x n pm − ∈jli ∈kir xl pr = ( ∈rji ∈kli + ∈jli ∈kri ) xl pr = ( δrk δ jl − δ jr δlk ) xl pr =∈ijk ∈ilr xl pr , ∂p j ∂ (∈klm xl pm ) ∂p j ∂ (∈klm xl pm ) − ∂x i ∂pi ∂pi ∂x i = −δij δil ∈klm pm = − ∈kjm pm ,

( p j , KOk ) =

MECHANICAL SYSTEMS, CLASSICAL MODELS

154 so that

( p j , pk ) = 0,(KOi , KOk ) =∈ijk KOi , ( p j , KOk ) =∈ijk pi , j , k = 1, 2, 3 .

(19.1.85)

If some of the above quantities are conserved (are first integrals), then also the other quantities have the same property (the Jacobi–Poisson theorem leads to new first integrals). For instance, if p1 , KO 2 and KO 3 are first integrals, then p2 , p3 and KO 1 are first integrals too; other first integrals cannot be obtained, the cycle being closed. If a particle P is acted upon by an elastic central force of intensity in direct proportion with the distance to the pole O , then the potential U is given by U = −(k / 2)(x12 + x 22 + x 32 ), k = const ; the second group of canonical equations (19.1.79'') reads p j = − kx j , j = 1, 2, 3 .

(19.1.79''')

As it can be easily verified, (19.1.84) are three first integrals of the canonical equations. The first equation of the first subsystem of equations (19.2.79'') and the first equation (19.1.79''') (for j = 1 ) give the first integral f1 = mx12 +

1 2 p . k 1

(19.1.86)

As well, Poisson’s bracket (KO 3 , f1 ) leads to the first integral f2 =

1 p p + mx1x 2 . k 1 2

(19.1.86')

One observes easily that one cannot obtain new first integrals starting from the first integrals (19.1.84), (19.1.86), and (19.1.86') and using the Jacobi–Poisson theorem; indeed, if one could obtain such a first integral, then the functions q j , p j , j = 1, 2, 3 , could be expressed by means of the six integration constants, the generalized momenta being thus constant (which is not possible, because the motion would be annulled). A first integral which (unlike the previous ones) contains explicitly the time is f3 = m ωx1 cos ωt − p1 sin ωt , ω =

k . m

A partial differentiation with respect to time leads to a new first integral f4 = f3 = − m ω 2 x1 sin ωt − p1 ω cos ωt ,

which is not distinct from the previous ones, because f 2 f32 + ⎛⎜ 4 ⎞⎟ = kf1 . ⎝ω⎠

(19.1.86'')

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155

As a matter of fact, the maximum number of first integrals in case of a free particle is six. 19.1.3.2 Problem of Two Particles Be means of the results in Sect. 18.3.2.3 and of the formulae (18.3.51), (18.3.51') for the kinetic energy T and for the simple potential U , respectively, we notice that we can use a formula of the form (19.1.81'); there results Hamilton’s function H =

(

)

fm1m2 1 1 1 pr2 + 2 pθ2 + 2 pϕ2 − . r 2m r r sin2 θ

(19.1.87)

Hence, the canonical equations read 1 1 1 pr , θ = pθ , ϕ = pϕ , m mr 2 mr 2 sin θ fm1m2 1 cot θ pθ2 + pϕ2 − pϕ2 , pϕ = 0 . , p θ = 2 r mr 2 sin θ sin θ r =

pr =

1 mr 3

(

)

(19.1.87')

We see that ϕ is a cyclic co-ordinate, so that pϕ = const is a first integral. We remain only with four canonical equations, determining thus the functions r = r (t ), θ = θ (t ), pr = pr (t ), pθ = pθ (t ) ; we obtain then the generalized angular velocity ϕ and, by a quadrature, we can calculate ϕ = ϕ (t ) too. 19.1.3.3 Motion of an Electrized Particle in an Electromagnetic Field Starting from the results obtained in Sect. 18.3.2.5, we find the generalized momenta pi = mvi + qAi , i = 1, 2, 3 ; the relation of definition (19.1.11) leads to H =

1 1 1 pi ( pi − qAi ) − ( pi − qAi ) ( pi − qAi ) − qAi ( pi − qAi ) + qA0 , m m 2m

where we took account (18.3.59). Finally, we can write Hamilton’s function in the form H =

1 1 ( p − qAi ) ( pi − qAi ) + qA0 = ( p p − 2qpi Ai + q 2 Ai Ai 2m i 2m i i

) + qA0

.

(19.1.88) One obtains thus the canonical equations νi = xi =

1 q ( p − qAi ) , pi = ( p j − qAj ) Aj ,i + qA0 , i = 1, 2, 3 . m i m

(19.1.88')

19.1.3.4 Motion of a Discrete System of Particles In case of a free discrete system of particles Pi of masses mi , i = 1, 2,..., n , the space of configurations is just E 3 n (we have s = 3n ). The kinetic energy is given by (we use the co-ordinates x (j i ) , j = 1, 2, 3, i = 1, 2,..., n )

MECHANICAL SYSTEMS, CLASSICAL MODELS

156 T =

(

1 n (i ) mi ⎡⎢ x1 ⎣ 2 i∑ =1

) + ( x2( i ) ) + ( x3( i ) ) 2

2

2

⎤. ⎥⎦

(19.1.89)

Assuming, for the sake of simplicity, that the given forces derive from a simple quasi-potential

(i ) U = U (x j ; t ) , we

obtain

Hamilton’s

function H = T − U .

Noting that p(j i ) = mi x (j i ) , j = 1, 2, 3, i = 1, 2,..., n , it results

H =

(

1 n 1 ⎡ (i ) ⎢ p1 2 i∑ = 1 mi ⎣

) + ( p2( i ) ) + ( p3( i ) ) 2

2

2

⎤. ⎥⎦

(19.1.89')

The canonical equations are written in the form (i )

x j

=

1 (i ) (i ) ∂U p , p j = ( i ) , j = 1, 2, 3, i = 1, 2,..., n . mi j ∂x j

(19.1.90)

Introducing the components of the momentum of the mechanical system Hl =

n

∑ pl( i ) , l

i =1

= 1, 2, 3 ,

(19.1.91)

and the components of the moment of momentum of the same system KOl =

n

∑ ∈jkl

i =1

x j pk , l = 1, 2, 3 , (i ) (i )

(19.1.91')

where we use the summation convention of the dummy indices for j , k = 1, 2, 3 , we can calculate Poisson’s brackets as in Sect. 19.1.3.1; we obtain thus the relations

( H j , H k ) = 0, ( KOj , KOk ) =∈jkl KOl , ( H j , KOk ) =∈jkl Hl , j , k = 1, 2, 3 ,

(19.1.92)

analogues to the relations (19.1.85). Obviously, we are led to analogous conclusions in what concerns eventual first integrals too. In particular, one can obtain all six first integrals previously considered in the case in which the mechanical system is isolated, hence it is subjected only to the action of internal forces Fij , i ≠ j , i , j = 1, 2,..., n ; the torsor of these forces is, obviously, equal to zero R = MO =

n

n

∑ ∑ Fij

i =1 j =1 n

n

∑ ∑ rij

i = 1 j =1

= 0,

(19.1.93)

× Fij = 0 .

(19.1.93')

Hamiltonian Mechanics

157

Mayer considered, in 1878, internal forces with a more general character that the Newtonian ones (definite in Sects. 1.1.1.11; see. 2.2.2.8 too), hence forces which verify only the global conditions (19.1.93), (19.1.93'), hence the conditions of equilibrium of a non-deformable mechanical system (as the mechanical system would be a rigid one at a given moment). In this context, we can state (we denote rij = rj − ri , i , j = 1, 2,..., n ) Theorem 19.1.10 (Mayer). The most general quasi-conservative internal forces which derive from a generalized quasi-potential U and verify the condition (19.1.93) are those for which

U = U ( r12 , r13 ,..., rn −1,n , r12 , r13 ,..., rn −1,n ; t ) .

(19.1.94)

Theorem 19.1.10' (Mayer). The most general quasi-conservative internal forces which derive from a generalized quasi-potential U and verify the condition (19.1.93) are those for which U = U ( r1 , r2 ,..., rn , r1 , r2 ,..., rn , r12 , r13 ,..., rn −1,n ; t ) .

(19.1.94')

From both theorems taken together, we get Theorem 19.1.11 (Mayer). The most general quasi-conservative internal forces which derive from a generalized quasi-potential U and the torsor of which vanishes (verifies the conditions (19.1.93), (19.1.93')) are those for which the quasi-potential is of the form (19.1.94). The motion of the mass centre is rectilinear and uniform and the theorem of areas can be applied in projection on the three planes of co-ordinates. We can also state Theorem 19.1.12 (Mayer). The necessary and sufficient condition of existence of the first integral of the mechanical energy, hence of the existence of a function of coordinates, velocities and time the total derivative with respect to time of which be n

n

∑ ∑ Fij

i = 1 j =1

⋅ vi

is U = 0 ; the internal forces must derive from a generalized potential which does not contain explicitly the time. The first integral is of the form T −U +

n

∂U

i =1

j

∑ v(j i ) ∂v( i )

= h , h = const .

(19.1.95)

If the potential U verifies the conditions of the Theorem 19.1.10 (is of the form (19.1.94) with U = 0 ), then the first integral reads T −U +

n

n

∂U

∑ ∑ rij ∂rij

i =1 j =1

= h , h = const .

(19.1.95')

MECHANICAL SYSTEMS, CLASSICAL MODELS

158

19.1.3.5 Motion of the Rigid Solid In case of a rigid solid S , free or subjected to constraints, we can start from the considerations made in Sect. 18.3.2.6. For instance, in case of the motion of a heavy homogeneous rigid solid of rotation, of mass M , which slides without friction on a fixed horizontal plane, we start from the kinetic energy (18.3.62) and from the potential energy (18.3.62'); the mechanical system being scleronomic and observing that p1 = M ρ1′ , p2 = M ρ 2′ , pϕ = I 3 (ϕ + ψ cos θ ), pψ = J ψ sin θ + I 3 (ϕ + ψ cos θ ) cos θ , pθ = ⎡⎣ Mf ′2 ( θ ) + J ⎤⎦ θ ,

we can express Hamilton’s function in the form

( pψ − pϕ cos θ ) pϕ2 ⎤ pθ2 1 ⎡ p12 + p22 + + + + Mgf ( θ ) . ⎢ 2 I 3 ⎥⎦ 2⎣ M Mf ′ ( θ ) + J J sin2 θ (19.1.96) 2

H = T ′ −U =

Hamilton’s equations read ρ1′ =

pϕ ( pψ − pϕ cos θ ) cos θ p1 p , ρ2′ = 2 , ϕ = , − M M I3 J sin2 θ ( pψ − pϕ cos θ )  pθ ψ = ,θ = , 2 2 J sin θ Mf ′ ( θ ) + J p1 = 0, p2 = 0, pϕ = 0, p ψ = 0, p θ =

+

Mf ′( θ ) f ′′( θ ) ′2

2

⎡⎣ Mf ( θ ) + J ⎤⎦

(19.1.96')

p02

1 ( pψ − pϕ cos θ )( pψ cos θ − pϕ ) − Mgf ′( θ ) . J sin 3 θ

We notice that ξ1′ , ξ2′ , ϕ and ψ are cyclic co-ordinates, four of the generalized momenta being thus constant; as well, H = h is the fifth first integral of the canonical system. In case of the plane motion of a rigid straight bar, we consider the formulation of the problem in Sect. 18.3.2.7. The kinetic energy is given by (18.3.64) and the potential of the forces is expressed by (18.3.64'), so that we can define the generalized momenta in the form p1 = M ρ1′ , p2 = M ρ2′ , pθ = I θ , obtaining Hamilton’s function H = T ′ −U =

1⎡ 1 1 p12 + p22 ) + pθ2 + k ( M ρ2′2 + I sin2 θ ) ⎤ . ( ⎢ ⎥⎦ 2 ⎣M I

We can write the canonical equation in the form

(19.1.97)

Hamiltonian Mechanics

159

1 1 1 p , ρ ′ = p , θ = pθ , M 1 2 M 2 I p1 = 0, p 2 = − kM ρ2′ , p θ = − kI sin θ cos θ . ρ1′ =

(19.1.97')

We notice that ρ1′ is a cyclic co-ordinate, the corresponding generalized momentum being constant; the constraints are scleronomic, so that Hamilton’s function will be a first integral of this system of equations too.

19.1.3.6 Double Pendulum The problem of the double pendulum has been treated in Sect. 18.3.2.8. In the frame of Lagrangian mechanics. The generalized co-ordinates being θ1 and θ2 , the corresponding momenta are

p1 = (I 1 + M 2 I 12 )θ1 + M 2 H 2 θ2 cos( θ2 − θ1 ), p2 = (I 2 + M 2 I 22 )θ2 + M 2 H 2 θ1 cos( θ2 − θ1 ) .

(19.1.98)

Starting from these results, we can express Hamilton’s function H = T ′ − U as a function on the canonical co-ordinates, obtaining then Hamilton’s equations.

19.1.3.7 Sympathetic Pendulums Let us take again the case of the sympathetic pendulums, considered in Sect. 18.3.2.9 in the frame of Lagrangian mechanics, Let be two identical physical pendulums, each one of mass M , of mass centres C 1 and C 2 , respectively, so that O1C 1 = O2C 2 = l , where O1 and O2 , are the poles through which pass the respective axes of suspension; if Q1 and Q2 are on O1C 1 and O2C 2 , respectively, with O1Q1 = O2Q2 = a , then we consider that the two pendulum are linked by the elastic spring Q1Q2 of elastic constant k (see Fig. 18.13 too). Choosing as generalized co-ordinates the angles θ1 and θ2 made by O1C 1 and O2C 2 , respectively, with the vertical line, the formulae (18.3.69) and (18.3.69') lead to T =

1 1 I ( θ12 + θ22 ) , U = 2 MgI − ( MgI + ka 2 2 2

)( θ12

+ θ22 ) + ka 2 θ1 θ2 .

(19.1.99)

Having to do with a scleronomic system, the generalized momenta will be p1 = I θ1 , p2 = I θ2 , and Hamilton’s function reads H =

1 1 ( p2 + p22 ) − 2MgI + 2 ( MgI + ka 2 2I 1

There result Hamilton’s canonical equations

)( θ12

+ θ22 ) − ka 2 θ1 θ2 .

(19.1.99')

MECHANICAL SYSTEMS, CLASSICAL MODELS

160

1 1 θ1 = p1 , θ2 = p2 , I I p1 = − MgI θ1 + ka 2 ( θ2 − θ1 ), p2 = − MgI θ2 + ka 2 ( θ1 − θ2 ) .

(19.1.99'')

Eliminating the generalized momenta, we obtain Lagrange’s equations 2 θ1 + ( ω2 + ω02 ) θ1 − ( ω2 + ω02 ) θ2 = 0, 2 θ2 − ( ω2 + ω02 ) θ1 + ( ω2 + ω02 ) θ2 = 0.

(19.1.100)

with the notation (18.3.71); we find thus again the solutions (18.3.72).

19.2 The Hamilton–Jacobi method After some results with a general character, including the Hamilton-Jacobi theorem, one considers the problem of the systems of equations with separate variables, putting in evidence Stäckel’s studies; some applications for the motion of a particle or of a system of two particles are then given.

19.2.1 General Results To can obtain first integrals of the system of 2s canonical equations, one establishes the Hamilton–Jacobi theorem, which allows to build up the solution of that system, in the hypothesis of the determination of the complete integral of the corresponding partial differential equation; then some properties concerning the integration constants are put in evidence and one deals with the particular case of the scleronomic constraints, as well as with the cases which admit cyclic co-ordinates. One makes also other considerations on the Hamilton–Jacobi method (Hamilton, W.R., 1890; Jacobi, C.G.J., 1882; Nordheim, L. and Fues, E., 1927; Pars, L., 1965).

19.2.1.1 The Hamilton-Jacobi Partial Differential Equation Let be a function S = S ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as , as + 1 ) ,

(19.2.1)

which depends on s generalized co-ordinates, on time and on s + 1 arbitrary constants (corresponding to the s + 1 independent variables); we calculate the partial derivatives ∂S S ≡ = p0 ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as , as + 1 ) , ∂t

(19.2.1')

∂S = p j ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as , as + 1 ) , j = 1, 2,..., s . ∂q j

(19.2.1'')

In this case, s + 1 of the s + 2 relations (19.2.1), (19.2.1'), and (19.2.1'') allow to determine the constants a j , j = 1, 2,..., s + 1 , as functions of qk , t , S , S and

Hamiltonian Mechanics

161

∂S ∂qk , k = 1,2,..., s ; eliminating these constants between the mentioned relations, we

obtain one or more relations of the form ∂S ∂S ∂S ⎞ f ⎛⎜ q1 , q2 ,..., qs ; t ; S , S , , ,..., = 0. ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.2.2)

If there results only one relation, this one is a partial differential equation of first order, non-linear in general, with respect to the function S ; this function, of the form (19.2.1), represents the complete integral of the partial differential equation (19.2.2). Let be, e.g., a partial differential equation of the form ∂S ∂S ∂S ⎞ f ⎛⎜ , ,..., = 0. ∂qs ⎟⎠ ⎝ ∂q1 ∂q2

(19.2.3)

The corresponding complete integral will be S = a1q1 + a2q2 + ... + as −1qs −1 + Cqs + as + 1 ,

(19.2.3')

f ( a1 , a2 ,..., as −1 ,C ) = 0 .

(19.2.3'')

with

In case of Clairaut's equation q1

∂S ∂S ∂S ∂S ∂S ∂S ⎞ + q2 + ... + qs + f ⎛⎜ , ,..., − S = 0, ∂q1 ∂q2 ∂qs ∂qs ⎟⎠ ⎝ ∂q1 ∂q2

(19.2.4)

we get the complete integral S = a1q1 + a2q2 + ... + as qs + f ( a1 , a2 ,..., as ) .

(19.2.4')

In case of the equation (19.2.3) (which does not contain the function S ), the arbitrary constant as + 1 appears in an additive form in the complete integral (19.2.3'), while, in case of the equation (19.2.4) (which contains the function S ), no one additive constant intervenes in the complete integral (19.2.4'). Let be thus a partial differential equation ∂S ∂S ∂S ⎞ f ⎛⎜ q1 , q2 ,..., qs ; t ; , ,..., = 0, ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.2.5)

which does not contain the function S ; the complete integral is thus of the form S = S ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as ) + as + 1 ,

(19.2.5')

one of the arbitrary constants being additive. We notice that in the relations (19.2.1'), (19.2.1'') intervene now only s arbitrary constants, called essential constants, which

MECHANICAL SYSTEMS, CLASSICAL MODELS

162

can be determined univocally from the relations (19.2.1''), assuming that the Hessian with respect to the generalized co-ordinates and to the arbitrary constants is non-zero ( S is a function of class C 2 ) ⎡ ∂2 S ⎤ det ⎢ ⎥ ≠ 0. ⎣ ∂q j ∂ak ⎦

(19.2.6)

As a matter of fact, the integral (19.2.5') is complete if this necessary and sufficient condition is fulfilled. Replacing in (19.2.1'), we see that the corresponding partial differential equation can be written in the form ∂S ∂S ∂S ⎞ S + ϕ ⎛⎜ q1 , q2 ,..., qs ; t ; , ,..., = 0; ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.2.7)

the complete integral which contains s essential constants and verifies the condition (19.2.6) is given by S = S ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as ) .

(19.2.7')

If we denote q 0 = t and if we take into account the notations (19.2.1'), (19.2.1''), then we can write the partial differential equation (19.2.5) in the form f ( q 0 , q1 , q2 ,..., qs ; p0 , p1 , p2 ,..., ps ) = 0 ;

(19.2.8)

the ordinary differential equations of the characteristics of the partial differential equation (19.2.8) read dq k ∂f dpk ∂f = , =− , q = qk ( τ ) , pk = pk ( τ ) , k = 0,1,2,..., s . dτ ∂pk dτ ∂qk k

(19.2.8')

We can say that the partial differential equation of first order (19.2.8) is associated to the system of ordinary differential equations of first order (19.2.8') and inversely. Let us take Hamilton’s function H = H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) as function ϕ in the equation (19.2.7); we obtain thus the partial differential equation ∂S ∂S ∂S ⎞ S + H ⎛⎜ q1 , q2 ,..., qs , , ,..., ;t = 0 , ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

where the generalized momenta

pj

(19.2.9)

are replaced by the partial derivative

∂S ∂q j , j = 1,2,..., s ; the complete integral of this equation, called the Hamilton–

Jacobi equation (or the equation in S ), is of the form (19.2.7'), with the condition (19.2.6). If we make τ = t in the system of ordinary differential equations (19.2.8') and use the notations which have been introduced, we get

Hamiltonian Mechanics

dq k ∂H dpk ∂H = = − , , k = 1, 2,..., s , ∂pk dt ∂qk dt dq 0 dp ∂H = 1, 0 = − . ∂t dt dt

163

(19.2.9')

The equations in the first row are just Hamilton’s canonical equations; the first equation in the second row is an identity and the second one leads to the relation dH / dt = H , which takes place along the trajectory of the representative point in the space Γ 2 s . Hence, the Hamilton–Jacobi equation is associated to Hamilton’s equations, the study of this equation being thus equivalent to the study of the canonical equations; but sometimes, the integration of the Hamilton–Jacobi equation can be realized on a simpler way or can give interesting suggestions for the integration of the canonical equations, being useful to solve the mechanical problem thus put.

19.2.1.2 The Hamilton-Jacobi theorem In connection with the Hamilton–Jacobi equation we can state Theorem 19.2.1 (Hamilton–Jacobi). If S ∈ C 2 is the complete integral (19.2.7') (which verifies the condition (19.2.6)) of the Hamilton–Jacobi equation (19.2.9), then the sequences ∂S ∂S = bj , bj = const, = p j , j = 1, 2,..., s , ∂a j ∂q j

(19.2.10)

determine the solutions q j = q j (t ) and p j = p j (t ) , j = 1,2,..., s , of the system of canonical equations (19.1.14). Starting from the first sequence of relations (19.2.10) (which is, in fact, a sequence of first integrals), taking into account the condition (19.2.6), we can determine the generalized co-ordinates q j = q j ( t ;a1 , a2 ,..., as , b1 , b2 ,..., bs ) , j = 1, 2,..., s , by means of the theorem of implicit functions; replacing then in the second sequence of the relation (19.2.10), we obtain the generalized momenta p j = p j ( t ; a1 , a2 ,..., as , b1 , b2 ,..., bs ) . The integration constants a j and bj of same index j , j = 1, 2,..., s , are called conjugate constants. Because the relations (19.2.10) must take place along the trajectory of the representative point P in the space Γ 2 s at any moment t , their total derivatives with respect to time must vanish; we will have ∂2 S ∂2S ∂2 S ∂2 S qk + = 0, qk + − p j = 0 . ∂qk ∂a j ∂t ∂a j ∂qk ∂q j ∂t ∂q j

(19.2.11)

We notice that, by introducing the complete integral (19.2.7') in the equation (19.2.9), the latter one must be identically verified in generalized co-ordinates and essential constants; differentiating this equation, successively, with respect to a j and to q j , we obtain

MECHANICAL SYSTEMS, CLASSICAL MODELS

164

∂2 S ∂H ∂2 S ∂2 S ∂H ∂2 S + = 0, + = 0. ∂a j ∂t ∂pk ∂a j ∂qk ∂q j ∂t ∂pk ∂q j ∂qk

Subtracting these relations one of the other and observing that S ∈ C 2 , we can write ∂2S ∂qk ∂a j ∂2S ∂q j ∂qk

∂H ⎞ ⎛ ⎜ qk − ∂p ⎟ = 0, ⎝ k ⎠

∂H ⎞ ⎛ ∂H ⎞ ⎛ ⎜ qk − ∂p ⎟ − ⎜ p j − ∂q ⎟ = 0, j = 1,2,..., s . ⎝ ⎠ j ⎠ k ⎝

(19.2.11')

Hence, these relations are equivalent to the relations (19.2.11). If the canonical equations take place, then – obviously – the relations (19.2.11') are also verified. Let us suppose now that the first sequence of the relations (19.2.11') holds; these relations can be considered as a homogeneous system of linear algebraic equations with a non-zero determinant (the Hessian (19.2.6)), which admits only trivial solutions, obtaining thus the first subsystem of the canonical equations. If we take into account this result, the second sequence of relations leads to the second subsystem of the canonical equations. The theorem enounced above is thus proved. Thus, the solution of the system of canonical equations (19.1.14) is reduced to the finding of a complete integral of the Hamilton–Jacobi equations (19.2.9). The two problems are equivalent, the Hamilton–Jacobi theorem being useful if we succeed to obtain such a complete integral. The respective method of calculation is known as the Hamilton-Jacobi method. We mention that the Hamilton–Jacobi theorem can be put in connection with the Theorem 19.1.19 of Liouville (see Sect. 19.1.2.4). We notice that a complete integral of the Hamilton–Jacobi equation is not the general integral of this equation; indeed a complete integral contains, in general, a smaller number of solutions than the general integral. But starting from a complete integral, we can find again the partial differential equation to which it corresponds, on the way shown in Sect. 19.2.1.1 for the function (19.2.5') with as + 1 = 0 . We mention that the equation S = S ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as ) = 0

(19.2.12)

represents the equation of a movable or deformable hypersurface in the configuration space Λs ; this equation is, in fact, a wave surface and pays an important rôle in the undulatory mechanics. The trajectory of the representative point in this space is orthogonal, at each point, to the mentioned surface.

19.2.1.3 Properties of the Integration Constants As it is known (see Theorem 19.1.4), Lagrange’s brackets formed by to integration constants of the system of canonical equations are constants along the trajectory of the representative point in the space Γ 2 s (see Sect. 19.1.2.1). We try now to specify the integration constants which are introduced by the Hamilton-Jacobi method. Let thus be

Hamiltonian Mechanics

165

[ a j , ak ] =

∂qi ∂pi ∂q ∂pi . − i ∂a j ∂ak ∂ak ∂a j

Differentiating the sequence (19.2.10) with respect to ak , we obtain ∂2 S ∂qi ∂2S + = 0, ∂qi ∂a j ∂ak ∂ak ∂a j

(19.2.13)

∂p j ∂2 S ∂ql ∂2S + = , j , k = 1, 2,..., s . ∂ql ∂a j ∂ak ∂ak ∂q j ∂ak

Inverting j by k we obtain other analogous relations. Replacing the partial derivative of the generalized momenta in Lagrange’s bracket considered above it results

[ a j , ak ] =

∂qi ∂ql ⎞ ∂2 S ⎛ ∂qi ∂ql ∂2 S ∂qi ∂2 S ∂ql . − + − ∂ql ∂qi ⎜⎝ ∂a j ∂ak ∂ak ∂a j ⎟⎠ ∂ak ∂qi ∂a j ∂a j ∂qi ∂ak

The first sum contains products of symmetric and antisymmetric factors with respect to the indices i and l , being thus equal to zero; taking into account the first relations (19.2.13) and observing that S ∈ C 2 , one sees that the other sums vanish too, so that

[ a j , ak ] = 0, j , k

= 1, 2,..., s .

(19.2.14)

Analogically,

[ bj ,bk ] =

∂qi ∂pi ∂q ∂p − i i . ∂bj ∂bk ∂bk ∂bj

Differentiating the sequences (19.2.10) with respect to bk , we get

∂p j ∂2 S ∂qi ∂2 S ∂ql , j , k = 1,2,..., s . = δ jk , = ∂qi ∂a j ∂bk ∂ql ∂q j ∂bk ∂bk

(19.2.13')

Inverting j by k , we obtain other relations, analogue to the above ones. Replacing the partial derivatives of the generalized momenta in this Lagrange bracket, we get

[ bj ,bk ] =

∂q ∂ql ⎞ ∂2 S ⎛ ∂qi ∂ql − i . ∂ql ∂qi ⎜⎝ ∂bj ∂bk ∂bk ∂a j ⎟⎠

This sum contains products of terms symmetric and antisymmetric with respect to the indices i and l , thus vanishing; it results

[ bj , bk ] = 0, j , k As well,

= 1, 2,..., s .

(19.2.14')

MECHANICAL SYSTEMS, CLASSICAL MODELS

166

[ b j , ak ] =

∂qi ∂pi ∂q ∂pi . − i ∂bj ∂ak ∂ak ∂bj

Replacing the partial derivative of the generalized momenta in the above Lagrange bracket, one obtains

[ b j , ak ] =

∂q ∂ql ⎞ ∂2 S ⎛ ∂qi ∂ql ∂2 S ∂qi − i + . ⎜ ⎟ ∂ql ∂qi ⎝ ∂bj ∂ak ∂ak ∂bj ⎠ ∂qi ∂ak ∂bj

Considerations analogue to those above show that the first sum vanishes; concerning the second sum, we take into account the first relation (19.2.12'). We obtain

[ bj , kk ] = δ jk , j , k

= 1, 2,..., s .

(19.2.14'')

Thus, all the Lagrange brackets which can be set up have been put in evidence. 19.2.1.4 Generalized Conservative Mechanical Systems. Cyclic Co-ordinates In case of a generalized conservative mechanical system, hence for which H = 0 , one obtains a remarkable form of the Hamilton–Jacobi equation; indeed, in this case we can write the first integral H = h , so that the equation in S reads S + h = 0 , wherefrom the integral S = − ht + S ( q1 , q2 ,..., qs ) , one of the integration constants being the energy constant h . One obtains thus the reduced Hamilton–Jacobi equation ∂S ∂S ∂S ⎞ ⎛ , ,..., =h, H ⎜ q1 , q2 ,..., qs , ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.2.15)

while the complete integral is given by

S = − ht + S ( q1 , q2 ,..., qs ; a1 , a2 ,..., as −1 , h ) .

(19.2.15')

The condition (19.2.6) becomes ⎡ ∂2S ⎤ det ⎢ ⎥ ≠ 0, as = h . ⎣ ∂q j ∂ak ⎦

(19.2.15'')

The solutions of the system of canonical equations will be given by the sequences of relations ∂S ∂S ∂S = bj , j = 1, 2,..., s − 1, = t + bs , = pk , k = 1, 2,..., s . ∂a j ∂h ∂qk

(19.2.15''')

The first s − 1 relations give the trajectory of the representative point in the configuration space Λs (hence, q j = q j (qs ), j = 1, 2,..., s − 1 ), the subsequent

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167

relation specifying the motion on this trajectory (after replacing the previous results, we get qs = qs (t ) and, returning, we obtain q j = q j (t ), j = 1,2,..., s − 1 ) . The last relations show that the function S is a simple potential for the generalized momenta, which are – in this case – conservative. As well, we see that the function S plays the rôle of a simple quasi-potential for the generalized momenta, which are thus quasi-conservative. Taking into account that the complete integral is not unique, it results that this quasi-potential is not uniquely determinate too (excepting an arbitrary constant, which appears in the usual case). In case of a generalized conservative mechanical system, the wave surface (19.2.12) is given by S = ht .

(19.2.16)

At a given moment t = t0 = const , one obtains S = const ; we have dS =

∂S dq j = p j dq j . ∂q j

If the constraints are scleronomic, the mechanical system being conservative (we have U = 0 too), then we can write 2T =

∂Tj ∂L q = q = − p j q j , ∂q j j ∂q j j

so that S =

P

∫P 2Tdt ,

(19.2.17)

0

the representative point P0 corresponding to the initial moment and the representative point P being an arbitrary point in the configuration space Λs . In this case, the function S is called action; it verifies the reduced Hamilton–Jacobi equation (19.2.15). In case of r cyclic co-ordinates q α , hence for which ∂H ∂q α = 0, α = 1, 2,..., r , the corresponding generalized momenta are constant ( pα = aα , α = 1, 2,..., r ); it results ∂S ∂q α = a α , so that the complete integral will be S =

r

∑ aαqα

α =1

+ S 0 ( qr + 1 , qr + 2 ,..., qs ; t ; a1 , a2 ,..., as ) ,

(19.2.18)

the conditions (19.2.6) being reduced to s

⎡ ∂2S 0 ⎤ ≠ 0. det ⎢ ⎥ ⎣ ∂q j ∂ak ⎦ j ,k =r + 1

(19.2.18')

MECHANICAL SYSTEMS, CLASSICAL MODELS

168 The Hamilton–Jacobi equation reads

∂S 0 ∂S 0 ∂S S0 + H ⎛⎜ qr +1 , qr + 2 ,..., qs , a1 , a2 ,..., ar , , ,..., 0 ; t ⎞⎟ = 0 , q q ∂ ∂ ∂qs ⎠ ⎝ r +1 r +2

(19.2.18'')

corresponding to a canonical system in 2(s − r ) variables (as it is known – as a matter of fact – from the study of Hamilton’s system of equations). The sequences of relations (19.2.10) become ∂S 0 ∂S = −q α + bα , α = 1, 2,..., r , 0 = bj , j = r + 1, r + 2,..., s , ∂aα ∂a j pα = a α , α = 1,2,..., r ,

∂S 0 = p j , j = r + 1, r + 2,..., s . ∂q j

(19.2.18''')

In case of a generalized conservative mechanical system, which has r cyclic co-ordinates, the function S will be of the form S = − ht +

r

∑ a αq α

α =1

+ S 0 ( qr + 1 , qr + 2 ,..., qs , a1 , a2 ,..., as −1 , h ) ,

(19.2.19)

and the reduced Hamilton–Jacobi equation becomes ∂S 0 ∂S 0 ∂S H ⎛⎜ qr + 1 , qr + 2 ,..., qs , a1 , a2 ,..., ar , , ,..., 0 ⎞⎟ = h . ∂qr + 1 ∂qr + 2 ∂qs ⎠ ⎝

(19.2.19')

There result ∂S 0 = −q α + bα , α = 1, 2,....r , ∂aα ∂S 0 = bj , j = r + 1, r + 2,..., s − 1, ∂a j ∂S 0 = t + bs , pα = aα , α = 1, 2,....r , ∂h ∂S 0 = p j , j = r + 1, r + 2,..., s . ∂q j

(19.2.19'')

19.2.1.5 Case of a Discrete System of Particles Let be a discrete system S of n particles, of masses mi , of velocities vi and of momenta Hi = m vi , i = 1, 2,..., n . Assuming that this system is not subjected to constrains, it results that s = 3n and that we can choose as space of configurations the representative space E 3n . Using the corresponding notations (see Sect. 18.1.1.2), we can introduce the generalized momenta Pk = M kVk = M k X k , k = 1, 2..., 3n .

(19.2.20)

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169

Assuming that the generalized forces Qk derives from a simple quasi-potential U = U ( X1 , X 2 ,..., X 3 n ; t ) , the equation of motion (18.1.38) reads Pk +

∂P dX j ∂U = Qk = , k = 1, 2,..., 3n ; ∂ d t Xk j =1 3n

∑ ∂Xkj

we observe that this equation can be written also in the form Pk +

⎤ ∂ ⎡ 3n 1 Pj2 − U ⎥ + ∑ ⎢ ∂Xk ⎣ j =1 2 M j ⎦

3n

1

⎛ ∂P

∑ M j Pj ⎜⎝ ∂Xkj j =1

−

∂Pj ⎞ = 0. ∂Xk ⎟⎠

If the generalized momenta derive from a quasi-potential S = S ( X1 , X 2 ,..., X 3 n ; t ) , then we find ∂ ⎡ ⎢S + ∂Xk ⎣⎢

2 ⎤ 1 ⎛ ∂S ⎞ ∑ 2M j ⎜⎝ ∂X j ⎟⎠ − U ⎥ = 0, k = 1, 2,..., 3n . j =1 ⎦⎥ 3n

It results that the function between the square brackets depends only on the time t , being of the form f (t ) . If, instead of the quasi-potential S , we use the quasi-potential

S − ∫ f (t )dt , then we notice that the 3n generalized momenta are not influenced;

hence, we can write the equation of motion 2

1 3 n 1 ⎛ ∂S ⎞ S + ∑ −U = 0 . 2 j =1 M j ⎜⎝ ∂X j ⎟⎠

(19.2.21)

As a matter of fact, observing that Hamilton’s function H = T − U is given by a relation of the form (19.1.87'), we can also write ∂S ∂S ∂S ⎞ , ,..., S + H ⎛⎜ X1 , X 2 ,..., X 3 n ; t ; = 0, ∂X1 ∂X 2 ∂Xn ⎟⎠ ⎝

(19.2.21')

obtaining thus the corresponding Hamilton–Jacobi equation. Let us build up the Pfaff form of Hamilton ω =

3n

∑ Pj dX j j =1

− Hd t .

This form is a total differential if and only if the relations ∂Pj ∂Pk ∂H ,− = = Pj , j .k = 1, 2,..., 3n , ∂Xk ∂X j ∂X j

hold. Taking into account the expression of Hamilton’s function, we can write

(19.2.22)

MECHANICAL SYSTEMS, CLASSICAL MODELS

170 Pj +

3n

∂P

1

∑ M k Pk ∂Xkj

= Pj +

k =1

3n

1

∂Pj

∑ M k Pk ∂Xk

k =1

=

∂U . ∂X j

But Pk M k = dXk dt , so that dPj dV j ∂U . = Mj = dt dt ∂X j

(19.2.23)

Hence, the existence of the quasi-potential S is connected to the existence of the quasi-potential U ; but this one is only a necessary condition for the existence of the function S , because some conditions concerning the initial state must be satisfied too. Thus, at the initial moment t = t0 we must have X j (t0 ) = X j0 , Vj (t0 ) = Vj0 , wherefrom Pj (t0 ) = Pj0 , U (t0 ) = U 0 and H (t0 ) = H 0 ; because, in a problem of Cauchy type, the equation of motion (19.2.23) must hold also at the initial moment, it results that the Pfaff form of Hamilton ω0 =

3n

∑ Pj0 dX j0 j =1

− H 0 dt 0

(19.2.22')

must be a total differential at the initial moment too. If this condition is also fulfilled, then one can use the field description by means of the Hamilton–Jacobi equation. Observing that dS =

∂S

3n

∑ ∂X j dX j j =1

+ Sdt ,

we can write 3n

∑ Pj dX j

dS =

j =1

− Hd t

(19.2.24)

too, where we take into account the Hamilton–Jacobi equation. Then ⎛ 1 3n 1 2 Pj + U dS = ⎜ ∑ ⎝ 2 j =1 M j

⎞ ⎛ 1 3n 2 ⎟ dt = ⎜ 2 ∑ M jVj + U ⎠ ⎝ j =1

⎞ ⎟ dt = ( T + U ) dt . ⎠

Introducing the kinetic potential (18.2.34), we obtain dS =

L dt .

(19.2.25)

We can write S1 − S 0 =

t1

∫t

0

L dt

(19.2.25')

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171

for a finite variation, integrating along the trajectory of the representative point in the space E 3 n . 19.2.1.6 Hamilton’s Principal Function Taking the final moment t1 as an arbitrary moment t , we can define the function S in the form S =

t

∫t

L (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; τ )dτ .

(19.2.26)

0

As a matter of fact, this quantity can be called Lagrangian action (shortly, action), corresponding – in a non-conservative case – to the quantity (19.2.17). If we take into account the equations of motion at the moment t , written in the form q j = q j (t ; q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ), p j = p j (t ; q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ), j = 1,2,..., s ,

then we get S = S (t ; q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ) . Hamilton proposed to eliminate the generalized momenta at the initial moment; there results the function S = S (t ; q1 , q2 ,..., qs , p1 , p2 ,..., ps ) ,

(19.2.27)

called the Hamilton principal function. We can calculate dS ∂S = q + S = L . dt ∂q j j

But ∂S ∂S ∂S ∂S q j − L = H (q1 , q2 ,..., qs , , ,..., ;t ) , ∂q j ∂q1 ∂q2 ∂qs

with the notation p j = ∂S ∂q j , j = 1, 2,..., s . Hence, Hamilton’s principal function must verify the equation in S . On the other hand, dS =

∂S dq j + Sdt = p j dq j − Hdt . ∂q j

(19.2.28)

If we take into account all the arguments in (19.2.27), then we can write dS = p j dq j − Hdt − p j0 dq j0 .

(19.2.28')

Putting the condition that this is a total differential, we find again the Hamilton-Jacobi equation, as well as the sequence ∂S ∂q j = p j , j = 1, 2,..., s ; the essential constants

172

MECHANICAL SYSTEMS, CLASSICAL MODELS

being q10 , q20 ,..., qs0 , we can write ∂S ∂q j0 = p j0 , j = 1, 2,..., s . The first subsystem of relations determines the motion of the representative point in the space Λs ; associating the second subsystem of relations, we obtain the motion of the representative point in the space Γ 2 s . Thus, Hamilton showed, in 1834, the possibility to obtain the final equation of motion at the moment t (the canonical co-ordinates as functions of time), by means of a complete integral of the equation in S . But the complete integral used by Hamilton is not arbitrary; indeed, the essential constants which have been used are the initial values of the canonical co-ordinates. Thus, one obtains a vicious circle: to obtain the final equations of motion, Hamilton’s principal function is necessary, while to set up this function, one must know the final equations of motion. The essential merit of Jacobi consists in showing how one can avoid this vicious circle; indeed, Jacobi showed, in 1837, that the final equations of motion can be obtained by means of an arbitrary complete integral of the Hamilton–Jacobi equation (19.2.9).

19.2.2 Systems of Equations with Separate Variables The setting up of a complete integral of the Hamilton–Jacobi partial differential equation can be realized, in many cases, in the form of a sum of s functions of only one generalized co-ordinate; in this case, we say that we have to do with a system of s equations with separate variables, which has interesting properties. We mention that the separability is a property connected both to the system and to the chosen independent co-ordinates. One puts thus the problem to determine the conditions of separability, as well as the form of the system which verifies such conditions; we consider thus natural systems with s degrees of freedom, the motion of which is described by means of s Lagrangian co-ordinates. There are a few cases (e. g., the problem of small oscillations) in which the separation leads to s independent equations, each one containing only one generalized co-ordinate. In general, one cannot isolate a particular generalized co-ordinate, studying its variation as in the case of a system with only one degree of freedom; but, in a certain sense, we can consider the variation of a single co-ordinate, neglecting the behaviour of the other co-ordinates. One cannot give the general form of the systems of equations with separate (or with separable) variables. Therefore, we will limit ourselves to the catastatic (hence holonomic and scleronomic) systems for which the kinetic energy is a positive definite quadratic form in the generalized velocities. In what follows, we will present firstly the case of mechanical systems with two degrees of freedom. After some particular cases of separability, we will study – further – the general case of mechanical systems with s degrees of freedom, in the frame of Stäckel’s theory. 19.2.2.1 Mechanical Systems with Two Degrees of Freedom We notice first of all that any scleronomic mechanical system with a single degree of freedom is separable, the complete integral of the corresponding Hamilton–Jacobi equation being of the form

Hamiltonian Mechanics

173 S (q ; t ; h ) = − ht + S (q ; h ) .

(19.2.29)

Passing to the scleronomic mechanical systems with two degrees of freedom, we will limit ourselves to the case of orthogonal systems, so that in the expression of the kinetic energy does not intervene the product of two generalized velocities; hence, T =

1 ( g q2 + g22q22 ) . 2 11 1

(19.2.30)

In this case, Hamilton’s function reads H =

1 11 2 ( g p1 + g 22 p22 ) − U , 2

(19.2.31)

where g 11 = 1/ g11 > 0, g 22 = 1/ g22 > 0 and U are functions of class C 1 in the generalized co-ordinates q1 and q2 ; we have supposed that the generalized forces derive from a simple potential. There results the Hamilton–Jacobi equation 2 2 1 ⎡ 11 ⎛ ∂S ⎞ 22 ⎛ ∂S ⎞ ⎤ + g g ⎜ ∂q ⎟ ⎜ ∂q ⎟ ⎥ − U = h , 2 ⎢⎣ ⎝ 1⎠ ⎝ 2 ⎠ ⎦

(19.2.32)

where the complete integral is of the form S = − ht + S (q1 , q2 ; a1 , h ) ; to realize a separation of the form S (q1 , q2 ,; a1 , h ) = S1 (q1 ; a1 , h ) + S2 (q2 ; a1 , h )

(19.2.33)

it is necessary to have g 11ϕ1 + g 22 ϕ2 = U + h , 1 ⎛ ∂Sk ⎞ , k = 1,2, a2 = h . 2 ⎜⎝ ∂qk ⎟⎠ 2

ϕk = ϕk (qk ; a1 , a2 ) =

(19.2.34)

Differentiating with respect to a1 and a2 , respectively, we obtain g 11ϕ1,1 + g 22 ϕ2,1 = 0, g 11ϕ1,2 + g 22 ϕ2,2 = 1 .

The coefficients g 11 and g 22 being positive, it results that the derivatives ϕ1,1 , ϕ2,1 and ϕ1,2 , ϕ2,2 , respectively, cannot be identically zero (for any generalized co-ordinates); as

well

⎡ ∂2 S ⎤ ⎡ ∂ (ϕ1 , ϕ2 ) ⎤ ∂S1 ∂S2 ϕ1,1ϕ2,2 − ϕ1,2 ϕ2,1 = det ⎢ det = ⎢ ⎥ ≠ 0. ⎣ ∂ (a1 , a2 ) ⎦⎥ ∂q1 ∂q2 ⎣ ∂q j ∂ak ⎦

MECHANICAL SYSTEMS, CLASSICAL MODELS

174 We can write

g 11 =

P1 P2 , g 22 = , Q1 + Q2 Q1 + Q2

the functions Pk = Pk (qk ), Qk = Qk (qk ), k = 1, 2 , being given by P1 =

ϕ1,2 ϕ2,2 1 1 . , P2 = − , Q1 = , Q2 = − ϕ1,1 ϕ2,1 ϕ1,1 ϕ2,1

(19.2.34')

The potential U must be given by U =

U1 + U 2 , Q1 + Q2

(19.2.35)

where the functions U k = U k (qk ), k = 1, 2 are U1 =

ϕ1 − hϕ1,2 ϕ2 − hϕ2,2 , U2 = − . ϕ1,1 ϕ2,1

(19.2.34'')

Resulting the kinetic energy T =

1 ( Q + Q2 2 1

) ⎛⎜

1 2 1 2 q1 + q P2 2 ⎝ P1

⎞, ⎟ ⎠

(19.2.35')

it is necessary that Hamilton’s function be of the form H =

1 P1 p12 + P2 p22 U 1 + U 2 − . Q1 + Q2 2 Q1 + Q2

(19.2.36)

The corresponding reduced Hamilton–Jacobi equation (19.2.15) is given by 1 Q1 + Q2

⎡ 1 ⎛ ∂S1 ⎞2 1 ⎛ ∂S2 ⎞2 ⎤ ⎢ P1 ⎜ ⎟ + 2 P2 ⎜ ∂q ⎟ − (U 1 + U 2 ) ⎥ = h . 2 q ∂ ⎝ 1 ⎠ ⎝ 2 ⎠ ⎣⎢ ⎦⎥

Choosing the function S in the form (19.2.33), this equation is decomposed in two equations of only one variable 2 2 1 ⎛ ∂S1 ⎞ 1 ⎛ ∂S2 ⎞ = + − = hQ2 + U 2 + a1 , P1 ⎜ hQ U a , P 1 1 1 2 ⎝ ∂q1 ⎟⎠ 2 2 ⎜⎝ ∂q2 ⎟⎠

where a1 is an arbitrary constant. Hence, the complete integral is given by

Hamiltonian Mechanics

175

S (q1 , q 2 ; t ;a1 , h ) = − ht +

2 ( hQ1 + U 1 − a1 ) dq1 + P1

∫

∫

2 ( hQ2 + U 2 + a1 ) dq 2 , P2 (19.2.37)

the condition of separability imposed above being sufficient too. Applying the Hamilton–Jacobi theorem, we obtain −

∂S = ∂a1

dS = dh p1 =

∫

d q1 − R1

∫

dq 2 = −b1 , R2

Q1 dq1 Q2 dq2 ∫ R1 + ∫ R2 = t − t0 ,

∂S 1 = ∂q1 P1

R1 , p2 =

∂S 1 = ∂q2 P2

R2 ,

(19.2.38)

(19.2.38')

where R1 (q1 ) = 2 P1 ( hQ1 + U 1 − a1 ) , R2 (q2 ) = 2 P2 ( hQ2 + U 2 + a1 ) .

(19.2.38'')

The equations (19.2.38) specify the motion of the representative point in the space Λ2 , being sufficient to solve the Lagrangian problem; by differentiation, these equations read d q1 dq 2 dt . = = Q1 + Q2 R1 R2

(19.2.39)

dt = dτ , Q1 + Q2

(19.2.39')

We denote

where τ can be considered as a conventional time in the motion of the representative point on the corresponding trajectory in the space Λ2 ; we notice that Q1 + Q2 > 0 , so that τ increases together with t . We obtain the equations 2

2

⎛ dq1 ⎞ = R , ⎛ dq2 ⎞ = R , ⎜ ⎟ ⎟ 1 ⎜ 2 ⎝ dτ ⎠ ⎝ dτ ⎠

(19.2.39'')

which can be considered each one as in the case of a mechanical system with only one degree of freedom (see Sect. 7.1.1.1); but the relation between t and τ depends on both generalized co-ordinates. However, the motion can be studied independently, in a certain measure. Obviously, the two motions would be independent if the sum Q1 + Q2 would be constant (certainly, we should, simultaneously, have Q1 = const and

MECHANICAL SYSTEMS, CLASSICAL MODELS

176

Q2 = const ); the motion would be only approximately independent if the sum Q1 + Q2 would vary not very much.

Let us assume that the sum Q1 + Q2 is bounded for all the values of the generalized co-ordinates (at least for all the values which are taken by these co-ordinates during the considered motion), so that 0 < Q1 + Q2 < N , N finite for any t . In this case τ → ∞ together with t → ∞ and we can make an appreciation of the general nature of the motion with respect to each generalized co-ordinate. If the generalized co-ordinate q1 , e.g., is – at the beginning – between two consecutive simple real zeros

of R1 (q1 ) ( α1 ≤ q1 ≤ β1 ) , then takes place an oscillatory motion in time, between these values, for any moment t ; the motion is periodical in τ , but is not periodical in t , yet it will be called motion of libration. Th sign of the radical will be taken + or – as q1 is increasing or decreasing, respectively. If q1 → γ1 , where γ1 is a double root of R1 , for t (and τ ) tending to infinite, then one can speak about an asymptotic (limit)

motion. If the sum Q1 + Q2 is not bounded, e.g. if Q1 + Q2 → ∞ together with t (which can take place if q1 and q2 tend to infinite with t ), then it can happen that the integral ∞

∫t

0

dt Q1 + Q2

be convergent; in this case lim τ = τ 0 for t → ∞ . If, in this case, we have α1 ≤ q1 ≤ β1 at the initial moment, then we have no more a motion of libration and lim q1 = l1 , α1 ≤ l ≤ β1 , for t → ∞ . As well, if q1 → γ1 , γ1 double root, then the

motion is pseudoasymptotic. Obviously, one can make analogous considerations for the generalized co-ordinate q2 . In the case in which librations can take place for each of the two generalized co-ordinates, the motion of the mechanical system is periodical if ν

v∫ dq2 R2

=

v∫ dq1 , ν ∈ _ . R1

(19.2.40)

Thus, if ν = n2 n1 , where n1 , n2 ∈ ` , have no one common factor, then, after n1 librations corresponding to the generalized co-ordinate q1 and after n2 librations corresponding to the generalized co-ordinate q2 , the mechanical system returns to the previous situation (the representative point P ∈ Λ2 returns to the same position), as well as in what concerns the position and the velocities. Obviously, the period will be T = n1 v∫

Q1 dq1 Q dq + n2 v∫ 2 2 . R1 R2

(19.2.40')

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177

If ν is an irrational number, then the motion is not periodical, while the trajectory of the representative point P ∈ Λ2 is contained in a rectangular interval ( αk ≤ qk ≤ βk , k = 1, 2 ), being an open curve; in this case the motion is quasi-periodical. We mention that we can choose the generalized co-ordinates corresponding to the mechanical system with two degrees of freedom in various modes; e.g., in case of the motion of a particle subjected to the action of a central force of Newtonian attraction one can use, in the plane of the motion, polar or parabolic or confocal co-ordinates etc. Correspondingly, one can obtain a separation of the variables in several modes for the same problem.

19.2.2.2 Mechanical Systems with s Degrees of Freedom. Particular Cases of Hamiltonians In case of a generalized conservative mechanical system with s degrees of freedom we search a complete integral (19.2.15'), trying to obtain a decomposition of the function S (q1 , q2 ,..., qs ; a1 , a2 ,..., as −1 , h ) in the form s

∑ S j ( q j ;a1 , a2 ,..., as −1 , h ) .

(19.2.41)

∂S j ∂S ∂S = = , j = 1, 2,..., s . ∂q j ∂q j ∂q j

(19.2.41')

S =

j =1

In this case pj =

The reduced Hamilton–Jacobi equation becomes ∂S ∂S ∂Ss ⎞ H ⎛⎜ q1 , q2 ,..., qs , 1 , 2 ,..., =h. ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.2.41'')

p j = f j ( q j ; a1 , a2 ,..., as −1 , h ) , j = 1, 2,..., s ,

(19.2.42)

Observing that

we get S = − ht +

s

∑ ∫ fj ( q j ;a1 , a2 ,..., as −1 , h ) dq j j =1

(19.2.43)

and the condition (19.2.6) reads ⎡ ∂f j ⎤ det ⎢ ⎥ ≠ 0. ⎣ ∂ak ⎦

(19.2.43')

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178

The generalized co-ordinates will be given by the system of algebraic equations ∂f j

s

∑ ∫ ∂ak j =1

dq j = bk , k = 1, 2,..., s − 1,

(19.2.42')

∂f j ∑ ∫ ∂ak dq j = t + bks , j =1 s

so that the generalized momenta will result in the form (19.2.42). The above integrals can be considered between the limits c j and q j , where c j are fixed constants, independent on the integration constants ak and bk . The most simple case in frame of the above considerations is that in which the variables are separated in the structure of Hamilton’s function, so that H = H ( g1 , g2 ,..., gs ) , g j = g j (q j , p j ),

∂g j ≠ 0, j = 1, 2,..., s . ∂p j

(19.2.44)

Imposing the conditions g j (q j , p j ) = a j , j = 1, 2,..., s , we can apply the theorem of implicit functions; it results p j = f j (q j ; a j ), j = 1, 2,..., s .

(19.2.45)

The reduced Hamilton–Jacobi equation leads to h = H (a1 , a2 ,..., as ) , so that the complete integral will be of the form S = − H ( a1 , a2 ,..., as ) t +

s

∑ fj (q j ;a j )dq j

.

j =1

We notice that ∂q j ∂p j

(

wherefrom ∂f j ∂a j = ∂g j ∂p j

)p = f

p j = fj

−1 j

∂f j = 1, ∂a j

≠ 0 ; but ∂f j ∂ak = 0, j ≠ k , so that j

⎡ ∂f j ⎤ det ⎢ ⎥ = ⎣ ∂ak ⎦

s

∂f j

∏ ∂a j

≠ 0,

j =1

the condition (19.2.43') being fulfilled. Hence, the sequences of algebraic relations which give the canonical co-ordinates will be (19.2.45) and

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179

−1 ∂f j ∂H ⎛ ∂g j ⎞ q t , j = 1, 2,..., s , d = dq j = b j + ∫ ∂a j j ∫ ⎜⎝ ∂p j ⎟⎠ ∂ aj p j = fj ( q j ;a j )

(19.2.45')

the problem being thus reduced to quadratures. If we choose Hamilton’s function in the form H = gs , g j = g j ( g j − 1 ; q j , p j ) ,

∂g j ≠ 0, j = 1, 2,..., s ; g 0 = 0 , ∂p j

(19.2.46)

then we can impose the conditions g j (a j −1 ; q j , p j ) = a j , j = 1, 2,..., s , a 0 = 0, as = h ; the theorem of implicit functions allows to write p j = f j (q j , a j −1 , a j ), j = 1, 2,..., s , a 0 = 0, as = h .

(19.2.47)

The reduced Hamilton–Jacobi equation is identically verified because we have taken gs = as = h ; the complete integral will be of the form S = − ht +

s

∑ ∫ f j ( g j ; a j − 1 , a j ) dq j . j =1

Taking into account considerations analogue to those in the preceding case, it results

(

∂f j ∂a j = ∂g j ∂p j

)p = f −1 j

≠ 0 ; we notice that ∂f j ∂ak = 0, j < k , too. As above, the j

condition (19.2.43') is verified, the value of the determinant being reduced to the product of its elements (all non-zero) of the principal diagonal. We notice that ∂g j + 1 ∂p j + 1

p j +1 = f j +1

∂f j + 1 ∂g j + 1 + = 0. ∂a j ∂a j

The trajectory of the representative point P ∈ Λs will be specified by the equations −1 ∂f j + 1 ∂f j ⎛ ∂g j ⎞ q + q = d d dq j ∫ ∂a j j ∫ ∂a j j +1 ∫ ⎜⎝ ∂p j ⎟⎠ p j = f j ( q j ;a j −1 ,a j )

−∫

∂g j + 1 ∂a j

−1

⎛ ∂g j +1 ⎞ dq j + 1 = bj , j = 1, 2,..., s − 1 , ⎜ ∂p ⎟ ⎝ j + 1 ⎠ p j +1 = fj +1 ( g j +1 ;a j ,a j +1 )

(19.2.47')

and the motion on this trajectory by the equation −1

⎛ ∂qs ⎞ ∫ ⎜⎝ ∂ps ⎟⎠ps = fs (qs ;a ,h ) dqs = t + bs . s −1

(19.2.47'')

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180

The generalized momenta are given by the relations (19.2.47), the problem being thus completely solved. Another interesting case is that in which the Hamiltonian is of the form H =

g1 + g2 + ... + gs g1 + g2 + ... + gs

, g j = g j ( q j , pj ),

(19.2.48)

∂g j ∂g j g j = g j ( q j , pj ) , −h ≠ 0, j = 1, 2,..., s . ∂p j ∂p j

The reduced Hamilton–Jacobi equation (19.2.15) reads ⎡

s

⎛

∂S ⎞ ⎟ − hg j ⎠

∑ ⎢⎣ g j ⎝⎜ q j , ∂q j j =1

∂S ⎞ ⎤ ⎛ ⎜ q j , ∂q ⎟ ⎥ = 0 . j ⎠⎦ ⎝

We can denote g j ( q j , pj

) − hg j ( q j , p j ) = a j , j

= 1, 2,..., s ,

s

∑aj j =1

= 0,

wherefrom, applying the theorem of implicit functions, we get p j = f j ( q j ; a j , h ) , j = 1, 2,..., s .

(19.2.49)

The complete integral of the Hamilton–Jacobi equation will be S = − ht +

s

∑ ∫ f j ( q j ; a j , h ) dq j . j =1

We notice that (we take into account the relation as = −(a1 + a2 + ... + as −1 ) , the s th essential constant being h ) ∂gk ⎞ ∂fk ∂f ∂g ∂gs ⎞ ∂fs ⎛ ∂gk = 1, ⎛⎜ 0 − h = −1, k = 0, k ≠ l , ⎟ ⎜ ∂p − h ∂p ⎟ ∂ps ⎠ ps = fs ∂ak ∂al ⎝ ∂ps ⎝ k k ⎠ pk = fk ∂ak ∂g j ⎞ ∂f j ⎛ ∂g j − g j = 0, j = 1, 2,..., s , k , l = 1, 2,..., s − 1 ; ⎜ ∂p − h ∂p ⎟ j ⎠ p = f ∂h ⎝ j j j

it results s ∂g j ⎞−1 ⎛ ∂g j ⎡ ∂f j ⎤ det ⎢ Δs , ⎥ = − ∏ ⎜ ∂p − h ∂p ⎟ ⎣ ∂ak ⎦ j j ⎠p = f j =1 ⎝ j j

where

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181

Δs =

1

0

0 ... 0

0

g1

0

1

0 ... 0

0

g2

0

0

1 ... 0

0

g3

... ... ... ... ... ...

...

.

0

0

0 ... 0

1

gs − 1

1

1

1 ... 1

1

− gs

Expanding after the first column (the second minor after the first line) we obtain Δs = 1 ⋅ Δs −1 + ( −1)s −1 ( −1)s − 2 g1 = Δs −1 − g1 ; analogously, we get Δs −1 = Δs − 2 − g2 a.s.o. Finally, we can write Δs = −(g1 + g2 + ... + gs ) ≠ 0 , the denominator of

the Hamiltonian cannot vanish, so that

det ⎡⎣ ∂f j ∂ak ⎤⎦ ≠ 0 . The generalized

co-ordinates will be given by the relations ∂fk ∂gk ⎞ −1 ∂fs ⎛ ∂gk q + q = − h d d s ⎜ k ∫ ∂ak ∫ ∂ak ∫ ⎝ ∂pk ∂pk ⎟⎠p = f (q ;a ,h ) dqk k k k k ∂gs ∂gs ⎞ −1 dqs = bk , k = 1, 2,..., s − 1, − ∫ ⎛⎜ −h ⎟ ∂ps ⎠ ps = fs (qs ;as ,h ) ⎝ ∂ps

(19.2.49')

the motion on the trajectory of the representative point P ∈ Λs being specified by s

∂f j

∑ ∫ ∂h j =1

dq j =

⎛ ∂g j

s

∑ ∫ g j ⎜⎝ ∂p j j =1

−h

∂g j ⎞ −1 dq j = t + bs , ∂p j ⎟⎠ p j = fj ( q j ;a j ,h )

(19.2.49'')

where as = −(a1 + a2 + ... + as −1 ) ; the relations (19.2.47) allow then the calculation of the generalized momenta. As a particular case, we can state Theorem 19.2.2 (Liouville). If the kinetic energy and the potential energy of the mechanical system are of the form s

T =

s

s

qk2

∑U k

1 Q j ∑ , V = −U = − k s=1 2∑ j =1 k = 1 Pk

∑Qj

,

(19.2.50)

j =1

respectively, where Q j = O j (q j ) , Pk = Pk (qk ) , U k = U k (qk ), j , k = 1, 2,..., s are functions of only one variable, then the equations of motion can be obtained by quadratures. Indeed, in this case Hamilton’s function is of the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

182 s

H =

s

s

∑ Pk pk2 ∑U k 1 2

k =1 s

−

∑Qj j =1

k =1 s

=

∑ Qj

∑ ⎡⎢⎣ 2 Pk (qk )pk2 1

k =1

− U k (qk ) ⎤ ⎥⎦

s

∑ Qj (q j )

j =1

,

(19.2.51)

j =1

hence a particular case of the function (19.2.48). We obtain thus the reduced Hamilton– Jacobi equation s ⎡ 1 ⎛ ∂S ⎞2 ⎤ ∑ ⎢ 2 Pk ⎜⎝ ∂qk ⎟⎠ − U k (qk ) ⎥ = h ∑ Qj (q j ), h = as , k =1 ⎣ j =1 ⎦ s

where we took into account (19.2.15'). If S = S1 (q1 ) + S2 (q2 ) + ... + Ss (qs ) , then, by separation of variables, we can decompose this system in the form 2

1 ⎛ ∂Sk ⎞ P = hQk (qk ) + U k − ak , k = 1, 2,..., s , a1 + a2 + ... + as = 0 . 2 k ⎜⎝ ∂qk ⎟⎠

We get, in this case, the complete integral (ensured by the above proof) S = − ht +

s −1

∑∫

k =1

2 ( hQk + U k − ak ) dq k Pk

2 ( hQs + U s + a1 + a2 + ... + as −1 ) d qs . Ps

+∫

The trajectory of the representative point P in the configuration space Λs will be given by the equation −

∂S = ∂ak

∫

dq k − Rk

∫

dqs = −bk , k = 1, 2,..., s − 1 , Rs

(19.2.52)

while the motion on the trajectory will be specifies by

∂S = ∂h

s

∑∫ j =1

Q j dq j Rj

= t − t0 ,

(19.2.52')

where we have denoted Rk (qk ) = 2 Pk (hQk + U k − ak ), k = 1, 2,..., s − 1, Rs (qs ) = 2 Ps (hQs + U s + a1 + a2 + ... + as −1 ).

There result then the generalized momenta

(19.2.52'')

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183

pj =

Rj ∂S = , j = 1, 2,..., s . ∂q j Pj

(19.2.52''')

As in the case s = 2 , the differential equations of motion may be written in a compact form dq1 = R1 (q1 )

dq 2 = ... = R2 (q 2 )

dqs dt = = dτ , Rs (qs ) Q1 (q1 ) + Q2 (q 2 ) + ... + Qs (qs ) (19.2.52iv)

where τ is a conventional measure of the time. In connection to these results one can make a discussion analogue to that in the case s = 2 . The mechanical systems characterized by the kinetic energy and the potential energy (19.2.50) are called Liouville’s systems. In the case s = 2 , as we have seen in the preceding subsection, the mechanical systems, which lead to systems of equations with separate variables, are just of such a form; but if s > 2 , then not any mechanical system, separable from the point of view of the canonical co-ordinates, is characterized by the formulae (19.2.50). 19.2.2.3 Stäckel’s Theorem As in the case s = 2 , we consider the orthogonal, scleronomic mechanical systems, studied by P. Stäckel in 1893–1895; in this case (Stäckel, P., 1908), T =

1 s 1 s g jj q2j = ∑ g jj p 2j , g jj = g jj (q1 ,q2 ,..., qs ) ≥ 0, ∑ 2 j =1 2 j =1 g jj =

1 , U = U (q1 ,q2 ,..., qs ) , g jj

(19.2.53)

where g jj , j = 1, 2,..., s , and U are functions of class C 1 . We can state Theorem 19.2.3 (Stäckel). The Hamilton–Jacobi equation built up by means of the kinetic energy and of the potential energy V = −U (19.2.53) is with separate variables if and only if there exists a non-singular square matrix λ ≡ ⎡⎣ λjk ⎤⎦ , λjk = λjk (q j ) , and a column matrix μ ≡ [ μj

]T ,

s

∑ g jj λjk

μj = μj (q j ), j , k = 1, 2,...s , so that to have

= δks , k = 1, 2,..., s ,

j =1

s

∑ g jj μj j =1

=U .

(19.2.54) (19.2.54')

Indeed, if the reduced Hamilton–Jacobi equation 2

1 s jj ⎛ ∂S ⎞ g ⎜ ⎟ − U = h , h = as , 2∑ ⎝ ∂q j ⎠ j =1

(19.2.55)

MECHANICAL SYSTEMS, CLASSICAL MODELS

184

is separable, then we can write a complete integral for which S =

s

∑ S j (qj ; a1 , a2 ,..., as ) .

(19.2.55')

j =1

Introducing in (19.2.55), this equation must be identically satisfied with respect to the essential constants a j ; differentiating with respect to these constants, we can write s

∂S

∂2S = 0, k = 1, 2,..., s − 1 , ∂ak ∂q j

∑ g jj ∂q j j =1

∂S

s

∑ g jj ∂q j j =1

∂2 S = 1. ∂as ∂q j

(19.2.56) (19.2.56')

Denoting λjk =

∂S j (q j ) ∂2 S j ∂S ∂2 S , = ∂q j ∂ak ∂q j ∂q j ∂ak ∂q j

we notice that the relations (19.2.56), (19.2.56') correspond to the relations (19.2.54); on the other hand, λjk = λjk (q j ), j , k = 1,2,..., s , and ⎡ ∂S j ∂2 S j ⎤ ∂S1 ∂S 2 ∂Ss ⎡ ∂2S j ⎤ det ⎡⎣ λjk ⎤⎦ = det ⎢ ... det ⎢ = ⎥ ⎥ ≠ 0, ⎣ ∂q j ∂ak ∂q j ⎦ ∂q1 ∂q2 ∂qs ⎣ ∂ak ∂q j ⎦

because the function S leads to a complete integral. Observing that s

∑ g jj λjs j =1

= 1,

the equation (19.2.55) allows to write U =

⎡ 1 ⎛ ∂S ⎞ ∑ g jj ⎢ 2 ⎜⎝ ∂q j ⎟⎠ − as λjs ⎢⎣ j =1 s

2

⎤ ⎥. ⎥⎦

Denoting 1 ⎛ ∂S ⎞ 1 ⎡ ∂S j (q j ) ⎤ μj = ⎜ − as λjs = ⎢ − as λjs , 2 ⎝ ∂q j ⎟⎠ 2 ⎣ ∂q j ⎦⎥ 2

2

we see that the relation (19.2.57) corresponds to the relation (19.2.54'), while μj = μ j (q j ), j = 1, 2,..., s . Hence, the conditions (19.2.54), (9.2.54') are necessary conditions of separability.

Hamiltonian Mechanics

185

Taking into account the relations of conditions (19.2.54), (19.2.54'), we can write the Hamilton–Jacobi reduced equation in the form 2

s s 1 s jj ⎛ ∂S ⎞ g ⎜ P ak − μ = j j ∑ ∑ ∑ 2 j =1 ⎝ ∂q j ⎟⎠ j =1 k =1

s

∑ g jj λjk j =1

or in the form s ⎡ 1 ⎛ ∂S ⎞2 ⎤ ∑ g ⎢ 2 ⎜⎝ ∂q j ⎟⎠ − ∑ ak λjk − μj ⎥ = 0 , ⎥⎦ j =1 k =1 ⎣⎢ s

jj

where a1 , a2 ,..., as −1 are arbitrary constants. To calculate a complete integral we use the sum (19.2.55'), where the functions of only one variable are given by the equations 2 ⎛ ∂S j ⎞ ⎛ s ⎞ ϕ j (q j ) = ⎜ = 2 ⎜ ∑ ak λjk + μj ⎟ , j = 1, 2,..., s . ⎟ ∂ q ⎝ k =1 ⎠ ⎝ j ⎠

(19.2.58)

It results

∫

S j (q j ) =

ϕ (q j ) dq j , j = 1, 2,..., s .

(19.2.58')

We obtain easily ⎡ ∂2 S j ⎤ ⎡ ∂2 S ⎤ det ⎢ det = ⎢ ⎥ = ⎥ ⎣ ∂ak ∂q j ⎦ ⎣ ∂ak ∂q j ⎦

s

∏ j =1

1 det ⎡⎣ λjk ⎤⎦ ≠ 0 , ϕj

so that the integral obtained on this way for the equation in S is a complete one. Hence, the conditions (19.2.54), (19.2.54') are sufficient conditions of separability too, Stäckel’s theorem being completely proved. The solution of Lagrange’s problem (the trajectory of the representative point P in the space of configurations Λs and the motion on this trajectory) is given by the Hamilton-Jacobi theorem in the form ∂S = ∂ak

s

∂S j

∑ ∂ak j =1

=

s

∑ ∫ λjk j =1

∂S = ∂as

s

∑ ∫ λjs j =1

dq j ϕ j (q j )

= bk , k = 1, 2,..., s − 1,

dq j ϕ j (q j )

(19.2.59) = t + bs .

The solution of Hamilton’s problem in the phase space Γ 2 s is then specified by the generalized momenta

pj =

ϕ j (q j ), j = 1,2,..., s .

(19.2.59')

MECHANICAL SYSTEMS, CLASSICAL MODELS

186

Thus, the considered problem is solved by quadratures. The obtained solution depends on the integration constants a j

and bj ,

j = 1, 2,..., s ; as one can notice, the constant as plays a special rôle, being the energy constant h . The formula (19.2.59') shows that the generalized momentum p j depends

only on the generalized co-ordinate q j ; in exchange, the generalized velocities

q j = g jj ϕ j (q j ), j = 1, 2,..., s ,

(19.2.60)

depend on all the generalized co-ordinates. The relation (19.2.60) allows to write

dq j

ϕj (q j )

= g jj dt = dτ j , j = 1, 2,..., s ,

(19.2.60')

too, where τ j , j = 1, 2,..., s , is a local measure of time, different for each co-ordinate; the sign of the radical is positive or negative as q j increases or decreases, respectively, during time, because g jj ≥ 0 . If g jj ≥ g jj > 0 and if

ϕ j (q j ) is a continuous

function, then the general character of the motion, corresponding to the co-ordinate q j , is specified by (19.2.60') It is interesting to see that, by differentiating the relation (19.2.59), we obtain s

∑ λjk j =1

q j ϕ j (q j )

= δks , k = 1, 2,..., s .

(19.2.61)

Comparing with the relation (19.2.54), we find again the equations (19.2.60'). The conventional time τ j tends to infinite together with t , the nature of the variation of the generalized co-ordinate q j depending on the zeros of ϕ j (q j ) . Thus, if qj

is between two consecutive simple zeros ϕ j (q j ) (with ϕ j (q j ) > 0 for

αj < q j < β j ), then we have a motion of libration in q j , while if q j → γ j for t → ∞ , where γ j is a double zero for ϕ j (q j ) , then we have an asymptotic (limit)

motion. We must notice that if a motion of libration takes place indefinitely between α j and β j , then this one is not, in general, periodical with respect to time. As a matter of fact, as in the case s = 2 , the discussion of the motion with respect to the generalized co-ordinates is of a particular interest. If Pj → 0 for t → ∞ , then it is possible that τ j does tend to a finite limit (not to an infinite one); in this case, the motion with respect to q j is pseudoasymptotic. If αj ≤ q j ≤ β j at the initial moment, then we do not obtain a motion of libration, while (eventually after a finite number of oscillations) lim q j = l j , α j ≤ l j ≤ β j for t → ∞.

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187

We mention that Stäckel’s theorem leads. at the same time, to s first integrals of Lagrange’s equations; thus, the relations (19.2.60) lead to 2 1 q j − βj = 2 Pj2

s

∑ ak λjk , j

k =1

= 1, 2,..., s ,

(19.2.62)

so that s

⎛ 1 q2

∑ λ jk ⎜⎝ 2 Pk2

k =1

k

⎞ − βk ⎟ = ak , j = 1, 2,..., s , ⎠

(19.2.62')

where λ jk is the normalized algebraic complement corresponding to the element λjk in the matrix λ , with λjk λlj = δkl . One see easily that for g jj =

Pj s

∑Ql

, Pj = Pj (q j ), Ql = Ql (ql ), j , l = 1, 2,..., s ,

(19.2.63)

l =1

and ⎡− 1 ⎢ P1 ⎢ ⎢ 0 ⎢ ⎢ ⎡⎣ λjk ⎤⎦ = ⎢ 0 ⎢ ⎢ ... ⎢ ⎢ 1 ⎢⎣⎢ P s Uj μj = ,Uj Pj

−

0

0

...

1 P2

0

...

...

1 P3 ...

1 Ps

1 Ps

0

−

... ... ...

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ Qs ⎥ Ps ⎥⎦⎥

Q1 P1 Q2 0 P2 Q3 0 P3 ... ... 0

1 Ps

= U j (q j ), j = 1, 2,..., s ,

(19.2.63')

(19.2.63'')

we find again Liouville’s system; taking into account the expansion of Δs , from the previous subsection, one can show that s

s

l =1

l =1

det ⎣⎡ λjk ⎦⎤ = ( −1)s −1 ∏ Pl −\1 ∑ Ql ≠ 0 .

(19.2.63''')

The conditions in Stäckel’s theorem can be satisfied if we take g 11 = ψ2 ψ3 ...ψs , g 22 = ψ3 ψ4 ...ψs ,..., g s −1,s −1 = ψs , g ss = 1 ,

(19.2.64)

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188

with ψl = ψl (ql ) , l = 1, 2,..., s ; in this case ⎡ −1 0 0 ⎢ ⎢ ψ2 −1 0 ⎢ 0 ψ −1 3 ⎢ ⎣⎡ λjk ⎤⎦ = ⎢ ... ... ... ⎢ 0 0 ⎢ 0 ⎢ 0 0 0 ⎢⎣

0⎤ ⎥ ... 0 0 ⎥ ... 0 0 ⎥ ⎥, ... ... ... ⎥ ⎥ ... −1 0 ⎥ ... ψs 1 ⎥⎥⎦

...

0

(19.2.64')

with det ⎡⎣ λjk ⎦⎤ = ( −1)s −1 ≠ 0 . The system is separable if the potential U is of the form U = μ1 ψ2 ψ3 ...ψs + μ2 ψ3 ψ4 ...ψs + μs −1 ψs + μs .

(19.2.64'')

Formally, taking into account the relations (19.2.54), (19.2.54'), we can express the kinetic energy and the simple potential in the form s

T =

s

s

1 g jj λjs ∑ gkk qk2 = 2∑ j =1 k =1

∑ g kk pk2

k =1 s

2 ∑ g jj λjs k =1

s

U =

∑ gkk μk

k =1 s

∑ g jj λjs

,

(19.2.65)

,

k =1

which has an affinity with Liouville’s formulation. But we must notice that Stäckel’s theorem does not cover the conditions of separability for an arbitrary Hamilton–Jacobi equation; such conditions could not yet be found. Let λ jk be the algebraic complement of the element λjk in the determinant Δ = det ⎡⎣ λjk ⎤⎦ ; in this case, by an expansion after the last column, we can write Δ = λjs λ js , so that λ js = ∂Δ ∂λjs . The normalized algebraic complement being

λ jk = λ jk Δ , j , k = 1, 2,..., s , it results

λjk λ js = δks , so that we can take

g jj = λ js , j = 1, 2,..., s in Stäckel’s theorem.

19.2.2.4 Levi-Civita’s Conditions Another attempt to the problem of the separation of variables in the Hamilton-Jacobi equation has been made by T. Levi-Civita in 1904. To do this, we will calculate the

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189

total derivative of the Hamilton–Jacobi equation (19.2.41'') with respect to q j in the form dH ∂H ∂H ∂pk = + = 0. dq j ∂q j ∂pk ∂q j

But the generalized momenta are given by (19.2.41'), so that pk = pk (qk ) , obtaining ∂H ∂H ∂p j + = 0(!), j = 1, 2,..., s . ∂q j ∂p j ∂q j

We must assume that ∂H ∂p j ≠ 0 ; otherwise, we should have ∂H ∂q j = 0 too, but we suppose that no one co-ordinate is cyclic. It results ∂H dp j ∂q j dp = − (!), k = 0, k ≠ j . ∂H dq j dq j ∂p j

(19.2.66)

We obtain thus the necessary and sufficient conditions which must be fulfilled by Hamilton’s function, so that the function (19.2.41) does correspond to a complete integral of the equation (19.2.41''). Other developments in this directions have been made by V. G. Imshenetskiĭ in 1869, by Burgatti in 1911, by M. S. Yarov-Yarovoi in 1963 and by others. Imposing also the conditions of integrability d ⎛ ∂pr ⎞ d ⎛ ∂pr ⎞ = , dqk ⎜⎝ ∂q j ⎟⎠ dq j ⎜⎝ ∂qk ⎟⎠

we are led to Levi-Civita’s conditions ⎛ d ⎜ ⎜ dq k ⎜ ⎜ ⎝

∂H ∂q j ∂H ∂p j

⎞ ⎟ ⎟ = 0, j ≠ k , j , k = 1,2,..., s . ⎟ ⎟ ⎠

(19.2.67)

Taking into account (19.2.66), we find s (s − 1)/ 2 relations (we notice that the relations are symmetric with respect to the indices j and k ) of the form ∂2 H ∂H ∂H ∂2 H ∂H ∂H − ∂qk ∂q j ∂p j ∂pk ∂q j ∂qk ∂p j ∂qk −

∂2 H ∂H ∂H ∂2 H ∂H ∂H + = 0, j ≠ k , j = 1, 2,..., s . ∂qk ∂p j ∂q j ∂pk ∂p j ∂pk ∂q j ∂qk

(19.2.67')

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190

We mention that these conditions, which must be fulfilled by Hamilton’s function so that the Hamilton-Jacobi equation be with separate variables, have a general character; but, unlike the conditions which appear in the less general case considered by Stäckel, these conditions can hardly be manipulate. As a matter of fact, they could be used only in the cases of two or three degrees of freedom; we mention that the case of two degrees of freedom has been considered by G. Morera in 1887. In 1944, Forbat completes Levi-Civita’s conditions with s condition of the form

∂H ∂2 H ∂H ∂2 H − = 0, j = 1,2,..., s ; ∂p j ∂qk ∂t ∂p j ∂p j ∂t

(19.2.67'')

the results can be thus applied to the case of rheonomous mechanical systems too. 19.2.2.5 A New Formulation of Hamilton–Jacobi Type One sees easily that, in the canonical system of equations (19.1.14), the generalized co-ordinates q j and the generalized momenta p j play a symmetric rôle, being sufficient to replace Hamilton’s function H by −H . In this order of ideas, the Hamilton–Jacobi partial differential equation (19.2.9) can be replaced by the partial differential equation ∂W ∂W ∂W , ,..., , p1 , p2 ,..., ps ; t ⎞⎟ = 0 . W − H ⎛⎜ ∂ ∂ ∂ p p p s ⎝ 1 ⎠ 2

(19.2.68)

If W ( p1 , p2 ,..., ps , a1 , a2 ,..., as ; t ) is a complete integral of this equation, the condition

⎡ ∂2W ⎤ det ⎢ ⎥ ≠0 ⎣ ∂p j ∂ak ⎦

(19.2.68')

taking place, then the canonical co-ordinates q j = q j (t ), p j = p j (t ), j = 1, 2,..., s , will be given by the sequences of relations ∂W ∂W = −bj , = q j , j = 1,2,..., s , ∂a j ∂p j

(19.2.68'')

where a j are essential constants, while bj are constants conjugate to these ones. Making a transformation of Legendre type and taking into account Donkin’s theorem (the formula (19.1.8)), we can write S + W = pj q j ;

(19.2.69)

thus, the new variable p j and the new function W will be given by pj =

∂S ∂S ,W = qj −S. ∂q j ∂q j

(19.2.69')

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191

Remaking the modality to pass from Lagrange’s equations to the canonical ones (Legendre’s transformation), we obtain qj =

∂S ∂W , S = pj − W , S = −W . ∂q j ∂q j

(19.2.69'')

We can write, as well, ∂S ∂W , =− ∂a j ∂a j

(19.2.70)

being thus led to the above results. Obviously, the modality to approach the problem is equivalent to the previous one, that we can make an analogous study concerning the possibility to separate the variables.

19.2.3 Applications First of all, we will consider the case of motion of a single particle; we pass then to the application of the Hamilton–Jacobi method in case of a system of two particles, as well as in case of a rigid solid. 19.2.3.1 Motion of a Particle The canonical co-ordinates are q j = x j , p j , j = 1,2, 3 in case of a free particle P (x1 , x 2 , x 3 ) (see Sect. 19.1.3.1); the Hamilton-Jacobi equation, which corresponds to the Hamiltonian (19.1.79'), reads 2 2 2 1 ⎡ ⎛ ∂S ⎞ ⎛ ∂S ⎞ + ⎛ ∂S ⎞ ⎤ = 2m (U + h ) . + S + ⎜ ∂x ⎟ ⎜ ∂x ⎟ ⎥ 2m ⎢⎣ ⎜⎝ ∂x1 ⎟⎠ ⎝ 2 ⎠ ⎝ 3 ⎠ ⎦

(19.2.71)

Assuming that the quasi-potential U is a potential (U = 0 ), we can take S = − ht + S ( x1 , x 2 , x 3 ) , h being the energy constant; the reduced Hamilton–Jacobi equation reads

( grad S )

2

2

2

2

⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎛ ∂S ⎞ =⎜ ⎟ + ⎜ ∂x ⎟ + ⎜ ∂x ⎟ = 2m (U + h ) . x ∂ ⎝ 1⎠ ⎝ 2 ⎠ ⎝ 3 ⎠

(19.2.71')

If S = − ht + S (x1 , x 2 , x 3 ; a1 , a2 , h ) is a complete integral, then we obtain the sequences of relations ∂S ∂S ∂S = b1 , = b2 , = t + b3 , ∂a1 ∂a2 ∂h

(19.2.72)

∂S ∂S ∂S . ,p = ,p = ∂x1 2 ∂x 2 3 ∂x 3

(19.2.72')

p1 =

MECHANICAL SYSTEMS, CLASSICAL MODELS

192

The first two equations of the sequence (19.2.67) specify the trajectory of the representative point P ∈ Λ3 , while the last equation gives the motion on this trajectory; the sequence of relations (19.2.67') completes the solution for the representative point P ∈ Γ 6 . If we take into account (19.1.79), we obtain the components of the velocity (we have p = grad S = grad S = m v ) x1 =

1 ∂S 1 ∂S 1 ∂S , , x = , x = m ∂x1 2 m ∂x 2 3 m ∂x 3

(19.2.72'')

the reduced Hamilton–Jacobi equation corresponding to the first integral of the mechanical energy. We notice that, at each point, the velocity v , of components x j , j = 1, 2, 3 , is normal to the wave surface S ( x1 , x 2 , x 3 ) = const ; hence, the trajectory of the representative point is normal to this surface. If we make the change of function S = K ln ψ in the reduced Hamilton–Jacobi equation (19.2.71'), then it results

(

K2 grad ψ 2m

)

2

− (U + h ) ψ2 = 0 ,

(19.2.71'')

getting thus a first form of Schrödinger’s equations. This result has been obtained in the frame of the research made to establish a differential equation from which must result the energetical levels of the electron of the atom of hydrogen; one takes U = e 2 / r , where m , e , h are the mass. the charge and the energy of the electron, respectively, and r is the distance electron-nucleus. Starting from the Hamilton function (19.1.80'), one obtains the Hamilton–Jacobi equation

( )

1 ⎡ ∂S S = 2m ⎢⎣ ∂r

2

+

( ) ( ) ⎤⎥⎦ − U (r , θ, z ;t ) = 0

1 ∂S r 2 ∂θ

2

+

∂S ∂z

2

(19.2.73)

in cylindrical co-ordinates ( q1 = r , q2 = θ , q 3 = z ); as well, if U = 0 , then we have S = − ht + S (r , θ , z ) , the reduced Hamilton–Jacobi equation being given by 2

2

2

1 ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎜ ∂r ⎟ + 2 ⎜ ∂θ ⎟ + ⎜ ∂z ⎟ = 2m (U + h ) . r ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(19.2.73')

A complete integral is S = − ht + S (r , θ , z ; a1 , a2 , h ) ; we can write a sequence of relations of the form (19.2.72), as well as the sequence ∂S ∂S ∂S = pr , = pθ , = pz . ∂r ∂θ ∂z

(19.2.74)

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193

The components of the velocity will be 1 ∂S  1 ∂S 1 ∂S . ,θ = , z = 2 ∂θ m ∂r m ∂z mr

r =

(19.2.74')

As well, in spherical co-ordinates ( q1 = r , q2 = θ , q 3 = ϕ ) we obtain the Hamilton– Jacobi equation

( )

1 ⎧ ∂S S + ⎨ 2m ⎩ ∂r

2

+

( )

1 ⎡ ∂S ⎢ r 2 ⎣ ∂θ

2

+

1 ⎛ ∂S ⎞2 ⎤ ⎫ ⎜ ⎟ ⎥ ⎬ − U (r , θ , ϕ ; t ) = 0 . sin2 θ ⎝ ∂ϕ ⎠ ⎦ ⎭

(19.2.75)

We can take S = − ht + S (r , θ , z ) if U = 0 ; it results the reduced Hamilton–Jacobi equation 2 2 2 1 ⎡ ⎛ ∂S ⎞ 1 ⎛ ∂S ⎞ ⎤ ⎛ ∂S ⎞ ⎥ = 2m (U + h ) . ⎜ ∂r ⎟ + 2 ⎢ ⎜ ∂θ ⎟ + ⎜ ⎟ sin2 θ ⎝ ∂ϕ ⎠ ⎦ r ⎣⎝ ⎝ ⎠ ⎠

(19.2.75')

The complete integral S = − ht + S (r , θ , ϕ ; a1 , a2 , h ) leads to a sequence of the form (19.2.72) , to the sequence ∂S ∂S ∂S = pr , = pθ , = pϕ ∂r ∂θ ∂ϕ

(19.2.76)

and to the components of the velocity 1 ∂S  1 ∂S 1 ∂S . ,θ = , ϕ = 2 2 2 m ∂r mr ∂θ mr sin θ ∂ϕ

(19.2.76')

U (x1 , x 2 , x 3 ) = U 1 ( x1 ) + U 2 (x 2 ) + U 3 (x 3 ) ,

(19.2.77)

r =

If

then the reduced Hamilton–Jacobi equation (19.2.71') correspond to a Hamilton’s function of the form (19.2.44), hence being with separate variables. We notice that g j ( x j , p j ) = p 2j / 2m − U j = a j , j = 1, 2, 3 ; there result the generalized momenta pj =

2m [U j ( x j ) + a j ], j = 1, 2, 3 .

Because H (a1 , a2 , a 3 ) = a1 + a2 + a 3 , while ∂g j / ∂p j = p j / m =

(19.2.77') 2(U j + a j )/ m ,

the co-ordinates x1 , x 2 , x 3 are given by (19.2.45'); we have

∫

dx j

U j (x j ) + a j

=

2 ( t + bj ) , j = 1, 2, 3 , m

(19.2.77'')

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194

the problem being thus reduced to three quadratures. Obviously, Hamilton’s first equations (19.2.79'') lead to the same result. If 1 U (r , θ , z ) = U r (r ) + U θ ( θ ) + U z ( z ) , 2

(19.2.78)

then Hamilton’s function (19.1.80') is of the form (19.2.46), with p 2θ − U θ ( θ ) = a1 , 2m g p2 gr = gr (g θ ; r , pr ) = θ2 − U r (r ) + r = a2 , 2m r pz2 H = gz = gz (gr ; z , pz ) = gr − U z (z ) + = a3 = h . 2m g θ = g θ ( θ , pθ ) =

One obtains the generalized momenta pθ =

2m [U θ ( θ ) + a j ],

a 2m ⎡U r (r ) − 12 + a2 ⎤ , ⎢⎣ ⎥⎦ r pz = 2m [U z (z ) + h − a2 ].

(19.2.78')

pr =

Taking into account the partial derivatives p ∂g θ = θ = m ∂pθ

2 (U θ + a1 ) ∂gr pr = = , m m ∂pr

pz ∂gz = = m ∂pz

(

)

a1 + a2 r2 , m

2 Ur −

2 (U z + h − a2 ) ∂gr 1 ∂gz = 2, = 1, , m ∂a1 r ∂a2

the formulae (19.2.47') give the trajectory of the particle P in the form

∫ ∫

dθ dr 2 b , −∫ = a1 m 1 2 U θ (θ ) + a j r U r (r ) − 2 + a 2 r dr dz 2 −∫ = b , a1 m 2 U z ( z ) + h − a2 U r (r ) − 2 + a 2 r

(19.2.78'')

while the formula (19.2.47'') specifies the motion on this trajectory by the equation

∫

dz = U z ( z ) + h − a2

2 b . m 2

(19.2.78''')

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195

The problem is thus reduced to quadratures. In case of a potential of the form (corresponding to the potential (18.3.46)) U ( r , θ , ϕ ) = U r (r ) +

1 r2

⎡U ( θ ) + 1 U ϕ (ϕ ) ⎤ , ⎢⎣ θ ⎥⎦ sin2 θ

(19.2.79)

Hamilton’s function (19.1.81') is of the form (19.2.46) too, with pϕ2 − U ϕ (ϕ ) = a1 , 2m pθ2 gϕ g θ = g θ (gϕ ; θ , pθ ) = + − U θ ( θ ) = a2 , sin2 θ 2m g pr2 H = gr = gr ( g θ ; r , pr ) = θ2 + − U r (r ) = a 3 = h . 2m r gϕ = gϕ (ϕ , pϕ ) =

The generalized momenta will be 2m [U ϕ (ϕ ) + a1 ],

pϕ =

a1 + a2 ⎤ , 2m ⎡U θ ( θ ) − ⎣⎢ ⎦⎥ sin2 θ a pr = 2m ⎡U r (r ) + 22 + h ⎤ . ⎣⎢ ⎦⎥ r

pθ =

(19.2.79')

We calculate the partial derivatives ∂g θ p = θ = ∂pθ m

2 (U θ + a1 ) ∂g θ p , = θ = ∂pθ m m

∂gr pr = = ∂pr m

(

(

2 Uθ −

)

)

a1 + a2 sin2 θ , m

a2 +h ∂g 1 ∂gr 1 r2 , θ = , = 2. 2 ∂a1 ∂ m a sin θ r 2

2 Ur −

As in the preceding case, the trajectory of the particle will be given by

∫ ∫

dϕ dθ − U ϕ (ϕ ) + a1 ∫ sin2 θ U ( θ ) − θ dθ U θ (θ ) −

a1 + a2 sin2 θ

−

∫

r2

= a1 + a 2 sin2 θ dr = a U r (r ) − 22 + h r

2 b , m 1 2 b , m 2

(19.2.79'')

the motion on this trajectory being specifies by

∫

dr a U r (r ) − 22 + h r

=

2 ( t + b3 ) . m

(19.2.79''')

MECHANICAL SYSTEMS, CLASSICAL MODELS

196

In this case too, the problem is reduced to quadratures. If U = k / r , r > 0 (case of forces of Newtonian attraction), then Hamilton’s function is given by (19.1.81') and the Hamilton–Jacobi equation reads

( )

1 ⎡ ∂S S + 2m ⎣⎢ ∂r

2

+

( )

1 ∂S r 2 ∂θ

2

+

2 1 ⎛ ∂S ⎞ ⎤ − k = 0 . ⎜ ⎟ ⎥ r 2 sin2 θ ⎝ ∂ϕ ⎠ ⎦ r

(19.2.80)

As a matter of fact, this case is contained in the previously considered one. The generalized momenta will be (U ϕ = U θ = 0, U r = k / r , while ϕ is a cyclic co-ordinate) pϕ = pθ = pr =

2ma1 = const,

( sina θ ), k a 2m ( h + − . r r )

2 m a2 −

1 2

(19.2.80')

2 2

The trajectory of the particle is given by dθ

ϕ 2 b , = − a1 a1 m 1 sin θ a2 − sin2 θ dθ dr 2 b2 , −∫ = a1 m r hr kr a + − 2 a2 − sin2 θ

∫

∫

2

(19.2.80'')

the motion on this trajectory being specified by

∫

rdr hr + kr − a2 2

=

2 ( t + b3 ) . m

(19.2.80''')

In the case of elliptic co-ordinates (see Sects. 18.3.2.1 and 19.1.3.1) let us suppose that the simple potential U is given by U =

U 1 (q1 ) U 2 (q2 ) U 3 (q 3 ) + + . (q1 − q2 )(q1 − q 3 ) (q2 − q 3 )(q2 − q1 ) (q 3 − q1 )(q 3 − q2 )

(19.2.81)

Taking into account Hamilton’s function (19.1.83) and the notations (18.2.37'), (18.3.39'), we can write the reduced Hamilton-Jacobi equation in the form 2

2

2

2 2 2 ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎛ ∂S ⎞ f (q1 ) ⎜ f (q2 ) ⎜ f (q 3 ) ⎜ − U1 − U2 ⎟ ⎟ ⎟ − U3 m m m ⎝ ∂q1 ⎠ ⎝ ∂q2 ⎠ ⎝ ∂q 3 ⎠ + + =h. (q1 − q2 )(q1 − q 3 ) (q2 − q 3 )(q2 − q1 ) (q 3 − q1 )(q 3 − q2 )

The identity

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197

2a1 + 2a2q1 + hq12 2a1 + 2a2q2 + hq22 2a1 + 2a2q 3 + hq 32 + + =h (q1 − q2 )(q1 − q 3 ) (q2 − q 3 )(q2 − q1 ) (q 3 − q1 )(q 3 − q2 )

is easily verified for any a1 and a2 ; denoting 2S j (q j ) = 2a1 + 2a2q j + hq 2j , j = 1, 2, 3 ,

(19.2.82)

the Hamilton–Jacobi equation takes the form 2

2

2

2 2 2 ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎛ ∂S ⎞ f (q1 ) ⎜ ⎟ − 2S1 m f (q2 ) ⎜ ∂q ⎟ − 2S 2 m f (q 3 ) ⎜ ∂q ⎟ − 2S 3 m ⎝ ∂q1 ⎠ ⎝ 2 ⎠ ⎝ 3 ⎠ + + = 0. (q1 − q2 )(q1 − q 3 ) (q2 − q 3 )(q2 − q1 ) (q 3 − q1 )(q 3 − q2 )

We find thus the complete integral

∫

S (q1 , q2 , q 3 ; a1 , a2 , h ) =

S1 (q1 ) dq1 + f (q1 )

∫

S 2 (q2 ) dq 2 + f (q2 )

∫

S 3 (q 3 ) dq 3 , f (q 3 )

which annuls each of the three ratios. The trajectory of the particle will be given by the equations

∫ ∫

dq1 S1 (q1 ) f (q1 ) q1 d q 1 S1 (q1 ) f (q1 )

+

∫

+∫

dq 2 S2 (q2 ) f (q2 ) q 2 dq 2 S 2 (q2 ) f (q2 )

+

+

∫ ∫

dq 3 S 3 (q 3 ) f (q 3 ) q 3 dq 3 S 3 (q 3 ) f (q 3 )

= b1 ,

(19.2.82') = b2 ,

the motion on the trajectory being specified by the equation

∫

q12 dq1 S1 (q1 ) f (q1 )

+

∫

q22 dq2 S2 (q2 ) f (q2 )

+

∫

q 32 dq 3 S 3 (q 3 ) f (q 3 )

= 2 ( t + b3 ) .

(19.2.82'')

If U 1 = U 2 = U 3 = 0 , then one obtains the equations of a straight line in elliptic co-ordinates. In case of an electrized particle in an electromagnetic field (see Sect. 19.1.3.3), the formula (19.2.88) leads to the Hamilton–Jacobi equation 1 S + ( S,i − qAi m

)( S,i

− qAi

) + qA0

= 0.

(19.2.83)

19.2.3.2 Mechanical Systems with Two degrees of freedom In the case of a mechanical system with two degrees of freedom (e.g., the motion of a particle on a plane or on a surface – a holonomic constraint) the Hamilton–Jacobi equations reads

MECHANICAL SYSTEMS, CLASSICAL MODELS

198

∂S ∂S ⎞ S + H ⎛⎜ q1 , q2 , , ;t = 0 . ∂q1 ∂q2 ⎟⎠ ⎝

(19.2.84)

By means of a complete integral S = S (q1 , q2 ; a1 , a2 ; t ) , the integrals of the canonical equations will be given by ∂S ∂S ∂S ∂S = b1 , = b2 , = p1 , = p2 . ∂a1 ∂a2 ∂q1 ∂q2

(19.2.84')

If the constraint is scleronomic, then we ca take S = − ht + S (q1 , q2 ; a1 , h ) , the reduced Hamilton–Jacobi equation being ∂S ∂S ⎞ ⎛ H ⎜ q1 , q 2 , , =h. q1 ∂q2 ⎟⎠ ∂ ⎝

(19.2.85)

The motion of the mechanical system is specified by ∂S ∂S ∂S ∂S = b1 , = t + b2 , = p1 , = p2 . ∂a1 ∂h ∂q1 ∂q2

(19.2.85')

The trajectory of the representative point (for various values of the constant b1 ) are normal to the curves S = const . In case of a heavy particle P , which is moving in a vertical plane, the reduced Hamilton-Jacobi equation reads 2 2 m ⎡ ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎤ +⎜ ⎢ ⎥ − gx 2 = h , ⎜ ⎟ ⎟ 2 ⎣ ⎝ ∂x1 ⎠ ⎝ ∂x 2 ⎠ ⎦

the Ox 1 -axis being horizontal, while the Ox 2 -axis is along the descendent vertical; because x1 is a cyclic co-ordinate, we search a solution of the form S = ax1 + ϕ (x 2 ) , being led to the equation m 2 ⎡a + ϕ ′2 (x 2 ) ⎤⎦ − gx 2 = h , 2⎣ 1

wherefrom S ( x1 , x 2 ; a1 , h ) = a1x1 +

∫

2h − a12 + 2 gx 2 dx 2 .

The trajectory of the particle is given by the equation x 1 − a1 ∫

dx 2 2h −

a12

+ 2 gx 2

= x1 −

a1 g

2h − a12 + 2 gx 2 = b1 ,

(19.2.86)

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199

being an are of parabola, tangent at the vertex to the straight line x 2 = (a12 − 2h )/ 2 g , and the motion on this trajectory is specified by

∫

dx 2 2h −

a12

+ 2 gx 2

=

1 2h − a12 + 2 gx 2 = t + b2 . g

(19.2.86')

From (19.2.86), (19.2.86') one obtains x1 = a1 (t + b2 ) + b1 , the motion of the projection of the particle on the horizontal being uniform (see Sect. 7.1.2.1 too). The curves S = const are the semi-cubical parabolas a1x1 +

1 3g

( 2h − a12

+ 2 gx 2

)3

= const ,

(19.2.86'')

which can be obtained one from the other by a translation along the Ox 1 -axis; these curves are normal to the straight line x 2 = (a12 − 2h )/ 2 g on which they have a cuspidal point. Varying the constant b1 , one obtains a family of parabolic trajectories (by a translation along the Ox1 -axis too), normal to the curves S = const , the general theoretical result being thus verified. In case of a plane motion of a particle, the Hamilton–Jacobi equation will of the form 2 2 1 ⎡ ⎛ ∂S ⎞ ∂S ⎞ ⎤ S + + ⎛⎜ ⎜ ⎟ ⎟ ⎢ ⎥ − U (x1 , x 2 ; t ) = 0 , 2m ⎣ ⎝ ∂x1 ⎠ ⎝ ∂x 2 ⎠ ⎦

(19.2.87)

in Cartesian co-ordinates; if U = 0 , then we take S = − ht + S (x1 , x 2 ) , being led to the reduced Hamilton–Jacobi equation 2

2

⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎜ ∂x ⎟ + ⎜ ∂x ⎟ = 2m (U + h ) . ⎝ 1⎠ ⎝ 2 ⎠

(19.2.87')

A complete integral S = − ht + S (q1 , q2 ; a1 , h ) leads to the sequences of relations ∂S ∂S ∂S ∂S = b1 , = t + b2 , = p1 , = p2 . ∂a1 ∂h ∂x1 ∂x 2

(19.2.87'')

The first equation specifies the trajectory of the particle in the plane Λ2 and the second equation corresponds to the motion on this trajectory; the last two equations complete the solution of the problem in the space Γ 4 . Let, e.g., be an elliptic oscillator, to which corresponds Hamilton’s function H =

1 k ( p 2 + p22 ) + 2 ( x12 + x 22 ) , 2m 1

(19.2.88)

MECHANICAL SYSTEMS, CLASSICAL MODELS

200

where k is an elastic constant. We notice that this function is of the form (19.2.44), where g j = p 2j / 2m + kx 2j / 2, j = 1, 2 ; denoting g j = a j , a j = const it results kx 2j ⎞ ⎛ 2m ⎜ a j − ⎟ , j = 1, 2 . 2 ⎠ ⎝

pj =

(19.2.88')

We obtain the complete integral 2

∑∫

S = − ( a1 + a2 ) t +

j =1

kx 2j ⎞ ⎛ 2m ⎜ a j − ⎟ dx j . 2 ⎠ ⎝

The Hamilton–Jacobi theorem allows to write the relations dx j 1 = t + bj , ω = ∫ ω Aj2 − x 2j

k ,A = m j

2a j , j = 1, 2 . k

Effecting the quadrature, we get x j = Aj sin ω ( t + bj ) , j = 1, 2 ,

(19.2.88'')

remaining to put the initial conditions. We obtain the Hamilton–Jacobi equation

( )

1 ⎡ ∂S S + 2m ⎢⎣ ∂r

2

+

( ) ⎤⎥⎦ − U ( r , θ;t ) = 0 ,

1 ∂S r ∂θ

2

(19.2.89)

in polar co-ordinates ( q1 = r , q2 = θ ). If U = 0 , then we have S = − ht + S (r , θ ) and the reduced Hamilton–Jacobi equation is given by 2

2

1 ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎜ ∂r ⎟ = 2 ⎜ ∂θ ⎟ = 2m (U + h ) . r ⎝ ⎝ ⎠ ⎠

(19.2.89')

Starting from the complete integral S = − ht + S (r , θ ; a1 , h ) , the trajectory and the motion on it will be given by ∂S ∂S = b1 , = t + b2 , ∂a1 ∂h

(19.2.89'')

and the generalized momenta by pr =

∂S ∂S ,p = . ∂r θ ∂θ

(19.2.89''')

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201

If U (r , θ ) = U r (r ) +

1 U θ (θ ) , r2

(19.2.90)

then Hamilton’s function is of the form (19.2.46), with pθ2 − U θ ( θ ) = a1 , 2m g pr2 H = gr = gr (g θ ; r , pr ) = θ2 − U r (r ) + = a2 = h . 2m r g θ = g θ ( θ , pθ ) =

One obtains the generalized momenta pθ = pr =

2m [U θ ( θ ) + a1 ],

(19.2.90')

a 2m ⎡U r (r ) + 12 + h ⎤ . ⎢⎣ ⎥⎦ r

Taking into account the partial derivatives ∂g θ p = θ = ∂pθ m ∂gr pr = = ∂pr m

(

2 (U θ + a1 ) , m

)

a1 +h ∂g 1 r2 , 1 = 2, ∂a1 m r

2 Ur −

the formulae (19.2.47') give the trajectory of the particle P in the form

∫

dθ dr −∫ = a1 2 U θ ( θ ) + a1 r U r (r ) − 2 + h r

2 b , m 1

(19.2.90'')

and the formula (19.2.47'') specifies the motion on this trajectory by the equation

∫

dr a U r (r ) − 12 + h r

=

2 ( t + b2 ) . m

(19.2.90''')

The problem is thus reduced to quadratures. If U = k / r , k > 0 (case of forces of Newtonian attraction), then the Hamilton-Jacobi equation is written in the form

( ) + ( r1 ) ( ∂∂Sθ ) ⎤⎥⎦ − kr = 0 .

1 ⎡ ∂S S + 2m ⎢⎣ ∂r

2

2

2

(19.2.91)

MECHANICAL SYSTEMS, CLASSICAL MODELS

202

As a matter of fact, this case is contained in the previously considered one. The generalized momenta are (U θ = 0,U r = k / r , k > 0 , and θ is a cyclic co-ordinate) pθ =

2ma1 ,

(

(19.2.91')

)

k a 2m h + − 12 . r r

pr =

The trajectory of the particle is given by

∫r

dr

hr + kr − a1 2

=

θ − a1

2 b , m 1

(19.2.91'')

the motion on the trajectory being specified by

∫

rdr hr + kr − a1 2

=

2 ( t + b2 ) . m

(19.2.91''')

Let us suppose that the particle P lies on a surface S for which the element of arc is given by a relation of the form

⎛ dq 2 dq 2 ⎞ ds 2 = ( Q1 + Q2 ) ⎜ 1 + 2 ⎟ , Pj = Pj (q j ),Q j = Q j (q j ), j = 1, 2 . P2 ⎠ ⎝ P1

(19.2.92)

In this case, one obtains the kinetic energy T =

1 ( Q + Q2 2 1

⎛ q12 q22 ⎞ P1 p12 + P2 p22 + . ⎟= ⎝ P1 P2 ⎠ 2 ( Q1 + Q2 )

)⎜

(19.2.92')

If the particle is subjected to no one given force (U = 0 ), then it results H = T and we can apply Liouville’s theorem. By separation of variables, hence by quadratures, one can obtain thus the possible trajectories (depending on the integration constants); in fact, thus are specified the geodesic lines of the surface. The surfaces which have this property are called Liouville’s surfaces. 19.2.3.3 Problem of Two Particles. Motion of Planets The problem of two particles has been considered in Sects. 19.3.2.3 and 18.1.3.2, assuming the action of forces of Hamiltonian attraction. The generalized system has three degrees of freedom and we can choose the generalized co-ordinates q1 = r , q2 = θ , q 3 = ϕ ; assuming that we study the relative motion of a particle with respect to the other one, we take the latter one as origin of the frame of reference. We can thus use the results obtained in Sect. 19.2.3.1 for the motion of a free particle (in spherical co-ordinates – the formulae (19.2.75'), (19.2.75''), and (19.2.75''')).

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203

We assume that the initial position of the particle in on the line of nodes ON , in the plane Ox1x 2 (Fig. 19.1); hence, for t = t0 we have θ = θ0 = π / 2, ϕ = ϕ0 . Imposing these conditions, it results b3 = −t0 , b1 =

m ϕ . 2a1 0

(19.2.93)

As far as we know, the trajectory of the particle is an ellipse of semi-major axis a and eccentricity e ; in this case, rmin = a (1 − e ) and rmax = a (1 + e ) . By differentiation of the equation (19.2.75'''), we obtain m dr = r 2 dt

hr 2 + kr − a2 .

The extreme values of the vector radius r must annul the quantity under the radical. From Viète’s relations, it results rmin + rmax = 2a = − k / h , rmin rmax = a 2 (1 − e 2 ) = −a2 / h , wherefrom h =

k −k < 0, a2 = a ( 1 − e 2 2a 2

)=

k p > 0. 2

(19.2.93')

Fig. 19.1 Motion of the Sun and of a planet

We notice that for a given colatitude θ one must have a2 − a1 / sin2 θ ≥ 0 , hence

θ ≥ arcsin a1 / a2 ; but θmin is just the complement of the angle i made by the plane of the orbit with the plane Ox1x 2 . Hence, a1 = a2 cos2 i =

k p cos2 i . 2

(19.2.93'')

MECHANICAL SYSTEMS, CLASSICAL MODELS

204

The trajectory will be given by (for θ we take the inferior limit π / 2 , corresponding to a point on the line of nodes, while for r we take the inferior limit rmin = a (1 − e ) , corresponding to the pericentre π ) θ

cos i dϑ

π /2

sin ϑ sin2 ϑ − cos2 i

∫

1 p − a

= ϕ − ϕ0 ,

sin ϑ dϑ

θ

∫

π / 2 sin ϑ

(19.2.94)

sin2 ϑ − cos2 i

r

dρ = + − ρ a e ρ ][ ρ − a (1 − e ) ] (1 ) [ a (1 −e )

∫

k b , m 2

while the motion on the trajectory will be specified by r

a

ρ dρ = [ a (1 + e ) − ρ ][ ρ − a (1 − e ) ]

∫

a (1 −e )

k ( t − t0 ) , m

(19.2.94')

At the moment in which the particle attains the pericentre π , the second equation (19.2.94) becomes θπ

∫

π /2

sin ϑ dϑ

sin ϑ dϑ

θπ

sin2 ϑ − cos2 i

=

∫

π /2

sin2 i − cos2 ϑ

= −χπ .

with χ π = − p k / mb2 . By integrating, we get cos θπ = χπ , sin i

(19.2.93''')

cos θπ = sin i sin χ π .

(19.2.93IV)

arcsin

wherefrom

Considering the relations between the elements of the spherical triangle N ππ ′ on a sphere with the centre O and of unit radius (the point π is the piercing point of the p radius with this sphere), we see that χ = N π . These results correspond to those π

obtained in Sect. 9.2.2.4 in the study of the orbit of a satellite. As a matter of fact, we can imagine that at the pole O is situated the Sun, the particle P being a planet; in this case, the plane Ox1x 2 is the plane of the ecliptic, while the point π is the perihelion. Thus, all six parameters ( a , e , i , ϕ0 , t0 and χ π ) which define the orbit of the planet are determined. We notice that

Hamiltonian Mechanics

205

dρ ∫ ρ [ a (1 + e ) − ρ ][ ρ − a (1 − e ) ] = a (1 −e ) r

ap

dρ

r

ap

∫

a (1 −e ) ρ

− ρ + 2a ρ − ap 2

r

= arcsin

−2ap + a ρ r −p π r −p = arcsin = − − arcsin er er 2 ρ 4a (a − p ) a (1 −e ) =−

( π2 − arcsin per− r ) ,

as well as sin ϑ dϑ

θ

∫

π /2

sin ϑ − cos i 2

2

= arcsin

cos θ = −χ , sin i

p corresponds to an arbitrary position on the orbit (the point P is where the arc χ = NP the intersection of the radius OP with the sphere of unit radius); hence,

χ − χπ = −

( π2 − arcsin per− r ) ,

so that cos ( χ − χ π ) = cos

( π2 − arcsin per− r ) = per− r .

Finally, we find the known equation of the trajectory in its plane r =

p , 1 + e cos ( χ − χ π )

(19.2.95)

corresponding to the equation (9.2.6). Starting from the equation (19.2.94') and using the constant h given by (19.2.93'), we find again the integral given in Sect. 9.2.1.3, which leads to Kepler’s equation (9.2.13). The first equation (19.2.94) specifies the motion in the plane of the ecliptic. Knowing that the trajectory is a plane curve, the problem can be treated in another way too, choosing the plane of the trajectory as plane Ox1x 2 . Considering, further, the relative motion of a planet with respect to the Sun, we can use polar co-ordinates as in Sect. 19.2.3.2 (the mechanical system formed by the two particles has two degrees of freedom in the plane of the motion); the Hamilton-Jacobi equation (19.2.91) leads thus to the classical equation of the trajectory. But we can make also a study in orthogonal Cartesian co-ordinates, after Jacobi. The reduced Hamilton–Jacobi equation will be 2

2

(

)

k ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎜ ∂x ⎟ + ⎜ ∂x ⎟ = 2m r + h . ⎝ 1⎠ ⎝ 2 ⎠

(19.2.96)

MECHANICAL SYSTEMS, CLASSICAL MODELS

206

Taking an arbitrary point P0 on the Ox 1 -axis, so that OP0 = r0 , and denoting

P0 P = ρ , we have r =

x12 + x 22 , ρ =

( x1 − r0 )2 + x 22 .

(19.2.97)

We will show that the function S =

σ

∫σ ′

k h + ds , s + r0 2

(19.2.96')

where σ ′ = r − ρ, σ = r + ρ

(19.2.97')

is a complete integral of the equation (19.2.96), r0 being an essential constant. Indeed, observing that σ and σ ′ depend on x1 and x 2 by means of r and ρ , it results x x −r ⎞ x x −r ⎞ ∂S h⎞ h⎞ ⎛ k ⎛ k = ⎛⎜ 1 + 1 + ⎟ − ⎛⎜ 1 − 1 + ⎟, m⎜ m⎜ ⎟ ⎟ ′ ∂x1 ⎝ r ρ ⎠ ρ ⎠ 2⎠ ⎝ r 2⎠ ⎝ σ + r0 ⎝ σ + r0 x x x x ∂S h⎞ h⎞ ⎛ k ⎛ k = ⎜⎛ 2 + 2 ⎟⎞ m ⎜ + ⎟ − ⎜⎛ 2 − 2 ⎟⎞ m ⎜ + ⎟, ′ ρ σ ρ σ ∂x 2 + + 2 2⎠ r r r r ⎝ ⎠ ⎠ ⎝ ⎠ ⎝ ⎝ 0 0

wherefrom, using the notation (19.2.97), we find (the terms which contain the product of the two radicals disappear) 2 x1 (x1 − r0 ) + 2 x 22 ⎤ ⎛ k ⎡ h ⎛ ∂S ⎞ ⎛ ∂S ⎞ + = + + ⎞⎟ 2 m ⎜ ⎜ ∂x ⎟ ⎜ ∂x ⎟ ⎢ ⎥ rρ ⎝ 1⎠ ⎝ 2 ⎠ ⎣ ⎦ ⎝ σ + r0 2 ⎠ 2 2 x (x − r0 ) + 2 x 2 ⎤ ⎛ k ⎡ h⎞ +m ⎢ 2 − 1 1 ⎥ ⎜ σ′ + r + 2 ⎟ . ρ r ⎠ 0 ⎣ ⎦⎝ 2

2

Taking into account the identities 2+

2 x1 (x1 − r0 ) + 2 x 22 σ 2 − r02 2r ρ + r 2 + ρ2 − r02 = = , rρ rρ rρ

2−

2 x1 ( x1 − r0 ) + 2 x 22 2r ρ − r 2 − ρ 2 + r02 σ ′2 − r02 = = , rρ rρ rρ

one sees that the equation (19.2.96) is verified. The equation of the trajectory and the motion on it will be given by ∂S / ∂r0 = K , K = const, ∂S / ∂h = t − t0 . We notice that x − r0 ∂σ ′ x − r0 ∂ρ ∂ρ ∂σ , , = =− 1 =− = − 1 ρ ρ ∂r0 ∂r0 ∂r0 ∂r0

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207

so that the first equation leads to −

x1 − r0 ρ

x − r0 k h + ⎞⎟ − 1 m ⎛⎜ ρ ⎝ σ + r0 2 ⎠ mk ds σ

−∫

σ′

k h + ⎞⎟ m ⎛⎜ ⎝ σ ′ + r0 2 ⎠

k h 2 ( s + r0 ) m ⎛⎜ + ⎞⎟ + 2⎠ s r ⎝ 0

=K.

Effecting the quadrature, we can also write x1 − r0 ⎞ x1 − r0 ⎞ h⎞ ⎛ h⎞ ⎛ ⎛ k ⎛ k ⎜1 − ⎟ m⎜ σ + r + 2 ⎟ − ⎜1 + ⎟ m ⎜ σ′ + r + 2 ⎟ = K . ρ ρ ⎝ ⎠ ⎠ ⎝ ⎠ ⎝ ⎝ ⎠ 0 0

Because 2r0 (x1 − r0 ) = r 2 − r02 − ρ2 , it results 1−

x1 − r0 2r ρ − r 2 + r02 + ρ2 ( σ + r0 ) ( σ ′ − r0 ) , = 0 = − ρ 2 ρr0 2 ρr0

1+

x1 − r0 2r ρ + r 2 − r02 − ρ 2 ( σ − r0 ) ( σ ′ + r0 ) , = 0 = − ρ 2 ρr0 2 ρr0

obtaining thus the equation of the trajectory in bipolar co-ordinates (of poles O and P0 ) in the form

( σ + r0 ) ( σ ′ − r0 )

k h k h + + ( σ − r0 ) ( σ ′ + r0 ) + = K′, σ + r0 2 σ ′ + r0 2 (19.2.98)

where K ′ = 2 ρr0 K / m is a new integration constant. We have ρ = 0 , r = r0 for the point P0 , hence σ = σ ′ = r0 ; taking into account the form of the constant K ′ , we see that the trajectory, which is a conic, passes through the point P0 . The genus of the conic depends on the sign of the constant h ; in case of the Solar system we have h < 0 , the conic being an ellipse. The motion on the orbit will be specified by t − t0 =

1σ 4 σ∫′

ds , k h⎞ ⎛ m⎜ + ⎟ ⎝ s + r0 2 ⎠

(19.2.98')

beginning from the point P0 for which t = t0 . If the orbit is parabolic ( h = 0 ), then the integration is immediate, obtaining Euler’s formula t − t0 =

1 ⎡ ( r + ρ + r0 )3 − 6 mk ⎣

( r − ρ + r0 )3 ⎤⎦ .

(19.2.98'')

MECHANICAL SYSTEMS, CLASSICAL MODELS

208

In the case in which h ≠ 0 (elliptic or hyperbolic orbits), analogous formulae have been given by Gauss. 19.2.3.4 Problem of the Two Centres We take again the problem of two centres, considered in Sect. 18.3.2.4. The kinetic energy is given by (18.3.55) and the simple potential reads (18.3.55'), obtaining Hamilton’s function μ2 ⎞ m 2 ⎛ λ2 λ − μ2 ) ⎜ 2 + 2 ( ⎟ 2 2 c − μ2 ⎠ ⎝λ −c 1 − 2 [ ( k1 + k2 ) λ − ( k1 − k2 ) μ ] , λ − μ2

H =

(19.2.99)

where λ and μ are elliptic co-ordinates in the plane; m is the mass of the particle, k1 and k2 characterize the centres of attraction (in inverse proportion to the squares of the distances), the distance between these centres being 2c . We find the generalized momenta p1 = m

( λ2

− μ2 ) λ

λ2 − c 2

, p2 = m

( λ2

− μ2 ) μ

c 2 − μ2

,

(19.2.100)

so that Hamilton’s function becomes H =

1 m ( λ2 − μ2 ) ⎣⎡ ( λ 2 − c 2 ) p12 + ( c 2 − μ2 ) p22 ⎦⎤ 2 1 − 2 [ ( k1 + k2 ) λ − ( k1 − k2 ) μ ] . λ − μ2

(19.2.99')

This Hamiltonian is of the form (19.2.51), hence of Liouville’s type, and one can make the separation of the variables. Thus, we obtain – obviously – the same relations (18.3.56), (18.3.56'), the trajectory and the motion on the trajectory being obtained by quadratures (elliptic integrals). The relations (19.2.100) complete then the solution in the phase space. This problem has been considered for the first time by Euler (in the plane case); it has been taken again in a three-dimensional treatment by Lagrange and then by Jacobi, who integrated it by his general method of calculation. This research has been an important impetus to enlarge the theory of elliptic integrals. We mention also the results obtained by Serret, Desboves and Andrade in this direction. G. Dincă studied, in 1965, the general motion of a particle subjected to the action of n central forces (which pass through n fixed centres), in a resistant medium; the problem is reduced to the above considerations, in some interesting particular cases. These results can be seen also as a first step in the study of the problem of three particles, which can be solved – in general – only by approximate methods of calculation; besides, the results can be used in the general problem of n particles too.

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209

19.2.3.5 Motion of the Rigid Solid The Hamilton–Jacobi method can be applied, as well, in the case of the free or the constrained rigid solid S . As in Sect. 19.1.3.5, we will consider a heavy, homogeneous rigid solid of rotation, of mass M , which slides frictionless on a fixed horizontal plane, having five degrees of freedom. Starting from Hamilton’s function (19.1.96), one obtains the canonical equations (19.1.96'). The generalized co-ordinates are ρ1′ , ρ2′ (which specify the position of the mass centre) and Euler’s angles ψ, θ and ϕ ; the generalized momenta p1 , p2 , pψ , pθ and pϕ are corresponding to them, I 3 and J are the moments of inertia with respect to the axis of rotation and to an axis normal to this one (with respect to the movable frame of reference R ), respectively, while the function f ( θ ) specifies the applicate of the mass centre with respect to the fixed plane.

The Hamilton–Jacobi equation reads 1⎧ 1 S + ⎨ 2 ⎩M

( )

⎡ ⎛ ∂S ⎞2 ⎛ ∂S ⎞2 ⎤ ∂S 1 ⎢ ⎜ ∂ρ ′ ⎟ + ⎜ ∂ρ ′ ⎟ ⎥ + 2 ⎝ 2 ⎠ ⎦ Mf ′ ( θ ) + J ∂θ ⎣⎝ 1 ⎠ +

2

+

1 ⎛ ∂S ⎞2 ⎜ ⎟ I 3 ⎝ ∂ϕ ⎠

2 1 ⎛ ∂S − cos θ ∂S ⎞ ⎫ + Mgf ( θ ) = 0 . ⎬ ⎜ ⎟ ∂ϕ ⎠ ⎭ J sin2 θ ⎝ ∂ψ

(19.2.101)

Because the constraints are scleronomic and the generalized co-ordinates ρ1′ , ρ2′ , ψ and ϕ are cyclic, we can choose a complete integral of the form S = − ht + a1 ρ1′ + a2 ρ2′ + a 3 ψ + a 4 ϕ + F ( θ ) ,

where h , a1 , a2 , a 3 and a 4 are integration constants. Replacing in the equation (19.2.101), we get the equation −h + +

( )

1⎡ 1 2 1 ∂F ( a + a22 ) + ′2 2 ⎢⎣ M 1 Mf ( θ ) + J ∂θ

2

1 2 1 a + ( a − a 4 cos θ )2 ⎥⎤ + Mgf ( θ ) = 0 , I 3 4 J sin2 θ 3 ⎦

which determines the function F ( θ ) ; we obtain thus the complete integral S = − ht + a1 ρ1′ + a2 ρ2′ + a 3 ψ + a 4 ϕ +

∫

Mf ′2 ( θ ) + J Θ ( θ ) dθ ,

(19.2.102)

where we have denoted Θ ( θ ) = 2 [ h − Mhf ( θ ) ] −

a12 + a22 a 42 ( a 3 − a 4 cos θ )2 , − − M I3 J sin2 θ

the five constants being essential constants.

(19.2.102')

MECHANICAL SYSTEMS, CLASSICAL MODELS

210

Differentiating the function S with respect to the initial constants, we find the equations of the trajectory of the representative point P ∈ Λ5 in the form ρ1′ −

a1 M

∫

Mf ′2 ( θ ) + J Θ(θ ) ψ−

ϕ−

∫

dθ = b1 , ρ2′ −

a2 M

Mf ′2 ( θ ) + J

∫

Θ(θ )

dθ = b2 ,

Mf ′2 ( θ ) + J a 3 − a 4 cos θ 1 dθ = b3 , ∫ J sin2 θ Θ(θ )

(19.2.103)

Mf ′2 ( θ ) + J ⎡ a 4 cos θ − ( a 3 − a 4 cos θ ) ⎤⎥ dθ = b4 , 2 ⎢ I ⎣ 3 J sin θ ⎦ Θ(θ )

the motion along this trajectory being specified by the equation t − t0 =

∫

Mf ′2 ( θ ) + J Θ(θ )

dθ .

(19.2.103')

Replacing the integral (19.2.103') in the first two equations (19.2.103), we obtain ρ ′ − b1 =

a1 a ( t − t0 ) , ρ ′ − b2 = 2 ( t − t0 ) ; M M

(19.2.103'')

we find thus again a known result, in accordance to which the projection of the mass centre on the fixed plane has a rectilinear and uniform motion. The components of the generalized momentum will be p1 = a1 , p2 = a2 , pψ = a 3 , pϕ = a 4 , pθ =

Mf ′2 ( θ ) + J

Θ(θ ) .

(19.2.103''')

Let be also the problem of the plane motion of a rigid straight bar, of mass M , attracted by elastic forces (of elastic constant k ) by the Ox1 -axis, considered in Sects. 18.3.2.7 and in 19.1.3.5. Hamilton’s function is given by (19.1.97), where ρ1′ and ρ2′ (the co-ordinates of the mass centre) and the angle of rotation θ are the generalized co-ordinates, while I is the central moment of inertia with respect to an axis normal to the plane of motion (see Fig. 18.12 too). The Hamilton-Jacobi equation will be

( )

1 ⎧ 1 ⎡ ∂S ⎞2 ⎛ ∂S ⎞2 ⎤ 1 ∂S S + ⎨ ⎢ ⎛⎜ +⎜ ⎟ ⎥+ 2 ⎩ M ⎣ ⎝ ∂ρ1′ ⎟⎠ ⎝ ∂ρ2′ ⎠ ⎦ I ∂θ

2

⎫ + k ( M ρ2′2 + I sin2 θ ) ⎬ = 0. ⎭ (19.2.104)

The constraints are scleronomic and ρ1′ is a cyclic co-ordinate, so that the complete integral is of the form S = − ht + a1 ρ1′ + S1 ( ρ2′ ) + S 2 ( θ ) .

Hamiltonian Mechanics

211

Replacing in the equation (19.2.104), we obtain −h +

1⎧ 1 ⎨ 2 ⎩M

⎡ 2 ⎛ ∂S1 ⎞2 ⎤ 2 ⎫ 2 ⎢ a1 + ⎜ ∂ρ ′ ⎟ ⎥ + kM ρ2′ ⎬ = −a2 , ⎝ ⎠ 2 ⎣ ⎦ ⎭ 1 ⎡ 1 ⎛ ∂S 2 ⎞2 2 ⎤ 2 ⎜ ⎟ + kI sin θ ⎥ = a2 , 2 ⎢⎣ I ⎝ ∂θ ⎠ ⎦

where a2 is the third essential constant; there results thus the complete integral S = − ht + a1 ρ1′ +

∫

R ( ρ2′ ) dρ2′ +

∫

Θ ( θ ) dθ ,

(19.2.105)

where we have denoted

R ( ρ2′ ) = 2 M ( h − a22 ) − a12 − kM 2 ρ2′2 , Θ ( θ ) = I ( 2a22 − kI sin2 θ ) .

(19.2.105')

The trajectory of the representative point P ∈ Λ3 is given by the equation dρ2′

ρ1′ − a1 ∫

∫

dρ2′ R ( ρ2′ )

−

I M

R ( ρ2′ )

∫

= b1 ,

(19.2.106)

b dθ = − 2 , 2 Ma2 Θ(θ )

while the motion on the trajectory is specified by 2M ∫

dρ2′ R ( ρ2′ )

= t − t0 .

(19.2.106')

Eliminating the integral between the first equation (19.2.106) and the equation (19.2.106'), we get ρ1′ =

a1 ( t − t0 ) + b1 , 2M

(19.2.107)

hence the projection of the mass centre on the Ox1 -axis has a uniform motion. The integral with respect to ρ2′ leads to ρ2′ = A cos ω ( t − t0 ) , A =

1 M k

2 M ( h − a22 ) − a12 , ω =

1 k; 2

(19.2.107')

the constants a1 and a2 must verify the positiveness of the quantity under the radical (we have k > 0, h > 0 too). The integral with respect to θ is an elliptic one. Obviously, these results correspond to those in Sect. 19.1.3.5, where one eliminates the

212

MECHANICAL SYSTEMS, CLASSICAL MODELS

generalized momenta p2 and pθ between (19.1.97) and (19.1.97') ( θ verifies the equation of the mathematical pendulum); moreover, the respective formulae allow then to calculate these generalized momenta.

Chapter 20 Variational Principles. Canonical Transformations We will present first the main variational principles, which allow a formulation of the motion problems of mechanical systems in the configuration space Λs , as well as in the phase space Γ 2s . Then, we consider the canonical transformations which let invariant the equations of motion in the space Γ 2s , as well as the variational principles from which they are deduced (their canonical form). A special attention is given to Noether’s theorem, obtaining thus first integrals for the equations of motion.

20.1 Variational Principles The differential principles of mechanics (with a local character) have been presented in Sect. 6.2.1.4 for a particle and in Sect. 11.1.2.10 for a discrete mechanical system of particles. New differential principles, in a global form, have been introduced in Sects. 12.1.1.2 and 20.1.2 for continuous mechanical systems and in Sects. 14.1.1.10 and 20.1.2.1 for the free rigid solid and for the rigid solid with constraints, respectively. In Sect. 18. the principles of Newton, d’Alembert, Gauss and Hertz are stated in the space E 3n , as well as various forms of the principle of virtual work (d’Alembert-Lagrange, Jourdain, Gauss a.s.o.) in the space Λs (Newton, I., 1686–1687; Alembert d’, 1743; Hertz, H., 1894). Unlike the differential principles, the variational principles of mechanics have a global character – being integral principles; they do not reply to certain causality conditions and do not correspond to an actual (present) state of the mechanical system, but contain in them the past as well the future of this one. In connection with the variational principles, we can mention the names of W.R. Hamilton, M.V. Ostrogradskiĭ, J.-L. Lagrange, P.-L. Moreau de Maupertuis, K.-G.-J. Jacobi, O. Hölder, A. Voss etc (Hamilton, W.R., 1890; Ostrogradskiĭ, M.V., 1946; Lagrange, J.-L., 1788; Jacobi, C.G.J., 1882; Voss, A., 1901). We notice that the basic principles of Newtonian mechanics (see Sect. 1.1.2.5) have a physical, mechanical meaning, their formulation being thus adequate. In the frame of Lagrangian and of Hamiltonian mechanics we are led to an analytical formulation of those principles, with a tendency of universality, even if we are situated – sometimes – in the frame of particular mechanical systems; obviously, this analytical formulation has also a pregnant mechanical substratum, even this one is not evident from the very beginning. The analytical principles of mechanics try, in general, to contain – besides the laws of Newtonian mechanics – also other natural laws (corresponding to a thermal

P.P. Teodorescu, Mechanical Systems, Classical Models, © Springer Science+Business Media B.V. 2009

213

MECHANICAL SYSTEMS, CLASSICAL MODELS

214

or electromagnetic field, or to some optical and acoustical phenomena a.s.o.); in fact, this is one of the tendencies of actual physics: unification of mechanical and physical phenomena in a unitary analytical formulation. The model used for these formulations is that of Fermat’s principle, governing geometrical optics, the variational method of calculus being essential. These analytical principles can have a differential character (as the principle of virtual work) or an integral character; in the last case, we study the behaviour of the mechanical system as a whole. We apply certain variations to the trajectories of the particles which constitute the mechanical system (or to the trajectory of the representative point), defining thus varied (admissible) paths; to a point on the true (real) trajectory will correspond, univocally, a point on the varied path. If the chronology of the motion is the same on both trajectories, then the variation is synchronous; otherwise, it is asynchronous. We pass in review some results having a general character (from a mathematical point of view, as well as concerning variational principles). We present then Hamilton’s and Maupertuis’s principles, and some applications of them to continuous mechanical systems; we consider also other variational principles in the frame of a general theory.

20.1.1 Mathematical Preliminaries We present in what follows some preliminaries having a mathematical character, concerning variational calculus, useful for the general form which can have the corresponding principles; the synchronous form as well as the asynchronous form can be put thus into evidence. 20.1.1.1 General Considerations In Sect. 7.2.1.4 we have made concise considerations concerning equations leading to the extremal curves in case of some particular functionals. In case of variational principles of mechanics, we take again the problem for the functionals of the form I ( x1 , x 2 ,..., xn ) ≡

t1

∫t

F ( t ; x1 ( t ) , x 2 ( t ) ,..., xn ( t ) , x1 ( t ) , x 2 ( t ) ,..., xn ( t ) ) dt ,

0

(20.1.1) where x j = x j ( t ) , j = 1, 2,..., n , are the parametric equations of a curve C in \n ; the co-ordinates x j can be, in particular, the co-ordinates of a particle in E 3 , the coordinates of a discrete mechanical system in E 3n , or the co-ordinates of a representative point in Λs or in Γ 2s . Corresponding to the definition of a functional, to every point P of C corresponds a real number I ( x1 , x 2 ,..., xn ) , by the integral (20.1.1); we admit that the functions x j ( t ) are of class C 2 on the interval [ t0 , t1 ] , the function F being also of class C 2 with respect to its arguments. In the basic problem one must determine the functions x j ( t ) for which I realizes an extremum (maximum or minimum). By convention, the geometric image of the functions x j ( t ) is called a real (true) curve C; starting from the point P of the curve C, we will define the point P ′ , hence

Variational Principles. Canonical Transformations

215

the varied path C ′ , using functions of the form x j′ = x j′ ( t ; ε ) , t ∈ [ t0 , t1 ] , of class

C 2 , the parameter ε varying in an interval containing the origin. We suppose that, along these varied paths, the conditions imposed in the problem, corresponding to the real (natural) curve, are not violated; if the latter one is obtained for ε = 0 , then we take x j′ ( t ;0 ) = x j ( t ) , so that x j′ ( t ) = x j ( t ) + εδx j ( t ) , j = 1, 2,..., n ,

(20.1.2)

where the variations δx j ( t ) are functions of class C 2 , while ε is a small parameter. It is convenient to assume that all varied paths C ′ are with fixed ends (they have the same origin P 0 and the same extremity P 1 ( x j′ ( t0 ) = x j ( t0 ) , x j′ ( t1 ) = x j ( t1 ) ) (see Fig. 20.1); hence, δx j ( t0 ) = δx j ( t1 ) = 0 , j = 1, 2,..., n .

(20.1.2')

If the points P and P ′ are in motion on the curve C and on the path C ′ , respectively, with the same chronology (the same time t ), we say that we have to do with a synchronous variation (denoted by δx j ); otherwise, the variation is asynchronous (denoted by Δx j ), the corresponding chronology being specified by the time variable

Fig. 20.1 Varied paths with fixed ends

t ′ = t + εΔt ,

(20.1.2'')

where Δt is a function of class C 2 , of time t ; we can write

x j′ ( t ′ ) = x j ( t ) + εΔx j ( t ) , j = 1, 2,..., n .

(20.1.2''')

Obviously, for Δt = 0 we obtain a synchronous variation. 20.1.1.2 Case of Synchronous Variations Introducing the variation of the function F , we can write F ( t ; x j′ , x j′

)

= F ( t ; x j , x j ) + εδF in case of a synchronous variation; but a development into a Taylor series allows us to write

MECHANICAL SYSTEMS, CLASSICAL MODELS

216 F ( t ; x j′ , x j′

n

⎛ ∂F

) = F ( t ; x j , x j ) + ε ∑ ⎜ ∂x j =1 ⎝

δx j +

j

∂F ⎞ δx + O ( ε2 ) , ∂x j j ⎟⎠

so that, using both expressions, we obtain

δF =

∂ F ( t ; x j′ , x j′ ) . ∂ε ε =0

(20.1.3)

It results also

δI ( x1 , x 2 ,..., xn ) =

∂ I ( x1′ , x 2′ ,..., xn′ ) . ∂ε ε =0

(20.1.3')

Observing that δ I ( x1 , x 2 ,..., xn ) = δ ∫ F ( t ; x j ( t ) , x j ( t ) ) dt t1

t0

=

∂ t1 F ( t ; x j′ ( t ) , x j′ ( t ) ) dt ∂ε ∫ t0

ε =0

=

t1

∫t

0

∂ F ( t ; x j′ ( t ) , x j′ ( t ) ) dt , ∂ε ε=0

we have δ I ( x1 , x 2 ,..., x n ) =

t1

∫ δF ( t ; x j ( t ) , x j ( t ) ) dt .

(20.1.3'')

t0

If the functional

I ( x1 , x 2 ,..., xn )

has an extremum, then the function

I ( x1′ , x 2′ ,..., xn′ ) , considered to be a function of the parameter ε (denoted f ( ε ) ), takes an extreme value for ε = 0 . If f (0) is an extremum, then – necessarily – f ′(0) = ∂I ( x1′ , x 2′ ,..., xn′ ) / ∂ε ε = 0 = 0 for any variation δx j ; taking into account

(20.1.3'), there results δ I ( x1 , x 2 ,..., x n ) = 0 .

(20.1.4)

We can thus state Theorem 20.1.1 If the functional I ( x1 , x 2 ,..., xn ) has an extremum for x j = x j ( t ) , then – necessarily – the relation (20.1.4) takes place for any virtual variation δx j ,

j = 1, 2,..., n , of the accepted class (20.1.2). If the condition (20.1.4) is verified, then we will say – by definition – that the functional I ( x1 , x 2 ,..., xn ) is stationary on the varied paths obtained starting from the functions x j = x j ( t ) , j = 1, 2,..., n . To can associate a sufficient condition of extremum, one takes into account also the variation of second order δ 2 I ( x1 , x 2 ,..., xn ) = ∂ 2 I ( x1′ , x 2′ ,..., x n′ ) / ∂ε2 ε = 0 . One can

Variational Principles. Canonical Transformations

217

prove that to realize a maximum or a minimum for the functional I ( x1 , x 2 ,..., xn ) it is sufficient to have δ 2 I ( x1 , x 2 ,..., xn ) < 0 or, δ 2 I ( x1 , x 2 ,..., xn ) > 0 , respectively, for any variation δx j ( t ) , j = 1, 2,..., n . If we differentiate the relation (20.1.2) with respect to time, then we obtain x j′ = x j + εdδx j / dt , so that δx j =

d δx ; dt j

(20.1.5)

there results the operator relation δ

d d = δ, dt dt

(20.1.5')

which takes place only in case of synchronous variations. This permutability relation is replaced by a transitivity relation ( δ d − dδ ≠ 0 ) in case of asynchronous variations. Let f ( t ) be a continuous function on the interval [ t0 , t1 ] , with real values. One supposes that there exists τ ∈ ( t0 , t1 ) , so that f ( τ ) ≠ 0 (to fix the ideas, f ( τ ) > 0 ); there exists an interval ( τ 0 , τ1 ) ⊂ [ t0 , t1 ] for which τ ∈ ( τ 0 , τ1 ) , and f ( t ) > 0 if t ∈ [ τ 0 , τ1 ] . Let also be a function η ( t ) of class C 1 on the interval [ t0 , t1 ] , with real values, defined by ⎧ 0 ⎪⎪ η ( t ) = ⎨ ( t − τ 0 )2 ( t − τ1 )2 ⎪ ⎪⎩ 0

for t0 ≤ t < τ 0 , for τ 0 ≤ t ≤ τ1 , for τ1 < t ≤ t1 .

We have t1

∫t

f ( t ) η ( t ) dt =

0

τ1

∫τ

f ( t ) ( t − τ 0 )2 ( t − τ1 )2 dt > 0 ;

0

by reducing ad absurdum, we can state Theorem 20.1.2 (the basic lemma of variational calculus; J.-L. Lagrange). Let f : [ t0 , t1 ] → \ be a continuous function. If t1

∫t

f ( t ) η ( t ) dt = 0

(20.1.6)

0

for any function η : [ t0 , t1 ] → \ of class C 1 , vanishing at the ends of the interval ( η ( t0 ) = η ( t1 ) = 0 ), then f ( t ) = 0 on the interval [ t0 , t1 ] . This result takes place also for η ∈ C 2 . In case of synchronous variations, the relation (20.1.3'') leads to

MECHANICAL SYSTEMS, CLASSICAL MODELS

218 δI ( x1 , x 2 ,..., x n ) =

t1 n

⎛ ∂F

δx j ⎜ ∫t ∑ j = 1 ⎝ ∂x j

+

0

∂F ⎞ δx dt , ∂x j j ⎟⎠

while the relation (20.1.5) allows to write (integration by parts) δI ( x1 , x 2 ,..., x n ) =

∂F ∑ ∂x j δx j j =1 n

t1

+

t1 n

⎡ ∂F

⎢ ∫t ∑ j = 1 ⎣ ∂x j

−

0

t0

d ⎛ ∂F ⎞ ⎤ δx d t ; dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

(20.1.7)

taking into account the conditions (20.1.2'), there results δI ( x1 , x 2 ,..., x n ) =

t1 n

⎡ ∂F

⎢ ∫t ∑ j = 1 ⎣ ∂x j 0

−

d ⎛ ∂F ⎞ ⎤ δx d t . dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

(20.1.7')

Imposing the stationarity condition (20.1.4), taking a non-zero variation δx j (all other variations vanishing), and using the basic lemma of the variational calculus, we can show that the parenthesis multiplying δx j must vanish; using this reasoning for any j = 1, 2,..., n , we obtain the Euler–Lagrange equations (denomination which puts in evidence the contribution of Euler (1744) and of Lagrange (1760) to the development of variational calculus) (Euler, L., 1955–1974)

∂F d ⎛ ∂F ⎞ − = 0 , j = 1, 2,..., n . ∂x j dt ⎜⎝ ∂x j ⎟⎠

(20.1.7'')

Corresponding to the results in Sect. 18.2.3.4, we notice that for F = 0 these equations admit the Jacobi’s first integral n

∂F

∑ ∂x j x j

−F =h,

(20.1.8)

j =1

where h is an integration constant. In general, we observe that in case of arbitrary synchronous varied paths (without fixed ends), we have δI ( x1 , x 2 ,..., x n ) =

∂F ∑ ∂x j δx j j =1 n

t1

,

(20.1.7''')

t0

along the integral curves of Euler–Lagrange equations. The problem of a brachistochrone curve, formulated by J. Bernoulli in 1696 and studied in Sect. 7.2.1.5, is the first problem of a variational nature which has been considered. 20.1.1.3 Case of Functionals Which Depend on Functions of Several Variables The results obtained above can be used successfully in case of discrete mechanical systems. To study continuous mechanical systems one must introduce a functional of the form

Variational Principles. Canonical Transformations

I (u ) =

t1

∫t

219

Fd t ,

(20.1.9)

0

where F =

∫Ω F ( t ; x1 , x 2 , x 3 , u , u , u1 , u2 , u3 ) dΩ ,

(20.1.9')

u = u ( x1 , x 2 , x 3 ; t ) being a function of class C 2 . Here Ω is a three-dimensional domain (eventually the support of a continuous mechanical system), which can be also two- or one-dimensional, dΩ = dx1 dx 2 dx 3 is the support element, while F is a density (corresponding to the unit of support); one has denoted u = ∂u / ∂t , ui = ∂u / ∂x i , i = 1, 2, 3 . Passing to a varied path, we introduce the admissible functions u ′ ( x1 , x 2 , x 3 ; t ) = u ( x1 , x 2 , x 3 ; t ) + δu ( x1 , x 2 , x 3 ; t ) , where δu are

functions of class C 2 , vanishing on the space frontier ∂Ω , as well as on the time one t0 , t1 . As in the previously considered synchronous case, we admit that the independent variables x1 , x 2 , x 3 , and t as well as their limits do not have variations. As in the previously subsection, we can prove Theorem 20.1.3 (the second basic lemma of the variational calculus). Let be a function f ( x1 , x 2 , x 3 ) , continuous on the domain Ω and with real values. If

∫ Ω f ( x1 , x 2 , x 3 ) η ( x1 , x 2 , x 3 ) dΩ = 0

(20.1.10)

for any function η of class C 1 , vanishing on Ω , then f ( x1 , x 2 , x 3 ) = 0 on the domain Ω . This result is valid also for η ∈ C 2 . Expanding into a Taylor series, we can write (we use permutability relations of the form (20.1.5')) δF =

∂F ∂F ∂ ∂F ∂ δu + δu + δu , ∂u ∂u ∂t ∂ui ∂x i

where a summation convention from 1 to 3 is used. We notice that t1

∫t

0

∂F ∂ ∂F ( δu ) dt = δu ∂u ∂t ∂u

∂F ∂ ( δu ) dΩ = ∂x i

∫Ω ∂ui

t1 t0

−

t1

∫t

0

∂ ⎛ ∂F ⎞ ⎜ ⎟ δudt , ∂t ⎝ ∂u ⎠

∂ ⎛ ∂F δu ⎞⎟ dΩ − ⎠

∫Ω ∂xi ⎜⎝ ∂ui

∂ ⎛ ∂F ⎞ ⎟ δudΩ . ⎠

∫Ω ∂xi ⎜⎝ ∂ui

If we apply a formula of flux-divergence type (see App., Subsec. 2.3.3) to the first integral in the second member of the second relation and if we notice that δu ∂Ω = δu t0 = δu t1 = 0 , then, finally, it follows

MECHANICAL SYSTEMS, CLASSICAL MODELS

220 δI =

t1

t1

∫ δF dt =

t1

⎡ ∂F − d ⎛ ∂F ⎞ − d ⎛ ∂F ⎞ ⎤ δudΩdt ; ⎜ ⎟ ∂u dt ⎝ ∂u ⎠ dx i ⎝⎜ ∂ui ⎠⎟ ⎦⎥

∫ ∫ δF dΩdt = ∫t ∫Ω ⎢⎣

t0 Ω

t0

0

we have introduced the derivatives d / dt and d / dxi , i = 1, 2, 3 , to notice that the differentiation is made taking into account the implicit dependence (by the agency of the function u ) of F on the independent variables x1 , x 2 , x 3 , t . Applying the second basic lemma of the variational calculus and by a reasoning similar to that of the previous subsection, we obtain the Euler–Ostrogradskiĭ equations in the form ∂F d ⎛ ∂F ⎞ d ⎛ ∂F ⎞ − ⎜ ⎟− ⎜ ⎟ = 0. ∂u dt ⎝ ∂u ⎠ dxi ⎝ ∂ui ⎠

(20.1.11)

Obviously, in case of a n-dimensional domain one obtains a similar result (summation from 1 to n). Also, in case of several dependent functions u α , α = 1, 2,...,m , there result the equations ∂F d ⎛ ∂F ⎞ d ⎛ ∂F ⎞ − = 0 , α = 1, 2,...,m . ⎜ ⎟− ∂u α dt ⎝ ∂u α ⎠ dx i ⎜⎝ ∂uiα ⎟⎠

(20.1.11')

20.1.1.4 Case of Asynchronous Variations We suppose that the variations which lead from the motion on the true curve to the motion on a varied path are such that the two motions are not synchronous; we will thus consider the functional (20.1.1), using the notations introduced in Sect. 20.1.1.1 for asynchronous variations. The quantities Δx j ( t ) play the rôle of possible displacements, of class C 2 , which satisfy linear constraint relations of the form n

∑ Akj Δx j j =1

+ Ak 0 Δt = 0 , k = 1, 2,..., m ,

(20.1.12)

where Akj = Akj ( x1 , x 2 ,..., xn ; t ) , j = 1, 2,..., n , and Ak 0 = Ak 0 ( x1 , x 2 ,..., xn ; t ) are functions of class C 1 . We consider also the finite form of these relations n

∑ Akj x j j =1

+ Ak 0 = 0 , k = 1, 2,..., m ;

(20.1.12')

eliminating the parameter Ak 0 , we find constraint relations of the form n

∑ Akj δx j j =1

= 0 , k = 1, 2,..., m , δx j = Δx j − x j Δt , j = 1, 2,..., n ,

(20.1.12'')

where δx j are virtual displacements. We notice that, expanding into a Taylor series, one can write

Variational Principles. Canonical Transformations

221

x j′ ( t + εΔt ) = x j′ ( t ) + εx j′ ( t ) Δt + O ( ε2

= x j ( t ) + εδx j ( t ) + εx j ( t ) Δt + O ( ε

2

)

),

taking into account (20.1.2); using also the relations (20.1.2''), (20.1.2''') and neglecting O ( ε2 ) , we find again the second relation (20.1.12''). The velocity on the varied path is given by d x j + ε Δx j dx j′ d d t d x j′ = = = x j + ε Δx j 1 − ε Δt + ... d dt ′ dt dt 1 + ε Δt dt d d = x j + ε Δx j − x j Δt + O ( ε2 ) . dt dt

(

)(

(

)

)

Observing that x j′ = x j + εΔx j and neglecting O ( ε2 ) , we obtain the operator relation Δx j =

d d Δx j − x j Δt , dt dt

(20.1.13)

which, in the asynchronous case, replaces the relation (20.1.5); this result corresponds to d ∂ x j + ε dt Δx j Δx j = d ∂ε 1 + ε Δt dt

(20.1.13') ε =0

and can be written also in the form Δx j =

d δx + xj Δt = δx j + xj Δt , dt j

(20.1.13'')

if we take into account (20.1.5) and (20.1.12''). By a change of variable, we may calculate the asynchronous variation of the functional (20.1.1) in the form ( t0′ = t0 + εΔt0 , t1′ = t1 + εΔt1 ) ∂ t1′ F ( t ; x j′ , x j′ ) dt ′ ∂ε ∫t0′ ε=0 ∂ ⎡ d dt F ( t + εΔt ; x j + εΔx j , x j + εΔx j ) 1 + ε Δt ⎤ dt ∂ε ⎣⎢ ⎦⎥ ε = 0 t1 ∂ d = ∫ dt , [ F ( t ) + εΔF ( t ) ] 1 + ε Δt t0 ∂ε dt ε=0 ΔI ( x1 , x 2 ,..., x n ) =

=

t1

∫t

0

{

(

(

)}

)

MECHANICAL SYSTEMS, CLASSICAL MODELS

222 so that ΔI ( x1 , x 2 ,..., x n ) =

t1

∫t

0

( ΔF + F ddt Δt ) dt .

(20.1.14)

This relation is reduced to the relation (20.1.3'') if Δt = 0 . Further, we may write ΔI ( x1 , x 2 ,..., x n ) =

=

n

∂F

∑ ∂x j Δx j j =1

t1 t0

t1

∫t

0

⎛ ⎜ F Δt + ⎝

∂F

n

j =1

j =1

t1

⎛ ⎞ ∂F x j − F ⎟ Δt −⎜∑ ⎝ j =1 ∂x j ⎠ n

+∫

⎡ ∂F

t1 n

t0

∑ ⎢⎣ ∂x j j =1

∂F

n

∑ ∂x j Δx j + ∑ ∂x j Δx j

−

+

⎡

n

+F

d ⎛ ∂F

x j ⎜ ∫t ⎣⎢ ∑ j = 1 dt ⎝ ∂x j t1 0

t0

⎞ d Δt ⎟ dt dt ⎠

⎤ ⎞ − F ⎟ + F ⎥ Δtdt ⎠ ⎦

d ⎛ ∂F ⎞ ⎤ Δx dt , dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

where we took into account the operator relation (20.1.13). But n

d ⎛ ∂F

∑ dt ⎜⎝ ∂x j x j j =1

⎞ − F ⎟ + F = ⎠

⎡ d ⎛ ∂F ⎞ ∂F ⎟− ⎠ ∂x j j =1

⎤ ⎥ x j , ⎦

n

∑ ⎣⎢ dt ⎝⎜ ∂x j

and we are thus led to t

⎡ n ∂F ⎛ n ∂F ⎞ ⎤1 ΔI ( x1 , x 2 ,..., x n ) = ⎢ ∑ Δx j − ⎜ ∑ x j − F ⎟ Δt ⎥ ⎣ j =1 ∂x j ⎝ j =1 ∂x j ⎠ ⎦t 0 +∫

t1 n

t0

⎡ ∂F

∑ ⎢⎣ ∂x j

−

j =1

d ⎛ ∂F ⎞ ⎤ δx d t , dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

(20.1.15)

on the basis of the relation (20.1.13); the second relation (20.1.12'') allows us to write t

⎡ n ∂F ⎤1 ΔI ( x1 , x 2 ,..., x n ) = ⎢ ∑ δx j + F Δt ⎥ + ⎣ j =1 ∂x j ⎦t 0

t1 n

⎡ ∂F

⎢ ∫t ∑ j = 1 ⎣ ∂x j

−

0

d ⎛ ∂F ⎞ ⎤ δx d t . dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

(20.1.15') If the functions Δx j (or δx j ), j = 1, 2,..., n , and Δt vanish at the ends of the interval (varied paths with fixed ends, for which the time is the same at both ends), it follows ΔI ( x1 , x 2 ,..., x n ) =

t1 n

⎡ ∂F

⎢ ∫t ∑ j =1 ⎣ ∂x j 0

−

d ⎛ ∂F ⎞ ⎤ δx d t , dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

(20.1.15'')

as in the synchronous case. Taking into account the constraint relations (20.1.12'') and using Lagrange’s multipliers λk , k = 1, 2,..., m , we may write

Variational Principles. Canonical Transformations

ΔI ( x1 , x 2 ,..., x n ) =

t1 n

⎡ ∂F

⎢ ∫t ∑ j = 1 ⎣ ∂x j

−

0

223

d ⎛ ∂F ⎞ + dt ⎝⎜ ∂x j ⎠⎟

⎤

m

∑ λk Akj ⎦⎥δx j dt .

k =1

By the usual method, one determines the multipliers λk , k = 1, 2,..., m , m < n , so that the m square brackets vanish; there remain n − m brackets which multiply n − m virtual displacements, now independent. As in the synchronous case, we prescribe the stationarity condition ΔI ( x1 , x 2 ,..., xn ) = 0 ;

(20.1.16)

applying the basic lemma of the variational calculus, we obtain the Euler–Lagrange equations with multipliers ∂F d ⎛ ∂F ⎞ − + dt ⎜⎝ ∂x j ⎟⎠ ∂x j

m

∑ λk Akj

= 0 , j = 1, 2,..., n ,

(20.1.16')

k =1

corresponding to the true curve C. In general, in case of arbitrary asynchronous varied curves (without fixed ends and for which the time is not the same at the ends of the interval), we have t

⎡ n ∂F ⎛ n ∂F ⎞ ⎤1 x j − F ⎟ Δt ⎥ ΔI ( x1 , x 2 ,..., x n ) = ⎢ ∑ Δx j − ⎜ ∑ ⎣ j =1 ∂x j ⎝ j =1 ∂x j ⎠ ⎦t 0 t1

⎡ n ∂F ⎤ = ⎢∑ δx j + F Δt ⎥ ⎣ j =1 ∂x j ⎦t0

(20.1.15''')

along the integral curves of the Euler–Lagrange equations with multipliers. Starting from the relations (20.1.7) and (20.1.15''), we obtain the relation connecting the asynchronous to the synchronous variation of a functional in the form

ΔI = δI + F Δt

t1 t0

.

(20.1.17)

If Δt0 = Δt1 = 0 (or, more general, if F ( t0 ) = F ( t1 ) and Δt0 = Δt1 ), there results ΔI = δI ,

(20.1.17')

the asynchronous and synchronous variations being equal; thus, the difference between the effects of those variations disappears. We notice also that, in case of varied paths with fixed ends ( Δx j ( t0 ) = Δx j ( t1 ) = 0 , j = 1, 2,..., n ) for which F = 0 (the function F does not depend explicitly on time), the Euler–Lagrange equations entail also Jacobi’s first integral (20.1.18). The relation (20.1.15) shows that one can write

MECHANICAL SYSTEMS, CLASSICAL MODELS

224

ΔI ( x1 , x 2 ,..., x n ) = − h ( Δt1 − Δt0 )

(20.1.18)

along the integral curves of Euler–Lagrange equations; to obtain ΔI = 0 , one must have Δt0 = Δt1 , hence the motion on the varied paths must have the same duration as on the true curve. This last condition concerning the asynchronous variations is somewhat restrictive; we can also write ΔI ( x1 , x 2 ,..., xn ) + h ( Δt1 − Δt0 ) t1

= Δ ∫ [ F ( x1 ( t ) , x 2 ( t ) ,..., x n ( t ) , x1 ( t ) , x2 ( t ) ,..., xn ( t ) ; t ) + h ] dt = 0 , t0

(20.1.18') a relation which can be used in various particular cases. We will estimate t1

∫t

ΔFdt =

0

t1

∫t

0

⎛ n ∂F ⎜ ∑ ∂x Δx j + ⎝ j =1 j

∂F x j Δt ∂ j = 1 x j n

−∑

t1

+

n

−

d ⎛ ∂F ⎞ ⎤ Δx j dt , dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥

d ⎛ ∂F

⎞

t1

t0

⎤

 ⎜ ∂x x j ⎟ + F ⎥ Δtdt ∫t ⎢⎣ ∑ d t j ⎝ ⎠ j =1 ⎦ t1

⎡ ∂F

⎢ ∫t ∑ j =1 ⎣ ∂x j

+

⎡

∂F ∑ ∂x j Δx j j =1 n

0

t0

t1 n

⎞ ∂F ∑ ∂x j Δx j + F Δt ⎟ dt = j =1 ⎠ n

0

using the variation (20.1.14) without the term Fd ( Δt ) / dt . We can write t1

∫t

0

⎛ ⎜ ΔF + ⎝

⎞ ∂F d ∑ ∂x j x j dt Δt − F Δt ⎟ dt = j =1 ⎠ n

+∫

⎡ ∂F

t1 n

t0

∑ ⎢⎣ ∂x j j =1

−

∂F ∑ ∂x j Δx j j =1 n

t1

t0

d ⎛ ∂F ⎞ ⎤ Δx j dt dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥

(20.1.19)

too. It follows t1

∫t

0

⎛ ⎜ ΔF + ⎝

⎞ ∂F d ∑ ∂x j x j dt Δt − F Δt ⎟ dt = j =1 ⎠ n

∂F ∑ ∂x j Δx j j =1 n

t1

(20.1.19') t0

along the integral curves of Euler–Lagrange equations. If the varied paths are with fixed ends ( Δx j ( t0 ) = Δx j ( t1 ) = 0 ), then we can state Theorem 20.1.4 (O. Hölder). In the case of asynchronous varied paths with fixed ends, the functional (20.1.1) verifies the relation t1

∫t

0

⎛ ⎜ ΔF + ⎝

n

∂F

d

⎞

∑ ∂x j x j dt Δt − F Δt ⎟ dt j =1

⎠

=0

(20.1.19'')

Variational Principles. Canonical Transformations

225

along the integral curves of the corresponding Euler–Lagrange equations. In case of some constraint relations of the form (20.1.12), one uses, as above, the methods of Lagrange’s multipliers.

20.1.2 The General Integral Principle Following the ideas of the previous section, we will state the general form of a variational principle and of the general integral principle; these results will be given for synchronous as well as for asynchronous variations. 20.1.2.1 Preliminary Considerations Let be a discrete mechanical system S of n particles and Pi a particle of this system, of position vector ri with respect to an inertial frame of reference, subjected to the action of the resultant Fi of the given forces (by given forces we intend external as well as internal ones); it is obvious that i = 1, 2,..., n . In the time interval [ t0 , t1 ] , the particle Pi describes the true curve C i of equation ri = ri ( t ) . Let us suppose that the particle Pi is displaced by εΔri , where ε is a small parameter, till the point Pi ′ of position vector ri′ = ri + εΔri , i = 1, 2,..., n .

(20.1.20)

Fig. 20.2 Varied paths

From the set of possible displacements Δri ( t ) of class C 2 , we choose those which, univocally, lead to the point Pi ′ , the locus of which is a varied path C i′ (see Fig. 20.2); the motion on this path corresponds to the proper time t ′ , given by (20.1.2''). One obtains thus an infinity of varied paths, the motion on those ones being, in general, asynchronous ( Δt ≠ 0 ). If Δt = 0 , the variations are synchronous, and ri′ = ri + εδ ri , i = 1, 2,..., n ,

(20.1.20')

MECHANICAL SYSTEMS, CLASSICAL MODELS

226

where δri are virtual displacements (synchronous variations). We assume that the possible displacements verify the linear constraint relations n

∑ αki

i =1

⋅ Δri + αk 0 Δt = 0 , k = 1, 2,..., m ,

(20.1.21)

where αki = αki ( r1 , r2 ,..., rn ; t ) , i = 1, 2,..., n , and αk 0 = αk 0 ( r1 , r2 ,..., rn ; t ) are functions of class C 1 . Considering also the finite form of the constraint relations n

∑ αki

i =1

⋅ ri + αk 0 = 0 , k = 1, 2,..., m ,

(20.1.21')

and eliminating the parameter αk 0 , we find constraint relations of the form n

∑ αki

i =1

⋅ δ ri = 0 , k = 1, 2,..., m , δ ri = Δri − vi Δt , i = 1, 2,..., n ,

(20.1.21'')

where, obviously, vi = ri are real velocities, while δ ri are virtual displacements. The velocity on the varied path is given by d dri′ vi + ε dt Δri d d vi′ = = = vi + ε Δri 1 − ε Δt + ... d ′ dt dt dt 1 + ε Δt dt d d = vi + ε Δr − vi Δt + O ( ε2 ) . dt i dt

(

(

)(

)

)

We notice that vi′ = vi + εΔvi and we neglect O ( ε2 ) ; the asynchronous variation of the velocity is thus given by Δ vi =

d d Δri − vi Δt , i = 1, 2,..., n . dt dt

(20.1.22)

Because d ( vi Δt ) / dt = vi dΔt / dt + ai dt , where ai is the acceleration of the particle Pi , we can write this relation also in the form Δ vi =

d δr + ai Δt , i = 1, 2,..., n , dt i

(20.1.22')

where we took into consideration the second relation (20.1.21''). If Δt = 0 , then one obtains the synchronous variation δ vi =

d δr , i = 1, 2,..., n , dt i

(20.1.23)

Variational Principles. Canonical Transformations

227

and the operator relation (20.1.5'). Hence, the relation between the two variations (asynchronous and synchronous) will be of the form dvi , i = 1, 2,..., n . dt

Δvi = δ vi + ai Δt , ai =

(20.1.22'')

The synchronous variation of the kinetic energy will be given by

1 n mi ( vi′2 − vi2 2 i∑ =1

εδT = T ′ − T =

n

1

) = 2 ∑ mi ⎡⎣ ( vi i =1

+ εδ vi )2 − vi2 ⎤⎦

n

= ε ∑ mi vi ⋅ δ vi + O ( ε2 ) , i =1

so that δT =

n

∑ mi vi

i =1

⋅ δ vi ;

(20.1.24)

analogously, we obtain the asynchronous variation ΔT =

n

∑ mi vi

i =1

⋅ Δ vi .

(20.1.24')

Performing a scalar multiplication of the relation (20.1.22'') by mi vi summing for all values of the index i , taking into account (20.1.24), (20.1.24'), and observing that d ( vi2 / 2 ) / dt = vi ⋅ ai , we can write the relation between the asynchronous and the synchronous variation of the kinetic energy in the form ΔT = δT +

dT Δt . dt

(20.1.25)

In case of quasi-conservative given forces, which derive from a quasi-potential U = U ( r1 , r2 ,..., rn ; t ) , we obtain the asynchronous variation ΔU =

n

∑ ∇iU ⋅ Δri

+ U Δt , U =

i =1

∂U ; ∂t

(20.1.26)

the synchronous variation is of the form δU =

n

∑ ∇iU ⋅ δri .

(20.1.26')

i =1

Observing that dU = dt

n

∑ ∇iU ⋅ vi

i =1

+ U

MECHANICAL SYSTEMS, CLASSICAL MODELS

228

and taking into account the second relation (20.1.21''), one obtains the relation between the asynchronous and the synchronous variations of the quasi-potential in the form ΔU = δU +

For the mechanical energy analogously,

dU Δt . dt

(20.1.27)

E = T − U = T + V , V = −U , one obtains,

ΔE = δE +

dE Δt , dt

(20.1.28)

while for Lagrange’s kinetic potential L = T + U = T − V we can write ΔL = δL +

dL Δt . dt

(20.1.28')

Generically, we denote by E a quantity of energetic nature, e.g., T ,U , E , L or any other linear combination (with numerical coefficients) of them; as a matter of fact, two of these quantities can be considered as basic ones (for instance T and L ), while the other quantities are expressed by means of them. We will thus have ΔE = δE +

dE Δt . dt

(20.1.28'')

Let us introduce the functional

A =

t1

∫t

Edt ,

(20.1.29)

0

called action; the quantity Edt will be the elementary action. We have thus to do with the kinetic action AT , corresponding to E = T , with the potential action AU , for E = U , with the Lagrangian action AL , corresponding to E = L , or with the mechanical action AE , for E = E . We mention that the magnitude 2 AT corresponds to the classical notion of action. Using the formula (20.1.14), we can express the asynchronous variation of the action A in the form t1

Δ A = Δ ∫ E dt = t0

t1

∫t

0

( ΔE + E ddt Δt ) dt ;

(20.1.29')

analogously, the corresponding synchronous variation is given by t1

δ A = δ ∫ E dt = t0

t1

∫t

δE dt ,

(20.1.29'')

0

where a permutation between the operator δ and the integral operator is possible. Taking into account (20.1.28''), we can express the asynchronous variation (20.1.29') in the form

Variational Principles. Canonical Transformations

Δ A = EΔt

t1 t0

+

t1

∫t

229

δE dt ;

(20.1.30)

0

hence, the relation between the asynchronous and the synchronous variations of an action is given by Δ A = δ A + EΔt

t1 t0

.

(20.1.30')

If Δt0 = Δt1 = 0 (or, more general, if E ( t0 ) = E ( t1 ) and Δt0 = Δt1 , the motion having the same duration on all varied paths), we obtain ΔA = δ A ,

(20.1.30'')

the asynchronous variation being equal to the synchronous one.

20.1.2.2 Two Forms of a General Integral Principle Starting from the principle of virtual work, which can be applied in case of ideal constraints (see Sect. 11.1.2.10), written in the form n

∑ Φi

i =1

⋅ δri = 0 , Φi = Fi − mi ai ,

(20.1.31)

where Fi are given forces (external and internal), and integrating with respect to time, between the limits t0 and t1 , we obtain t1 n

∫t i∑=1 Φi

⋅ δri dt = 0 .

(20.1.31')

0

Inversely, starting from (20.1.31') and using the basic lemma of Lagrange (the virtual displacements δri are independent or verify the constraint relations (20.1.21''), case in which one uses Lagrange’s multipliers λk , k = 1, 2,..., m , leading to the constraint Ri =

forces

m

∑ λk αki ,

i = 1, 2,..., n ),

we

find

k =1

Φi = 0

or

Φi + Ri = 0 ,

i = 1, 2,..., n , hence Newton’s equations of motion. The relations (20.1.31) and (20.1.31') are thus equivalent. We can calculate t1 n

mi ai ∫ t i∑ =1

⋅ δri dt =

0

=

n

t1

i =1

t0

∑ mi vi ⋅ δri

−

n

∑ ∫t

t1

i =1

mi

0

t1 n

∫t

0

t1 n d d ( vi ⋅ δ ri ) dt − ∫ ∑ mi vi ⋅ ( δ ri )dt t dt dt 0 i =1

∑ mi vi ⋅ δ vi dt =

i =1

n

t1

i =1

t0

∑ mi vi ⋅ δri

−

t1

∫t

where we took into account (20.1.23) and (20.1.24); on the other hand

0

δTdt ,

MECHANICAL SYSTEMS, CLASSICAL MODELS

230 n

∑ Fi

i =1

⋅ δri = δW ,

(20.1.32)

where δW is the virtual work of the external and internal given forces. Finally, the relation (20.1.31') leads to t1

∫t

( δW + δT ) dt =

n

∑ mi vi

i =1

0

⋅ δri

t1 t0

;

(20.1.33)

using only asynchronous variations, one can also write t1

∫t

0

(

)

t

1 n ⎡ ΔW + ΔT − P + dT Δt ⎤ dt = ⎡ m v ⋅ Δr − 2T Δt ⎤ , i ⎢∑ i i ⎥ ⎥⎦ dt ⎣⎢ ⎣ i =1 ⎦t 0

(20.1.33')

where we took into account (20.1.21'') and (20.1.25), where ΔW = δW + P Δt =

n

∑ Fi

i =1

⋅ Δri

(20.1.32')

is the asynchronous variation of the work of external and internal forces and where P =

dW = dt

n

∑ Fi

i =1

⋅ vi

(20.1.34)

is the power of the same given forces. In case of synchronous varied paths with fixed ends, the relation takes the form (20.1.33), hence t1

∫t

( δW + δT ) dt = 0 ,

(20.1.35)

0

equivalent to the relation (20.1.31'), that is to the principle of virtual work (20.1.31). Following V. Vâlcovici, we will call this relation the general integral principle, because – starting from (20.1.35) – we can obtain the most important variational (integral) principles, in case of synchronous variations (Vâlcovici, V., 1969, 1973). In the asynchronous case, we start from the relation (20.1.33') and consider asynchronous varied paths with fixed ends, for which Δt0 = Δt1 = 0 ; we obtain thus t1

∫t

0

(

)

⎡ ΔW + ΔT − P + dT Δt ⎤ dt = 0 . ⎥⎦ dt ⎣⎢

(20.1.33'')

We notice that the relation (20.1.33') can be written also in the form t1

∫t

0

(

)

⎡ ΔW + ΔT + dT − P Δt + 2T d Δt ⎤ dt = ⎢⎣ ⎥⎦ dt dt

n

∑ mi vi

i =1

⋅ Δri

t1

; t0

(20.1.33''')

Variational Principles. Canonical Transformations

231

one can obtain in this case a general integral principle, corresponding to asynchronous varied paths with fixed ends ( Δri ( t0 ) = Δri ( t1 ) = 0 , i = 1, 2,..., n ) in the form t1

∫t

0

(

)

⎡ ΔW + ΔT + dT − P Δt + 2T d Δt ⎤ dt = 0 . dt dt ⎣⎢ ⎦⎥

(20.1.35')

As it can be easily seen, in the above calculus an important rôle is played by the operator Δ = δ + Δt

d , dt

(20.1.36)

which is applied for the position vector ri and the velocity vi , as well as for all the quantities of energetical nature or of the nature of an action which occur (even the relation (20.1.30') can be obtained by applying this operator). As well, the operator can be used to pass from a synchronous to an asynchronous form of an integral or of a variational principle. The permutation relation of the form (20.1.5') does no more take place in this case; one must use the transitivity relation (see relation (20.1.22)) d d d d ( Δt ) . Δ−Δ = dt dt dt dt

(20.1.36')

20.1.3 Hamilton’s Principle We will present, in what follows, Hamilton’s, Hölder’s and Voss’s principles, corresponding to synchronous or asynchronous variations of true trajectories, respectively; some methods of calculus are thus put into evidence.

20.1.3.1 Case of Synchronous Variations We can write δW =

n

∑ ∇iU ⋅ δri

= δU

(20.1.37)

i =1

in case of quasi-conservative forces and of synchronous variations; taking into account the definition relation of the kinetic potential ( L = T + U ), the general integral principle (20.1.35) leads to t1

∫t

δL dt = 0 .

(20.1.38)

0

Using the permutability property (20.1.3'') between the operator δ and the integral, we have t1

δ AL = δ ∫ L dt = 0 . t0

(20.1.38')

MECHANICAL SYSTEMS, CLASSICAL MODELS

232

We can thus state that along the varied paths of a discrete mechanical system subjected to ideal constraints and to quasi-conservative given forces, the Lagrangian action AL is stationary (necessary condition) in case of synchronous variations; we notice that the mechanical system must be holonomic, corresponding to this type of variations. Let us assume that the motion of the mechanical system is described by means of generalized co-ordinates, so that L = L ( q1 , q 2 ,..., qs , q1 , q2 ,..., qs ; t ) ; in this case, starting from the relation (20.1.38) and using the results of Sect. 20.1.1.2 (Euler– Lagrange equations (20.1.7'')), we find again Lagrange’s equations of second kind (18.2.38). If the given forces do not derive from a quasi-potential, then n

∑ Fi

i =1

⋅

∂ri = Q j , j = 1, 2,..., s , ∂q j

and, starting from (20.1.35), we obtain (we use again the results of Sect. 20.1.1.1 and the summation convention for dummy indices) t1

∫t

0

d ⎛ ∂T ⎞ ⎡ ∂T ⎢ ∂q − dt ⎜ ∂q ⎟ + Q j ⎣ j ⎝ j ⎠

⎤ ⎥ δq j dt = 0 ; ⎦

(20.1.39)

applying the basic lemma of Lagrange, we find again the equations (18.2.29). In case of non-holonomic constraints, we use Lagrange’s multipliers method as in Sect. 20.1.1.4; there results (we assume to have the constraint relations (18.2.10)) t1

∫t

0

d ⎛ ∂T ⎞ ⎡ ∂T ⎢ ∂q − dt ⎜ ∂q ⎟ + Q j + ⎣ j ⎝ j ⎠

m

⎤

∑ λk akj ⎦⎥ δq j dt

k =1

= 0.

(20.1.39')

By the same reasoning as in the above mentioned subsection, we find again Lagrange’s equations with multipliers (18.2.32), (18.2.33). Hence, the relation (20.1.38') is sufficient (not only necessary) to specify the motion in the frame of the mentioned conditions. So, we state Theorem 20.1.5 (W.R. Hamilton). Among all possible motions of a mechanical system subjected to holonomic and ideal constraints and acted upon by quasi-conservative given forces on synchronous varied paths with fixed ends, only and only the motion (of the representative point on the true curve) for which the Lagrangian action is stationary takes place. We mention that the most times – not always – the Lagrangian action realizes a minimum for the actual motion (on the true curve). We have obtained this result in the form of a theorem, starting from the principle of the virtual work; but we can interpret this result also as a variational principle (Hamilton’s principle), which leads to Lagrange’s equations of motion. This principle was stated in 1834 by W.R. Hamilton for scleronomic constraints; it was extended by M.V. Ostrogradskiĭ in 1848 to the case of rheonomous constraints. One observes that, unlike differential principles (in which, to establish the motion at a given moment, one considers only the motion in a vicinity of this one), in case of variational principles the

Variational Principles. Canonical Transformations

233

motion of the mechanical system at a given moment is specified by its motion in the whole (finite) interval of time. Often, Hamilton’s principle is considered to be the basic principle of mechanics; indeed, classical mechanics as well as the relativistic one can be deduced starting from such a principle. As well, the theory of the electromagnetic field can be obtained from an analogous principle, as other theories of physical nature. Hence, the importance of those principles consists also in their expression in a unified form, corresponding to the tendency of extremum of phenomena in the nature. So, the variational principles of Hamilton’s type lead to “equations of motion” which can be Newton’s, Maxwell’s or Schrödinger’s equations; using a variational principle as basis of formulation, all these domains of physical nature have, in a certain measure, a structural analogy. If experimental results impose the change of physical matter in the theory corresponding to a certain domain, then this analogy can show similar changes in another domain. For instance, experiments made at the beginning of this century have put into evidence the necessity to quantify the electromagnetic radiation as well as the elementary particles; indeed, starting from the development of quantification methods for the mechanics of particles, one could construct a quantum electrodynamics. Although its importance from the point of view of the philosophy of mechanical systems, we must notice that Hamilton’s principle can – sometimes – lead to more intricate calculations from a practical point of view as a direct study based on Newton’s equation. If the given forces are not quasi-conservative, then one cannot introduce the function L , and if the mechanical system is not holonomic, then the operator δ cannot permute with the integral. In this case, in general, the varied paths are no more possible curves, and we must use the relation (20.1.35); the problem is no more a problem of variational calculus. We observe that we may distinguish between an integral principle and a variational one, the first one having a more general character. Indeed, by convention, an integral principle is deduced from the general integral principle (20.1.35), without the intervention of the variation of a functional (the variation of an action), this aspect appearing only under the integral sign; if the variation of a functional occurs, then we have to do with a variational principle. Perhaps, this distinction is conventional; however, we must mention that a variational principle is linked to a problem of extremum of a functional, allowing thus to use a rich methodology of calculus. If the constraints are non-holonomic but the given forces are quasi-conservative, then Hamilton’s principle is an integral one and has the form (20.1.38) (it is true that the virtual displacements are not independent), while if the constraints are holonomic, the given forces being quasi-conservative or not, then we find again the general integral principle (20.1.35), which – in fact – represents a more general integral form of Hamilton’s principle in a synchronous case. Thus, as one can see, Hamilton’s principle has a sufficient general character, even in the synchronous case. However, the difference (given by the non-holonomic or holonomic character of the ideal constraints) between the forms (20.1.38) and (20.1.38') of Hamilton’s principle is clearly put in evidence. We notice also that, in case of quasi-conservative forces deriving from a generalized potential, the virtual work of the given forces can be expressed in the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

234 δW =

=

n

∑ Fi

i =1

d ⎛ ∂U ⎞ ⎤ ⎡ ∂U δq ⋅ δri = Q j δq j = ⎢ − q t ⎝⎜ ∂q j ⎠⎟ ⎦⎥ j ∂ d ⎣ j

∂U d ⎛ ∂U d ⎛ ∂U ⎞ ∂U d ⎞ δq − δq + δq = δU − δq ; dt ⎜⎝ ∂q j j ⎟⎠ ∂q j j dt ⎜⎝ ∂q j j ⎟⎠ ∂q j dt j

in case of synchronous variations, for which the operator relation (20.1.5') is valid, the general integral principle becomes t1

∫t

( δU + δT ) dt =

0

∂U δq ∂q j j

t1

. t0

Further, assuming variations with fixed ends ( δq j ( t0 ) = δq j ( t1 ) = 0 , j = 1, 2,..., s ), there results that Hamilton’s variational principle maintains its form for any natural system (we can permute the operator δ with the integral sign). The property of extremum expressed by Hamilton’s principle is an intrinsic property; by way of consequence, it can be expressed in any system of co-ordinates. We will thus pass from the co-ordinates q j , u j = q j , j = 1, 2,..., s , in the space Λs , to the coordinates q j , p j , j = 1, 2,..., s , in the space Γ 2s ; a Legendre transformation permits it (see Sects. 19.1.1.2 and 20.1.1.3). But we must have some care, because the variations δq j and δu j are not independent; indeed, to given variations δq j there correspond variations of the generalized velocities of the form δu j =

In

this

case,

let

us

d δq , j = 1, 2,..., s . dt j

consider

the

Lagrangian

(20.1.40) action

for

which

L = L ( q1 , q 2 ,..., qs , u1 , u2 ,..., us ; t ) ; corresponding to Hamilton’s theorem, this

action must be stationary along the curves satisfying the differential equations q j = u j , j = 1, 2,..., s .

(20.1.40')

The co-ordinates q j are fixed at the ends, while the co-ordinates u j do not possess this property. Because we deal only with arcs of curve which verify the differential equations (20.1.40'), the variations being synchronous, it is sure that the conditions (20.1.40) are verified. Using the method of Lagrange’s multipliers, we obtain δ∫

t1 t0

[L

( q1 , q 2 ,..., qs , u1 , u2 ,..., us ; t ) + λj ( q j − u j

) ] dt

= 0,

where the functions λj = λj ( t ) , j = 1, 2,..., s , remain to be determined. The equations of the extremal curves (the Euler–Lagrange equations) will be given by dλ j ∂L ∂L − = 0, − + λj = 0 , j = 1, 2,..., s ; ∂q j dt ∂u j

Variational Principles. Canonical Transformations

235

obviously, by eliminating the multipliers λj , we find again Lagrange’s classical equations. We denote H ( q1 , q 2 ,..., qs , p1 , p2 ,..., ps ; t ) =

L o o ∂q u −L , ∂u j j

where the linear transformation p j = ∂L / ∂u j , j = 1, 2,..., s , allows to pass from the 2s-dimensional space of co-ordinates q j , u j , j = 1, 2,..., s , to the phase space Γ 2s , the cap specifying the fact that the transformation has been effected (as in Sect. 19.1.1.3); we assume that the condition ⎡ ∂2L ⎤ det ⎢ ⎥ ≠0 ⎣ ∂u j ∂qk ⎦

is verified. We can thus state Theorem 20.1.6 (G.H. Livens). The stationarity condition of Theorem 20.1.5 (Hamilton’s theorem) is written in the form δ ALΓ = δ ∫

t1

t0

[ p j q j

− H ( q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ] dt = 0 ,

(20.1.41)

in the phase space. The form taken by Hamilton’s principle in the phase space (given by Livens in 1919) is called also the canonical form of Hamilton’s principle. We notice that δ∫

t1 t0

( p j q j

− H ) dt =

=

t1

∫t

0

∫ t δ ( pj q j t1 0

∂H ⎡⎛ ⎢ ⎜ q j − ∂p j ⎣⎝

− H ) dt =

t1

∫t

0

∂H ∂H ⎛ ⎜ p j δq j + q j δ p j − ∂q δq j − ∂p δ p j j j ⎝

∂H ⎞ ⎛ ⎟ δ p j − ⎜ p j + ∂q j ⎠ ⎝

⎞ ⎟ δq j ⎠

⎤ ⎥ dt + p j δq j ⎦

t1 t0

⎞ ⎟ dt ⎠

= 0,

where we took into account the operator relation (20.1.5'). In the hypothesis of fixed ends (for generalized co-ordinates) the integrated part vanishes. Applying the basic lemma of the variational calculus and observing that the variations δq j , δ p j , j = 1, 2,..., s , are arbitrary, we find again Hamilton’s canonical equations (19.1.14); however, if we write Euler-Lagrange equations for (20.1.41), hence the equations d (p dt j

⎛

∂H ⎞ ⎛ ∂H ⎞ ⎟ = 0 , −q j − ⎜ − ∂p ⎟ = 0 , j = 1, 2,..., s , j ⎠ j ⎠ ⎝

) − ⎜ − ∂q ⎝

we find again the same equations (19.1.14). The time and the generalized co-ordinates are fixed at the ends of the interval ( δq j ( t0 ) = δq j ( t1 ) = 0 ), but the generalized momenta do not verify a similar

MECHANICAL SYSTEMS, CLASSICAL MODELS

236

property ( δ p j ( t0 ) ≠ 0 , δ p j ( t1 ) ≠ 0 ); indeed, the conditions at the ends (bilocal) for

p j do not furnish any information, taking into account that the derivatives p j do not appear in the integrand expression. Of course, the two integrals which occur in (20.1.38') and (20.1.41) have the same value in the actual motion. But Hilbert observed that, in the formulation (20.1.38'), one can attain a minimum, while in the formulation (20.1.41) – for the same problem – it is possible to obtain neither a maximum nor a minimum (hence, a saddle point). We can put in evidence this observation by a simple example: the rectilinear motion of a particle subjected to its own weight (a field of uniform acceleration g ), for which x = gt 2 / 2 + vt , t0 = 0 , x (0) = 0 , x ( t1 ) = gt12 / 2 + vt1 . In this case,

L = mx 2 / 2 + mgx , while H = p 2 / 2m − mgx . Let us compare the actual motion with a motion on a varied path, given by x = gt 2 / 2 + vt + α ( t ) , α ( t ) ∈ C 2 , α ( 0 ) = α ( t1 ) = 0 ; we have

AL + δ AL = =

t1

∫0

( m2 x

2

)

+ mgx dt

(

)

⎡ m ( v + gt + α )2 + mg 1 gt 2 + vt + α ⎤ dt . ⎢⎣ 2 ⎥⎦ 2

t1

∫0

There results ⎡ m α 2 + m ( v + gt ) α + mg α ⎤ dt ⎣2 ⎦ t1 m 2 t1 m 2 t1 = ∫ α dt + m ( v + gt ) α 0 = ∫ α dt ; 0 2 0 2 δ AL =

t1

∫0

we state that δ AL > 0 , excepting the case in which α = 0 , hence α ( t ) = const , so that α ( t ) ≡ 0 (taking into account the conditions at the ends of the interval). Hence, the motion on the true curve corresponds to a minimum of the Lagrangian action. Using the canonical form of Hamilton’s principle, we can calculate

ALΓ =

t1

∫0

( − 21m p

2

)

+ mgx + px dt =

t1

∫0

⎡ − 1 ( p − mx )2 + m x 2 + mgx ⎤ dt . 2 ⎣⎢ 2m ⎦⎥

For the actual motion we have x = gt 2 / 2 + vt , p = m ( v + gt ) , while for the varied motion

we

can

take

x = gt 2 / 2 + vt + α ( t ) ,

p = m ( v + gt ) + β ( t ) ,

α ( t ) , β ( t ) ∈ C , α ( 0 ) = α ( t1 ) = 0 , without any conditions for β ( t ) at the ends of the interval. Analogously, we obtain 2

δ ALΓ =

t1

∫0

(

m⎡ 2 β − α α − 2 ⎢⎣ m

) ⎤⎥⎦ dt . 2

We observe that δ ALΓ > 0 for β = m α and α non-identical zero, while δ ALΓ < 0 for α ≡ 0 and β non-identical zero; the functional ALΓ is stationary on the true curve and has neither a maximum, nor a minimum (saddle point).

Variational Principles. Canonical Transformations

237

Let us remember that we have introduced the Lagrangian action in Sect. 19.2.1.6 in the form of a function S (t ) which verifies the Hamilton-Jacobi equation, as one can see by a total differentiation with respect to time; starting from this function, one has obtained Hamilton’s principal function. So, we can write the relation

dS = p j dq j − H ( q1 , q 2 ,..., qs , p1 , p2 ,..., ps ; t ) dt , which corresponds to Hamilton’s Pfaff form (19.1.45).

20.1.3.2 Case of Asynchronous Variations. Hölder’s Principle. Voss’s Principle. Another Form of Hamilton’s Principle In case of asynchronous variations, occurs the time variation Δt as a function of time. If Δt0 = Δt1 = 0 it follows that the initial and the final times are the same, the duration of the motion being also the same on all varied paths; besides, we can make the latter affirmation also in the more general case in which Δt0 = Δt1 ≠ 0 . We mention that these conditions are sufficiently restrictive. Corresponding to the results of Sect. 20.1.1.4, we can write an identity of the form t1

∫t

0

∂L d ∂L ⎛ ⎞ ⎜ ΔL + q j ∂q dt Δt − LΔt ⎟ dt = ∂q Δq j j j ⎝ ⎠

t1 t0

t1 ⎡ ∂L d ⎛ ∂L ⎞ ⎤ +∫ ⎢ − Δq dt t0 ∂q j dt ⎝⎜ ∂q j ⎠⎟ ⎦⎥ j ⎣ (20.1.42)

for Lagrange’s kinetic potential L = L ( q1 , q 2 ,..., qs , q1 , q2 ,..., qs ; t ) , where asynchronous variations are put into evidence. Along the integral curves of Lagrange’s equations there results t1

∫t

0

∂L d ∂L ⎛  ⎞ ⎜ ΔL + q j ∂q dt Δt − LΔt ⎟ dt = ∂q Δq j j j ⎝ ⎠

t1

;

(20.1.42')

t0

in case of varied curves with fixed ends ( Δq j ( t0 ) = Δq j ( t1 ) = 0 ) and in case of catastatic constraints (the set of possible displacements coincides with the set of virtual displacements), we can state Theorem 20.1.7 (Hölder). Among all possible motions of a mechanical system subjected to holonomic, catastatic, ideal constraints, and to quasi-conservative given forces on asynchronous varied paths with fixed ends, only and only the motion (of the representative point on the true curve) for which one has the relation t1

∫t

0

∂L d ⎛  ⎞ ⎜ ΔL + q j ∂q dt Δt − LΔt ⎟ dt = 0 j ⎝ ⎠

(20.1.42'')

takes place. Indeed, in this case, the possible displacements Δq j , j = 1, 2,..., s , are independent, and we find again Lagrange’s equations; but, in case of non-holonomous constraints

MECHANICAL SYSTEMS, CLASSICAL MODELS

238

one can use the method of multipliers, finding again Lagrange’s equations with multipliers. The constraints are catastatic, so that ri = 0 , i = 1, 2,..., n , for all particles of the mechanical system; so, a particular configuration of this system is represented by the same point in the space Λs , for all values of the time t . As in the case of the Theorem 20.1.5, the results remain valid for all quasi-conservative given forces which derive from a generalized quasi-potential. This theorem can be used also as a principle: Hölder’s principle, stated in 1896; obviously, this principle is an integral one. If Δt = 0 , then we find again Hamilton’s principle. Using the results of Sect. 20.1.1.4, we can calculate the asynchronous variation of the Lagrangian action in the form t

t1 ⎡ ∂L ⎤1 Δ AL = Δ ∫ L dt = ⎢ Δq j − EΔt ⎥ + t0 ⎣ ∂q j ⎦t 0

where

t1

∫t

0

d ⎛ ∂L ⎞ ⎤ ⎡ ∂L ⎢ ∂q − dt ⎜ ∂q ⎟ ⎥ δq j dt , ⎣ j ⎝ j ⎠⎦ (20.1.43)

E is the generalized mechanical energy E =

∂L q − L . ∂q j j

(20.1.43')

Assuming that we have to do with varied paths with fixed ends and that Δt0 = Δt1 = 0 , there results

Δ AL =

t1

∫t

0

d ⎛ ∂L ⎞ ⎤ ⎡ ∂L ⎢ ∂q − dt ⎜ ∂q ⎟ ⎥ δq j dt . j ⎣ ⎝ j ⎠⎦

(20.1.43'')

We are thus led to Theorem 20.1.8 (A. Voss). Among all possible motions of a mechanical system subjected to ideal constraints, acted upon by quasi-conservative given forces, on asynchronous varied paths with fixed ends (in space and time), only and only the motion (of the representative point on the true curve) which makes stationary the Lagrangian action Δ AL = Δ ∫

t1

t0

L dt = 0

(20.1.44)

takes place. This result is valid in case of holonomic constraints (one obtains Lagrange’s equations), as well as in case of non-holonomic ones (one obtains Lagrange’s equations with multipliers); it remains valid also in case of quasi-conservative forces which derive from a generalized quasi-potential. As above, one can consider this theorem to be a new variational principle: Voss’s principle (1900). It is an extension of Hamilton’s principle in case of asynchronous variations; besides, one can easily obtain this result, taking into account the relation (20.1.30'), which links the asynchronous to the synchronous variation of an action, and making Δt0 = Δt1 = 0 .

Variational Principles. Canonical Transformations

239

Along the extremal curves to which lead Lagrange’s equations, the relation t

⎡ ∂L ⎤1 Δq j − EΔt ⎥ , = ⎢ ⎣ ∂q j ⎦t 0

Δ AL

(20.1.43''')

corresponding to the Poincaré-Cartan integral invariant (see Sect. 21.1.2.4) takes place. We mention also the connection between this relation and the relation (19.2.28') concerning Hamilton’s principal function. As well, in case of varied paths with fixed ends, we can write Δ AL + EΔt

t1 t0

= 0.

(20.1.43iv)

Comparing Hölder’s and Voss’s principles, we see that the second one is a variational principle (an advantage), and that the mechanical system is not subjected to catastatic constraints; in compensation, the varied paths must be with fixed ends also for time, a sufficient severe restriction, not implied by Hölder’s principle. On the other hand, in the corresponding demonstration the principle of virtual work was not used, but only the connection with Lagrange’s equations; thus, one must not have L = T + U (as it was supposed, mentioning the action of quasi-conservative given forces), but we can consider mechanical systems which admit a Lagrangian L which verifies the condition (18.2.34'''). Starting form the general integral principle (20.1.35'), which corresponds to asynchronous variations with fixed ends and imposing catastatic constraints ( dT / dt = P ), we can write t1

∫t

0

( ΔW + ΔT + 2T ddt Δt ) dt = 0 .

(20.1.45)

If the given forces are quasi-conservative, then we have ΔW = ΔU − U Δt , so that we obtain ( L = T + U ) t1

∫t

0

( ΔL + 2T ddt Δt − U Δt ) dt = 0 .

(20.1.45')

Assuming that the quasi-conservative forces derive from a simple quasi-potential and observing that, in case of catastatic constraints, we have q j

∂L ∂T = q j = 2T2 = 2T , T = 0 , ∂q j ∂q j

we find again Hölder’s principle. If the given forces are quasi-conservative, then the relation (20.1.35') becomes ( ΔW = ΔU − U Δt , L = T + U and E = T − U ) t1

Δ ∫ L dt + t0

t1

∫t

0

( ddTt − P − U ) Δtdt + ∫

t1 t0

E d( Δt ) = 0 ;

MECHANICAL SYSTEMS, CLASSICAL MODELS

240 integrating by parts, we can write also t1

Δ ∫ L dt + t0

t1

∫t

0

( ddTt − P ) Δtdt − ∫ ( ddEt + U ) Δtdt + E Δt t1

t0

t1 t0

= 0.

In case of catastatic constraints we have dE = dT − dU = dW − dU = −Udt , so that Δ∫

t1 t0

L dt + E Δt tt10 = 0 ;

(20.1.46)

we find thus a variational principle of Hamilton type, in the same conditions Δt0 = Δt1 = 0 . In fact, this is Voss’s principle, stated for L = T + U in the restricted frame of catastatic constraints, case in which the generalized mechanical energy is reduced to the mechanical energy ( E = E ). The condition Δt0 = Δt1 = 0 is very inconvenient. To eliminate it, we can start from the relation (20.1.43iv), which can be written also in the form t1

Δ ∫ ( L + h ) dt + ( E − h ) Δt t0

t1 t0

= 0,

where h is constant with respect to time. If L = 0 , then we obtain Jacobi’s first integral (or of Jacobi type) E = h ; so we can state Theorem 20.1.9 Among all possible motions of a generalized conservative mechanical system, subjected to ideal constraints, on asynchronous varied paths with fixed ends, only and only the motion (of the representative point on the true curve) for which t1

Δ ∫ ( L + h ) dt = 0 , h = const , t0

(20.1.47)

takes place. Also this result can be used as a principle; in fact, it is a generalization of Hamilton’s principle for which there are not imposed restrictions concerning time variations (at the ends of the time interval). If E = E = h in case of catastatic constraints, we can state the same result for a conservative mechanical system; starting from the relation (20.1.46), one obtains the same principle. As well, one can obtain the same result using Hölder’s principle.

20.1.3.3 Changes of Independent Variable. Conformal Mappings As we have seen, one of the important properties of the variational principles is the possibility to use any system of co-ordinates; the generalized form (20.1.47) of Hamilton’s principle, corresponding to a generalized conservative mechanical system, allows to make a step further, i.e. to introduce a change of independent variable. Let be dt = udθ ,

(20.1.48)

Variational Principles. Canonical Transformations

241

where u = u ( q1 , q 2 ,..., qs ) is a given positive function of class C 1 ; we may consider θ as an artificial time, measured in the motion of the representative point in the space Λs as in Sects. 19.2.2.1, 19.2.2.3, and 19.2.2.3. Denoting by L ′ = L ′ ( q1 , q2 ,..., qs , q1′ , q2′ ,..., qs′ ) , q j′ = dq j / dθ , j = 1, 2,..., s , the new form of the function L by introducing the new generalized velocities, we can write the principle (20.1.47) in the form θ1

Δ ∫ Λ dθ = 0 , Λ = u ( L ′ + h ) . θ0

(20.1.47')

The corresponding Euler–Lagrange equations

d ⎛ ∂Λ ⎞ ∂Λ − = 0 , j = 1, 2,..., s , dθ ⎜⎝ ∂q j ⎟⎠ ∂q j

(20.1.49)

hold only for generalized conservative mechanical systems ( L ′ = dL ′ / dθ = 0 and E = h ); one obtains thus the trajectory of the representative point in the phase space, but not the direct connection between the position of the point on the orbit and the time. If we denote by “prime” the change of independent variable in a certain quantity, we can write T2 = θ2T2′ = T2′ / u 2 , T1 = θT1′ = T1′ / u , so that Λ = u (T2 + T1 + T0 + U + h ) =

T2′ + T1′ + u (T0 + U + h ) ; u

(20.1.50)

the corresponding first integral of Jacobi type leads to the condition T2′ − (T0 + U ) = h . u2

(20.1.50')

In particular, let be a mechanical system with two degrees of freedom for which T2 =

1 ( g q 2 + 2g12q1q2 + g22q22 ) , T1 = g1q1 + g2q2 ; 2 11 1

(20.1.51)

by a convenient transformation we can pass to isothermic (isometric) co-ordinates for which (by convention, we can use the same notation for the generalized co-ordinates) T2 =

1 u ( q12 + q22 ) , 2

T1 = α1q1 + α2q2 .

(20.1.51')

By a change of independent variable of the form (20.1.48), we obtain Λ=

1 2 ( q ′ + q2′2 ) + α1q1′ + α2q2′ + β , β = u (T0 + U + h ) ; 2 1

there results the equations of motion

(20.1.51'')

MECHANICAL SYSTEMS, CLASSICAL MODELS

242 q1′′ + γq 2′ =

∂α ∂α ∂β ∂β , q 2′′ − γq1′ = , γ = 1 − 2. ∂q1 ∂q 2 ∂q 2 ∂q1

(20.1.52)

The isoenergetic orbits (of energy h ) are specified by the system (20.1.52), satisfying the relation 1 2 ( q ′ + q2′2 2 1

) = β.

(20.1.52')

We obtain thus the normal form of the equations of motion of a generalized conservative mechanical system with two degrees of freedom; these are the equations of the plane motion of a particle subjected to a field of conservative forces of potential β and to a gyroscopic force γ v . The method of calculation presented above may be used successfully also for Liouville type systems, considered in Sects. 18.3.1.2 and 19.2.2.2, obtaining similar results. Let be a mechanical system with two degrees of freedom, of Lagrangian 1 2

L = ( q12 + q22 ) + U ( q1 , q2 ) .

(20.1.53)

In case of a conformal mapping z = q1 + iq 2 = f ( ζ ) , ζ = ξ1 + iξ2 , where f is a regular function in a convenient domain for ζ , we may write dz = M dζ , M = f ′ ( ζ ) , so that 1 2

L = M 2 ( ξ12 + ξ22 ) + W ( ξ1 , ξ2 ) , W ( ξ1 , ξ2 ) = U ( q1 , q2 ) .

(20.1.53')

Applying the method of change of independent variable, we find Λ=

1 2 ( ξ ′ + ξ2′2 ) + M 2 (W + h ) , 2 1

(20.1.54)

with a condition of energetical nature 1 2 ( ξ ′ + ξ2′2 2 1

) = M 2 (W

+ h ).

(20.1.54')

For instance, the conformal mapping z = ζ 2 corresponds to the study of the Keplerian motion.

20.1.4 Maupertuis’s Principle. Other Variational Principles In what follows, we will present Maupertuis’s principle in various forms, in the general asynchronous case; as well, we will put into evidence the possibility to enounce also other variational principles (for instance, the principle of the least potential action), equivalent to Hamilton’s and Maupertuis’s principles.

Variational Principles. Canonical Transformations

243

20.1.4.1 Maupertuis’s Principle We will introduce the kinetical action in the form

AT =

t1

∫t

Td t =

0

1 t1 ∂L q dt , 2 ∫t0 ∂q j j

(20.1.55)

in the hypothesis of a conservative mechanical system and of catastatic constraints. We mention that, often, one uses the quantity A = 2 AT , called action. We admit also that the mechanical energy is constant along any varied path; if E = h on the true curve, then it results E + ΔE = h + Δh on a varied path. Starting from the relation (20.1.42') and using the condition L = 0 (we have  T = U = 0 ), we find t1

∫t

0

∂L ∂L d ⎛ ⎞ ⎜ ΔL + ∂q q j dt Δt ⎟ dt = ∂q Δq j j j ⎝ ⎠

t1

. t0

Applying the operator Δ to the relation (20.4.43'), we can write ⎛ ∂L ⎞ ΔE = Δ ⎜ q j ⎟ − ΔL = Δh , ⎝ ∂q j ⎠

so that ∂L ∂L d ⎡ ⎛ ∂L ⎞ ⎤ ⎢ Δ ⎜ ∂q q j ⎟ − Δh + ∂q q j dt Δt ⎥ dt = ∂q Δq j j j ⎣ ⎝ j ⎠ ⎦

t1

∫t

0

t1

; t0

using the relation (20.1.14), we get Δ AT =

t1 ∂L 1 1 ∂L Δ q dt = Δq j 2 ∫t0 ∂q j j 2 ∂q j

t1 t0

+

1 ( t − t0 ) Δh . 2 1

(20.1.55')

Besides, starting from the relation (20.1.43'') which takes place along the solutions of Lagrange’s equations, and making E = h , we have Δ AL = Δ ∫

t1 t0

Ldt =

∂L Δq j ∂q j

t1

− h ( Δt1 − Δt0 ) ;

t0

by means of the relation (20.1.43'), we obtain Δ∫

t1

t0

∂L ∂L q j dt − Δ [ h ( t1 − t0 ) ] = Δq j ∂q j ∂q j

t1 t0

− h ( Δt1 − Δt0 )

MECHANICAL SYSTEMS, CLASSICAL MODELS

244

and find again the relation (20.1.55'), in the same conditions. Assuming that we have to do with varied paths with fixed ends, for which Δh = 0 , the constants being the same on all these paths (isoenergetic paths), it follows t1

Δ AT = Δ ∫ Tdt = 0 .

(20.1.56)

t0

As a consequence of Lagrange’s equations, in the above mentioned conditions, this relation represents a necessary condition for the motion on the true curve. To put into evidence the sufficiency of this relation, we must show that, starting from the relation (20.1.56), we obtain Lagrange’s equations. To do this, we must solve a problem of extremum with constraints, of the form t1

Δ ∫ Fd t = 0 , F = T + λ ( E − h ) , E = T − U , t0

where λ = λ ( t ) is a Lagrange multiplier. Using the relation (20.1.15) and equating to zero the displacements Δq j , j = 1, 2,..., s , at the ends of the time interval, we may write t1 ∂F ⎛ ⎞ Δ ∫ Fd t = ⎜ F − q j ⎟ Δt t0 ∂q j ⎝ ⎠

t1 t0

+

t1

∫t

0

d ⎛ ∂F ⎞ ⎤ ⎡ ∂F ⎢ ∂q − dt ⎜ ∂q ⎟ ⎥ δq j dt . ⎣ j ⎝ j ⎠⎦

For the sake of simplicity, without loosing the generality, we will suppose that the initial moment at P0 is the same for all varied paths ( Δt0 = 0 ), Δt1 being not fixed; but we fix the co-ordinates q j ( t1 ) , j = 1, 2,..., s , so that we have (we observe that T −U = h ) F −

∂F q = − ( 1 + 2λ )T = 0 ∂q j j

for t = t1 . Using the basic lemma of the variational calculus, we obtain Euler– Lagrange equations in the form d dt

∂T ⎤ ∂T ∂U ⎡ ⎢ ( 1 + λ ) ∂q ⎥ − ( 1 + λ ) ∂q = − λ ∂q , j = 1, 2,..., s , j ⎦ j j ⎣

or in the form ⎡ d ⎛ ∂T ⎞

∂T

∂U ⎤

∂T

∂U

(1 + λ ) ⎢ ⎜ − = − λ − ( 1 + 2λ ) ; ⎟− ∂q j ⎦⎥ ∂q j ∂q j ⎣ dt ⎝ ∂q j ⎠ ∂q j

multiplying by q j and summing for all indices j , we have ⎡ d ⎛ ∂T

⎞

dT

dU ⎤

∂T

dU

(1 + λ ) ⎢ ⎜ − = − λ , q j ⎟ − q − ( 1 + 2λ ) ∂q j j dt ⎦⎥ dt ⎣ dt ⎝ ∂q j ⎠ dt

Variational Principles. Canonical Transformations

245

where we took into account that U = T = 0 , corresponding to the catastatic constraints and to the condition L = 0 . Because the kinetic energy is a quadratic form, we can write (1 + λ )

d (T − U ) dU ; = −2T λ − ( 1 + 2λ ) dt dt

taking into account that T − U = h , we remain with −2T λ − ( 1 + 2λ )

dU d = − [ ( 1 + 2λ )T ] = 0 . dt dt

Hence, ( 1 + 2λ )T is constant along the true curve; the condition imposed for t = t1 shows that this constant vanishes, so that λ = −1/ 2 at any moment. With this value of λ , we see that the above Euler–Lagrange equations are just Lagrange’s equations of motion. One can thus state Theorem 20.1.10 (Maupertuis). Among all possible motions of a mechanical system subjected to catastatic ideal constraints, acted upon by conservative given forces, on asynchronous varied isoenergetic paths with fixed ends, only and only the motion (of the representative point on the true curve) which makes stationary the kinetic action (20.1.56) takes place. As in the cases considered in the previous Section, we can consider that this result is a variational principle: the principle of the least action (Maupertuis’s principle). This principle played an important rôle in the development of mechanics, being the first variational principle of mechanics, enounced for the first time by P.L. Moreau de Maupertuis in 1740 in a quite obscure form; this formulation led to a passionate controversy, especially with S. Koenig, a controversy in which we mention also the intervention of François Marie Arouet dit Voltaire (an adversary of Maupertuis). The problem has been taken again by L. Euler (1744) who gave the formulation of the principle in case of central forces; Lagrange (1760) and then Jacobi (1847) have given its general form. Sometimes, the principle of least action is called also the MaupertuisLagrange principle. Obviously, for Δt = 0 we can enounce an analogous synchronous principle. As we have seen, in case of catastatic constraints the relation (20.1.35') is reduced to (20.1.45); comparing with Maupertuis’s principle, written in the developed form t1

∫t

0

( ΔT + T ddt Δt ) dt = 0 ,

and eliminating the term Td ( Δt ) / dt , we can state Theorem 20.1.11 Among all possible motions of a mechanical system, subjected to catastatic ideal constraints, on asynchronous varied paths with fixed ends, only and only the motion (of the representative point on the true curve) which makes stationary the kinetic action (20.1.56) with the condition t1

∫t

0

ΔTdt =

t ∫t ΔLdt 1

0

(20.1.57)

MECHANICAL SYSTEMS, CLASSICAL MODELS

246

takes place. In case of quasi-conservative given forces, deriving from a simple quasi-potential, we have ΔL = ΔU − U Δt , so that the condition (20.1.57) is reduced to

∫t ( ΔE + U Δt ) dt t1

= 0;

(20.1.57')

0

if the forces are conservative, then there result E = h and U = 0 , the condition (20.1.57') being identically verified. We find again Maupertuis’s principle. We observe that the Theorem 20.1.11 represents a generalization (larger conditions) of the Theorem 20.1.10; obviously, we can affirm the same thing for the corresponding variational principles. Eliminating U between L = T + U and T − U = h , it follows L = 2T − h ; from (20.1.38') and (20.1.55) we obtain, in this case,

AL = 2 AT − h ( t1 − t0 ) = A − h ( t1 − t0 ) ,

(20.1.58)

a relation connecting the kinetic and the Lagrangian actions. If t0 = 0 and t1 = t , we find again the relation (19.2.15'), connecting the function S (t ) of the Hamilton-Jacobi theory (to which corresponds AL ) to the function S (t ) , which appears in case of generalized conservative mechanical systems (to which corresponds A = 2 AT , according to the formula (19.2.17)).

20.1.4.2 Other Forms of the Least Action Principle We can express the action

A = 2 AT also in the form

A =

t1 n

mi vi2 dt ∫t i∑ =1 0

for a mechanical system S of n particles Pi , i = 1, 2,..., n ; Maupertuis enounced the least action principle in the synchronous form δ∫

S1 n S0

∑ mi vi dsi

i =1

= 0,

where si is the curvilinear abscissa on the trajectory of the particle Pi ,

(20.1.59)

S0 and S1

being the states of the system S at the time t0 and t1 , respectively. Hence, the elementary action 2Tdt is equal to the elementary work of the momenta of the considered mechanical system S. In case of a single particle P , we obtain δ∫

P1 P0

mv ds = 0 ;

(20.1.59')

Variational Principles. Canonical Transformations

247

if on the particle does not act any given force, its motion will be uniform (as it results from the study of the motion in an intrinsic frame of reference), so that v = const , and the relation (20.1.58') is reduced to δ∫

P1 P0

ds = 0 .

(20.1.59'')

The trajectory of the particle P will be, in this case, a “geodesic line” in the constraint conditions imposed to the motion. If the particle moves on a given fixed surface, then the trajectory is a geodesic line on the respective surface, passing through the points P0 and P1 . If there are not constraint relations, then the geodesic is a straight line and we find again the principle of inertia. Using the conservation theorem of mechanical energy, Jacobi eliminates the time in the least action principle. Starting from

2T =

n

∑ mi vi2 =

i =1

n

∑ mi

i =1

( ), dsi dt

2

we find δA = δ ∫

S1 S0

2 T (U + h )dt = δ ∫

= δ∫

S1 S0

2 (U + h )

n

∑

i =1

S1 S0

n

2T

mi dsi2

∑ mi dsi2

i =1

= 0.

(20.1.60)

This synchronous form of the least action principle represents Jacobi’s principle and has a pure geometric character; we observe that the integral A , in which the time disappeared, can be calculated by means of any other parameter. As well, the condition that the motion be isoenergetic on all varied paths becomes irrelevant; this is the great advantage of this formulation. One observes that, using a Lagrangian of the form Λ = 2 T (U + h ) , one obtains s differential equations of Euler–Lagrange type, which are not independent; but they only allow to obtain the trajectory of the representative point in the space Λs . If we impose the condition T − U = h , then the velocity on this trajectory is also determined. In generalized co-ordinates, we have n

∑ mi dsi2

i =1

= gij dqi dq j ,

so that

A =

S1

∫S

2 (U + h ) gij dqi dq j .

(20.1.60')

0

Replacing the mechanical system S by a representative point P in the Lagrangian space Λs , it is convenient to endow this space with the metrics

MECHANICAL SYSTEMS, CLASSICAL MODELS

248

dσ 2 = gij dqi dq j .

(20.1.61)

In the case in which the mechanical system is not subjected to any given force (U = const ), we can neglect a multiplicative constant, remaining the variational condition δ∫

S1 S0

dσ = 0 ;

(20.1.61')

thus, we state Theorem 20.1.12 (Jacobi). Let be given a discrete mechanical system S subjected to holonomic, ideal constraints, which is acted upon by no one given force; the trajectory of the representative point P in the space Λs , endowed with the metrics (20.1.61), is a geodesic line of this space. As in Sect. 7.2.1.6, we can write the differential equations of the geodesic lines in the form g jk qk′′ + [ kl , j ]qk′ql′ = 0 , j = 1, 2,..., s ,

(20.1.62)

where [ kl , j ] is Christoffel’s symbol of the first kind; the differentiation is made with respect to an arbitrary parameter, which can be even the curvilinear abscissa along the trajectory of the representative point. Multiplying by the normalized algebraic complement, summing and using Christoffel’s symbol of the second kind, there result the equations of the geodesic lines in the normal form ⎧⎪ j ⎫⎪ q j′′ + ⎨ ⎬ qk′ ql′ = 0 , j = 1, 2,..., s ; ⎪⎩ k l ⎪⎭

(20.1.62')

we notice the analogy which, obviously, does exists between these equations and the normal form (18.2.47') of Lagrange’s equations, in which we make Q j∗ = 0 , j = 1, 2,..., s . If the given forces are non-zero, hence if the potential U is not constant, then we introduce a manifold Λs of metrics dσ 2 = 2 (U + h ) dσ 2 = gij dqi dq j , gij = 2 (U + h ) gij , i , j = 1, 2,..., s ,

(20.1.63)

so that δA = δ∫

S1 S0

dσ = 0 .

(20.1.63')

We can thus state Theorem 20.1.12' (Jacobi). Let be given a discrete mechanical system S subjected to holonomic, ideal constraints and acted upon by conservative given forces; the trajectory of the representative point P , corresponding to the manifold Λs and endowed with the metrics (20.1.62) is a geodesic line of this manifold.

Variational Principles. Canonical Transformations

249

After obtaining the geodesic lines, the law of motion is given by

dt =

dσ dσ . = ( 2 U +h) ( ) 2 U +h

(20.1.64)

The Theorems 20.1.12 and 20.1.12' put in evidence the character of minimum of the considered variational principle, as well as the denomination given to it. In fact, if the functional is not minimized, it remains – at least – stationary.

20.1.4.3 The Analogy with Fermat’s Principle From an optical point of view, a material medium is characterized by the refraction index n =

c , v

(20.1.65)

where c is the velocity of light in vacuum, while v is its velocity in the respective medium. This index is a function of point ( n = n ( x1 , x 2 , x 3 ) ) in a non-homogeneous medium; in an anisotropic medium, it is function also of direction. Let us consider a medium, in general, non-homogeneous; the duration of light propagation is given by (we take t0 = 0 , by convention) t =

s

∫s

0

dσ 1 s = ∫ nd σ , v c s0

where dσ is the arc element, while s is the curvilinear co-ordinate along the path of the light photon. Corresponding to Fermat’s principle, among all paths which join two points of a material medium, the light photon follows the shortest optical path, so that s

δ ∫ ndσ = 0 , s0

(20.1.66)

in a synchronous formulation. We find thus a formal analogy between this optical principle and Maupertuis’s mechanical principle. Using (20.1.60), we can write, for a single particle, δ∫

P P0

2 (U + h )ds = 0 ,

neglecting the factor m . The trajectory of the light photon coincides with the trajectory of the particle if U = n 2 / 2 − h . If we assume that, in the vicinity of the Earth surface, the refraction index decreases linearly with the altitude z , we can write n = n 0 ( 1 − kz / H ) , where H is the height of the atmosphere, n0 is the refraction index at the surface of the Earth, while k is a constant. Neglecting ( z / H )2 with respect to z / H , we obtain

MECHANICAL SYSTEMS, CLASSICAL MODELS

250 U = c − gz , c =

n2 1 2 n0 − h , g = k 0 , H 2

(20.1.67)

hence the potential of a gravitational force in the proximity of the Earth surface, with a conventional gravitational acceleration. Hence, if the refraction index n has a linear variation with the altitude, then the light is propagated along a parabola with a vertical axis. The refraction index will have the form n = n ( q1 , q 2 ,..., qs ) in the space Λs , so that we obtain for the photon a geodesic line in a manifold Λs of metrics dσ = ndσ . But the complete analogy between the two principles has been realized in 1924 by Louis de Broglie in the frame of undulatory mechanics; he replaces the phase velocity v by a group velocity u , so that uv = const . In this case, the principle (20.1.66) takes the form s

δ ∫ udσ = 0 , s0

(20.1.66')

analogous to the form (20.1.59') of Maupertuis’s principle for a particle.

20.1.4.4 Plane Motion in a Field of Conservative Forces In a plane motion of a particle on a curve C, we put in evidence the generalized coordinates s (the curvilinear co-ordinate, starting from the fixed point P0 ) and θ (the angle made by the external normal n to the curve C with the fixed axis Ox (Fig. 20.3)). We assume that s = s ( θ ) is an increasing function; if the curve is closed, then it is also convex.

Fig. 20.3 Plane motion in a field of conservative forces

Let be the integral I =

P1

∫P

2 (U + h )ds ;

0

if we pass from the curve C to the varied path C ′ by a displacement δn = δn (s ) along the external normal, we can write

Variational Principles. Canonical Transformations

δI =

P1

∫P

0

δU ds + 2 (U + h )

251 P1

∫P

2 (U + h )δ ds .

0

We have δU = N δn , where N is the component of the field of forces along the external normal, while δ ds = δndθ = δnds / ρ , ρ being the curvature radius at the point P . We obtain thus δI =

P1

∫P

0

W 2 (U + h ) . δnds , W = N + ρ 2 (U + h )

(20.1.68)

Writing the equation of motion along the external normal and taking into account the mechanical energy conservation theorem, there results N = − mv 2 / ρ = −2 (U + h ) / ρ , and we find again Jacobi’s form of the least action principle. Let be a convex and closed simple curve C, and C an analogous curve which contains the first one. We can state Theorem 20.1.13 (Whittaker). If a particle P which is in motion in the interior of a ring formed by two conex and closed simple plane curves C (internal) and C (external) and is acted upon by conservative forces for which the potential U has not singularities in this ring, then it admits at least a periodic orbit if W < 0 on C and W > 0 on C .

20.1.4.5 Other Variational Principles We will introduce, following V. Vâlcovici, a kinetic-potential energy Eα ,β = αT + βU

(20.1.69)

of ( α, β ) kind and a corresponding kinetic-potential action

Aα ,β =

t1

∫t

0

Eα , β d t

(20.1.69')

of the same kind. To establish the conditions in which one can enounce an asynchronous variational principle of the form t1

Δ Aα ,β = Δ ∫ Eα ,β dt = t0

t1

∫t

0

( ΔE

α ,β

+ Eα , β

)

d Δt dt = 0 , dt

we start from the general integral principle (20.1.35'), corresponding to the principle of virtual work, the varied paths having fixed ends; subtracting the relation (20.1.35') from the relation t1

∫t

0

we find the condition

( αΔT + βΔU + ( αT + βU ) ddt Δt ) dt = 0 ,

252

MECHANICAL SYSTEMS, CLASSICAL MODELS

t1

{ α − 1 ΔT + βΔU − ΔW + ( P − ddTt ) Δt + [ α − 2 T + βU ] ddt Δt } dt = 0 ,

∫t

(

0

)

(

)

(20.1.70) which allows the determination of the function Δt . In case of catastatic constraints ( P = dT / dt ) and of quasi-conservative given forces ( ΔW = ΔU − U Δt ), we obtain a much more simple form, i.e. t1

∫t

0

{ α − 1 ΔT + ( β − 1 ) ΔU + U Δt + [ α − 2 T + βU ] ddt Δt } dt = 0 . (

)

(

)

(20.1.70') If the given forces are conservative (U = 0 ), then we can write a mechanical energy conservation theorem ( E = T − U = h ), so that the condition (20.1.70') becomes t1

∫t

0

{ α − 1 Δh + ( α + β − 2 ) ΔU + [ α − 2 h + ( α + β − 2 )U ] ddt Δt } dt = 0 , (

)

(

)

(20.1.70'') where we assumed that along a varied path we have E + ΔE = h + Δh . Supposing further that we have to do with isoenergetic varied paths ( Δh = 0 ), this latter condition has the form t1

( α + β − 2 ) Δ ∫ Udt + ( α − 2 ) h ( Δt1 − Δt0 ) = 0 t0

(20.1.71)

or the form t1

Δ ∫ [ ( α + β − 2 )U + ( α − 2 ) h ] dt = 0 . t0

(20.1.71')

We can thus state Theorem 20.1.14 Among all possible motions of a mechanical system, subjected to catastatic, ideal constraints and acted upon by conservative given forces, on asynchronous varied isoenergetic paths with fixed ends, only and only the motion (of the representative point on the true curve) which makes stationary the action Aα ,β with the condition (20.1.71') takes place. Obviously, in this case too, starting from this theorem, one can obtain a corresponding variational principle. In particular, for α = 2 and β = 0 we find again the Theorem 20.1.10 of Maupertuis. As well, for α = β = 1 and Δt0 = Δt1 we find a particular form of Voss’s principle; but we observe that, if we make α = β = 1 in the condition (20.1.70'), then we arrive at the form considered in Sect. 20.1.3.2. Taking α = 0 and β = 2 , as well as Δt0 = Δt1 , we obtain an asynchronous least potential principle stated first by V. Vâlcovici and afterwards, in general conditions, by P.P. Teodorescu) (the same duration on all varied paths)

Variational Principles. Canonical Transformations

253

t1

Δ AU = Δ ∫ U dt = 0 .

(20.1.72)

t0

To can formulate a least mechanical action principle in the form t1

Δ AE = Δ ∫ E dt = 0 ,

(20.1.73)

t0

we must have E ≠ const ; starting from the condition (20.1.70') and making α + β = 0 , we find ( E = T − U , L = T + U ) α∫

t1

t0

( ΔE + E ddt Δt ) dt + ∫ (U Δt − ΔL − 2T ddt Δt ) dt = 0 . t1

t0

The first variation vanishes, so that we can have a least mechanical action principle if the condition t1

∫t

U Δt dt =

0

t1

∫t

0

( ΔL + 2T ddt Δt ) dt

(20.1.73')

is verified; we find a more intricate relation (specifying Δt as a function of t ) as in the previous case.

20.1.4.6 Larmor’s Principle If some ignorable co-ordinates which not appear in the kinetic potential L do exist (see Sect. 18.2.3.6), we can state a variational principle, in the frame of which appears a functional which becomes stationary on a certain class of varied paths. We have thus to do with a holonomic mechanical system with s degrees of freedom, in which the first r generalized co-ordinates are ignorable. Let thus be the functional t ⎛ ∫t ⎜⎝ L 1

0

−

d r ∂L ⎞ qk dt . ∑ dt k =1 ∂qk ⎟⎠

Using the relation (20.1.43'''), we obtain the variation of this functional in the form t

r ⎡ ∂L ⎤1 ⎡ ∂L ⎛ ∂L ⎞ ⎤ 1 Δ q E Δ t Δ q q Δ − − + j ⎜ ∂q ⎟ ⎥ ∑ k k ⎢ ∂q ⎥ ⎢ ⎝ k ⎠ ⎦t 0 ⎣ j ⎦t0 k =1 ⎣ ∂qk t

t

1 r ⎡ s ∂L ⎛ ∂L ⎞ ⎤ = ⎢ ∑ Δq j − EΔt − ∑ qk Δ ⎜ . ⎟ ⎥ ⎝ ∂qk ⎠ ⎦t k =1 ⎣ j =r + 1 ∂q j 0

The varied motion will be subjected to the restrictions (18.2.79'), with the same constant as in the true motion. In this case (we calculate the derivative d / dt under the integral sign) t

r ∂L t1 ⎛ ⎡ s ∂L ⎤1 ⎞ Δ∫ ⎜ L − ∑ qk ⎟ dt = ⎢ ∑ Δq j − EΔt ⎥ . t0 ⎝ ⎠ k = 1 ∂qk ⎣ j = r +1 ∂q j ⎦t 0

MECHANICAL SYSTEMS, CLASSICAL MODELS

254

By means of the relations (18.2.79') and by the normal procedure, we can express the above integrand as a function of the constants C 1 ,C 2 ,...,C r and of the generalized velocities qr +1 , qr + 2 ,..., qs , obtaining thus the function R of Routh, which depends on the palpable co-ordinates qr +1 , qr + 2 ,..., qs . Fixing the palpable co-ordinates at the ends of the time interval (not also the ignorable co-ordinates) and fixing the initial and the final times ( Δt0 = Δt1 = 0 ), there results r t1 t1 ⎛ ∂L ⎞ Δ AR = Δ ∫ Rdt = Δ ∫ ⎜ L − ∑ qk ⎟ dt = 0 . t0 t0 ⎝ q ∂  k ⎠ k =1

(20.1.74)

We may state Theorem 20.1.15 (Larmor). Among all possible motions of a generalized conservative mechanical system, subjected to ideal constraints, acted upon by quasi-conservative given forces, on asynchronous varied paths with fixed ends (in space, for the palpable co-ordinates, and in time), only and only the motion (of the representative point on the true curve) which makes stationary the Routhian action (20.1.74) takes place. This result can be considered as a generalization of the Theorem 20.1.8 and may be used as a variational principle: Larmor’s principle. Starting from this principle, we obtain – in the usual way – Routh’s equations.

20.1.5 Continuous Mechanical Systems After some results with a general character, we consider some particular mechanical systems; we will thus present the basic equations in the motion of strings, of straight bars and of linear elastic solids. We used till now analytical methods of calculus only to study discrete mechanical systems (with a finite number of degrees of freedom); in case of an infinite number of degrees of freedom, the problems impose a specific treatment. For instance, in case of vibrations of an elastic solid, every one of its point participates at this motion; the motion may be described only if all parameters which indicate the position of each point are involved.

20.1.5.1 Lagrangian Formalism To put into evidence the treatment of the problem, we will start from a continuous mechanical system, which will be studied – at the beginning – by discrete methods in a Lagrangian formalism. Let be a linear elastic, isotropic and homogeneous straight bar (see Sect. 12.1.1), subjected to longitudinal small motions; it can be approximated (mathematically modelled) by a discrete system of particles Pi of equal masses m along the axis, separated at equal distances a (in a position of equilibrium) by springs of elastic constants k (Fig. 20.4a); during the motion, the particle Pi has a displacement ui along the bar axis, so that ui = ui ( t ) (Fig. 20.4b). As a matter of fact, we study thus a linear polyatomic molecule (case in which the number of particles is finite). We obtain thus the Lagrangian

Variational Principles. Canonical Transformations

L = T +U =

255

1 1 mui2 − ∑ k ( ui + 1 − ui )2 , 2∑ 2 i i

the forces which act on those particles (because of the springs at left and right) being Fi = − k ( ui − ui −1 ) + k ( ui +1 − ui ) . We may also write

L =

∑ aLi

, Li =

i

(

u − ui 1 ⎡m 2 ui − ka i +1 ⎢ a 2⎣a

)

2

⎤; ⎥⎦

there result Lagrange’s equations u −u u −u m u − ka i +1 2 i + ka i 2 i −1 = 0 . a i a a

In a process of passing to limit, m / a tends to the density μ , ( ui +1 − ui ) / a tends to the linear strain ε , Fi tends to the force F , while k tends to the rigidity of axial forces EA , where E is the longitudinal elasticity modulus and A is the area of the cross section ( F = EAε ). In this case, the kinetic potential will be (for a bar of length l)

Fig. 20.4 Discrete model of a linear elastic, isotropic, homogeneous straight bar l

∫ 0 L dx =

( ) ⎤⎥⎦ dx ,

1 ∞ ⎡ 2 ∂u μu − EA ⎢ ∫ −∞ ∂x 2 ⎣

2

(20.1.75)

where u = u ( x ; t ) and x is the abscissa along the bar axis; observing that, by passing to the limit,

( ) − ( ∂∂ux )

EA ⎡ ∂u a ⎣⎢ ∂x

x

x −a

⎤ ⎦⎥

leads to EA ( ∂ 2 u / ∂x 2 ) , we find the equation of motion μu − EA

∂ 2u = 0. ∂x 2

(20.1.75')

So, characteristic aspects of passing from discrete to continuous systems are put in evidence. As one can see, the position co-ordinate x is not a generalized one; it plays

MECHANICAL SYSTEMS, CLASSICAL MODELS

256

the rôle of a continuous index, replacing the numerable index i . As to every index i corresponds a generalized co-ordinate ui , so to every x corresponds a generalized coordinate u ( x ) ; taking into account also the dependence on time, we must have u = u ( x ; t ) . In the above conditions, L is – in fact – a Lagrangian density; in what follows we will use the notation L (as for the Lagrangian), because only the Lagrangian density plays a rôle in describing the motion of the mechanical system. The motion of a three-dimensional continuous mechanical system, which occupies the domain Ω , will be described – in general – by means of the Lagrangian

∫Ω L dΩ , L

=

L ( u , u1 , u2 , u3 , u ; x1 , x 2 , x 3 ; t ) ,

(20.1.76)

where L is the Lagrangian density, while ui = ∂u / ∂x i , i = 1, 2,..., n , u = ∂u / ∂t , x1 , x 2 , x 3 , t being independent variables. Hamilton’s principle in a synchronous case is written in the form t1

δ ∫ dt ∫ t0

Ω

L dΩ = 0 .

(20.1.76')

We notice that, in this case, the parameters x1 , x 2 , x 3 , t have not variations, the variation process having no influence neither on the integration domain Ω , nor on the integration limits with respect to time; thus, δu will vanish on ∂Ω as well as for t0 and t1 . On the basis of the results in Sect. 20.1.1.3, we can write Euler-Ostrogradskiĭ equations d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L − = 0, ⎜ ⎟+ dt ⎝ ∂u ⎠ dx i ⎜⎝ ∂ui ⎟⎠ ∂u

(20.1.77)

where we use the summation convention of dummy indices. Instead of s differential equations of second order for a discrete mechanical system, in case of a continuous one we find a partial differential equation in four variables: x1 , x 2 , x 3 , and t . In general, we may have several types of displacements u α = u α ( x1 , x 2 , x 3 ; t ) , α = 1, 2,...,m ; for instance, in case of vibrations of an elastic solid we have m = 3 . One obtains the equations of motion d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L − = 0 , α = 1, 2,...,m . ⎜ ⎟+ dt ⎝ ∂u α ⎠ dx i ⎝⎜ ∂uiα ⎠⎟ ∂u α

(20.1.78)

Introducing the functional derivative (the variational derivative – an extension of the Euler–Lagrange derivative) δ ∂ d ∂ = − δu α ∂u α dx i ∂uiα

(20.1.79)

and observing that δL / δu α = ∂L / ∂u α , α = 1, 2,...,m , we can write the equations of motion also in the form

Variational Principles. Canonical Transformations

257

d ⎛ δL ⎞ δL = 0 , α = 1, 2,...,m , ⎜ ⎟− dt ⎝ δu α ⎠ δu α

(20.1.79')

similar to that of Lagrange’s equations. 20.1.5.2 Hamiltonian Formalism To put in evidence the possibility of using the Hamiltonian formalism, we take again the example considered at the preceding subsection by discretization. We introduce thus the generalized momenta ∂L ∂L =a i ∂ui ∂ui

pi =

and build up the Hamiltonian H =

∑ pi ui

∂L

∑ a ∂uii ui

−L =

i

−L =

i

∂L

∑ a ⎛⎜⎝ ∂uii ui i

− Li ⎞⎟ . ⎠

Using the same process of passing to limit and introducing the Lagrangian density L , we obtain the Hamiltonian H =

∞

⎛ ∂L u − L ⎞ dx ; ⎟ ∂u ⎠

∫ −∞ ⎜⎝

we are thus led to use further the notion of density. In the motion of a three-dimensional continuous mechanical system which occupies the domain Ω and the position of which is specified by m generalized co-ordinates u α = u α ( x1 , x 2 , x 3 ; t ) , α = 1, 2,...,m , we introduce the generalized momenta density πα =

∂L , ∂u α

(20.1.80)

hence, the Hamiltonian density

H =

m

∑ πα u α − L

,

(20.1.81)

α =1

of the form H = H ( u α , u1α , u2α , u3α , u α , π α ; x1 , x 2 , x 3 ; t ) . Hamilton’s function is thus given by H =

so that

∫Ω H

dΩ =

⎛

m

α α ∫Ω ⎜⎝ α∑=1 π u

⎞ − L ⎟ dΩ , ⎠

(20.1.81')

MECHANICAL SYSTEMS, CLASSICAL MODELS

258 dH =

⎡

∂H

⎛ α ∫Ω ⎢⎣ α∑=1 ⎜⎝ ∂u α du m

+

∂H ∂H dπ α + α duiα ⎞⎟ + ∂π α ∂ui ⎠

H dt ⎤⎥ dΩ . ⎦

Integrating by parts and using the same procedure as in Sect. 20.1.1.3, we obtain dH =

⎧

∂H

∂H d ⎛ ∂H ⎞ α ⎤ ⎫ dπ α + du + H dt ⎬ dΩ dx i ⎝⎜ ∂uiα ⎠⎟ i ⎦⎥ ∂π α ⎭ ∂ H ⎡ m ⎛ ∂H ⎤ = ∫ ⎢ ∑ ⎜ α du α + α dπ α ⎞⎟ + H dt ⎥ dΩ , Ω⎣ ∂π ⎠ ⎦ α = 1 ⎝ ∂u

⎡ α ∫Ω ⎨⎩ α∑=1 ⎢⎣ ∂u α du m

+

where we used the notation (20.1.79) and observed that δ H / δπ α = ∂H / ∂π α . If we differentiate, starting from the second form (20.1.81'), we can write (we use the functional derivative) dH =

δL δL ⎡ m ⎛ α α ⎤ α α α α ⎞ ∫Ω ⎢⎣ α∑=1 ⎜⎝ u dπ +π du − δu α du − δu α du ⎟⎠ − Ldt ⎥⎦ dΩ ;

the notation (20.1.80) and the equations (20.1.79'), written in the form δL δL  α , α =π α , α =π δu δu

(20.1.82)

lead to dH =

⎡

m

α α ∫Ω ⎢⎣ α∑=1 ( − π du

⎤ + u α dπ α ) − Ldt ⎥ dΩ . ⎦

Comparing the two expressions of dH , we find the equations of motion u α =

δH δH , π α = − α , δπ α δu

(20.1.83)

analogue to Hamilton’s equations, as well as the relation

H = − L .

(20.1.83')

In a developed form, we have u α =

∂H d ⎛ ∂H ⎞ ∂H , π α = − α + , dx i ⎜⎝ ∂uiα ⎟⎠ ∂π α ∂u

(20.1.83'')

obtaining a form which has no more a symmetric character. The canonical form of Hamilton’s synchronous principle will be δ∫

t1

t0

⎡ m α α ⎤ α α α α α ∫Ω ⎢⎣ α∑=1 π u − H ( u , u1 , u2 , u3 , π ; x1 , x 2 , x 3 ; t ) ⎥⎦ dΩdt = 0 .

(20.1.84)

Variational Principles. Canonical Transformations

259

We may calculate dH = dt

⎡

m

∂H

⎛ α ∫Ω ⎢⎣ α∑=1 ⎜⎝ ∂u α u

+

∂H α ⎞ ⎤ π ⎟ + H ⎥ dΩ ; ∂π α ⎠ ⎦

with the aid of the equations of motion (20.1.83) we observe that, along the corresponding integral curves, we obtain dH = dt

∫Ω H dΩ .

(20.1.85)

Hence, the Hamiltonian H is conserved in time if the Hamiltonian density H does not depend explicitly on time ( H = 0 ). In general, let be the functional F =

∫Ω F dΩ ,

(20.1.86)

where F = F ( u α , u1α , u2α , u3α , u α , π α ; x1 , x 2 , x 3 ; t ) . We can write dF = dt

⎡ m ⎛ ∂F α ∂F α ⎞ ⎤ ∫Ω ⎢⎣ α∑=1 ⎜⎝ ∂u α u + ∂πα π ⎟⎠ + F ⎥⎦ dΩ ;

taking into account the equations of motion (20.1.83), there results dF = dt

⎡

m

δF δ H δF δ H ⎞ ⎤ ⎟ + F ⎥ dΩ . α − δπ δπ α δu α ⎠ ⎦

⎛ ∫Ω ⎢⎣ α∑=1 ⎜⎝ δu α

(20.1.87)

We observe thus that we obtain a quantity similar to the Poisson bracket; using an analogous notation, we may write dF = dt

∫Ω ⎡⎣ ( F , H

) + F ⎤⎦ dΩ .

(20.1.87')

We may thus affirm that F is conserved in time if simultaneously ( F , H ) = 0 and

F = 0 ; one can also say that F is a constant of motion. The conservation theorems for integral quantities can be obtained as in the previous chapter, and one can develop analogous methods of calculation. But we must observe that, besides the theorems concerning “macroscopic” constants of integration (for integral quantities), one can state theorems for “microscopic” such constants (for densities); the latter results are of a special interest in mechanics of continuous deformable media. We remark that all the above considerations may be used also for a field description with the aid of variational principles (even in the absence of a mechanical system). 20.1.5.3 Motion of Linear Elastic Solids To determine the density L of the Lagrangian in case of a linear elastic solid, we must calculate the densities T and V of the kinetic energy, and of the potential energy, respectively, so that

MECHANICAL SYSTEMS, CLASSICAL MODELS

260 T =

∫Ω TdΩ , V

=

∫ Ω Vd Ω ;

(20.1.88)

to do this, we need some preliminary considerations. Let f0 ( x10 , x 20 , x 30 ; t ) be a function of class C 1 , expressed by means of material coordinates; using spatial co-ordinates, we obtain the function in the form f ( x1 , x 2 , x 3 ; t ) (see Sect. 12.1.1.1). The partial derivative of first order with respect to one of the variables, e.g. the variable x10 , is written in the form

∂u ⎞ ∂f ∂f ∂x k ∂f ⎛ ∂f ∂u2 ∂f ∂u3 1 + 10 ⎟ + , = = + ∂x k ∂x10 ∂x1 ⎜⎝ ∂x10 ∂x1 ⎠ ∂x 2 ∂x10 ∂x 3 ∂x10

(20.1.89)

where we took into account (12.1.3''), u j , j = 1, 2, 3 , being the components of the displacement vector; neglecting the displacement gradient (the tensor αij = ∂u j / ∂x10 ,

i , j = 1, 2, 3 ) with respect to unity, one has ∂f0 ∂f . = 0 ∂x1 ∂x1

(20.1.89')

Hence, in this case, we obtain the same results, in the same form, either we use material or spatial co-ordinates. The symmetric part of the tensor αij is the strain tensor ( εij = u( i , j ) = ( ui , j + u j ,i ) / 2 = α( i , j ) ); if i = j one obtains the linear strains, while if i ≠ j , γij = 2 εij , i , j = 1, 2, 3 , represent the angular strains. The hypothesis of neglecting

the displacement gradient with respect to unity includes thus the case of infinitesimal deformations, case in which the relations between strains and displacements are linear. The antisymmetric part of the tensor αij is the local rotation of rigid body tensor ( ωij = u[ j ,i ] = ( u j ,i − ui , j ) / 2 = α[ i , j ] ) is also neglected with respect to unity (as the displacement gradient). We may thus speak about a linear theory from a geometric point of view (case in which relations of the form (20.1.89') hold). Taking into account the notions introduced in Sect. 12.1.1.1, from relations (12.1.2'') and (12.1.3'') we obtain the velocity and the acceleration (in spatial description) v =

du ∂u ∂u dv d2 u ∂v ∂v = = 2 = vj + , a= vj + ∂x j ∂t dt ∂ ∂t x dt dt j

(20.1.90)

for a point of the solid; the vector r0 (corresponding to the initial moment) is constant. Neglecting the non-linear terms, we remain with the displacement velocity (neglecting the spatial derivative, the total derivative is reduced to the time one) v =

∂u = u ∂t

(20.1.90')

Variational Principles. Canonical Transformations

261

and with the displacement acceleration a =

∂v ∂2 u  . = v = 2 = u ∂t ∂t

(20.1.90'')

In this case, the kinetic energy density will be given by 1 2

T = μui ui ,

(20.1.91)

where μ = μ ( r ; t ) is the density. The work effected by the external given forces is stored by the elastic solid and given back by unloading and bringing to the initial form as a work of internal forces (or internal work); it is called also deformation work or strain energy. We denote by w the unit deformation work (the deformation work density or the elastic potential), that is the work corresponding to the unit volume in the vicinity of a point of the body, and which the state of stress (the totality of stresses around a point) yields by the state of strain (the totality of strains around a point) of the solid at the very same point. We call elementary deformation work (elementary strain energy) the work corresponding to a volume element dΩ of the domain Ω occupied by the solid; we have dWint = wdΩ .

(20.1.92)

The internal work is then given by Wint =

∫Ω wdΩ ,

(20.1.92')

1 σ ε 2 ij ij

(20.1.93)

where w =

is the elastic potential. We have introduced the stress tensor σij , i , j = 1, 2, 3 (if i = j we obtain the normal stresses, while if i ≠ j we obtain the tangential stresses). If we take into account the linear constitutive law of R. Hooke

(

)

1 σij = λ ⎡ εll δij − 2 − ε ⎤ , i , j = 1, 2, 3 , ⎢⎣ ν ij ⎥⎦

(20.1.94)

in case of a homogeneous, isotropic body, where λ is Lamé’s elastic constant and ν is the transverse contraction coefficient of Poisson (Poisson’s ratio), we may write also w =

(

)

1 ⎡ 1 λ ( ε )2 − 2 − ε ε ν ij ij 2 ⎢⎣ ll

⎤. ⎥⎦

(20.1.93')

MECHANICAL SYSTEMS, CLASSICAL MODELS

262

Finally, the potential energy density is given by

V = w − F ⋅u,

(20.1.95)

where F is the volumic force (the external given force on the unit volume). We find thus the kinetic potential density of Lagrange

(

)

L = μu j u j − λ ⎡⎢ (u j , j )2 − 1 − (ui , j + u j ,i )(ui , j + u j ,i ) ⎤ + F ⋅ u ; ⎥⎦ 2 2 ⎣ 2 2ν 1

1

1

1

(20.1.96) Euler–Ostrogradskiĭ equations will be of the form (we remind observations in Sect. 20.1.1.3) d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L + − = 0 , j = 1, 2, 3 . dt ⎜⎝ ∂u j ⎟⎠ dx i ⎜⎝ ∂u j ,i ⎟⎠ ∂u j

(20.1.96')

Finally,

(

)

∂ 1 ( μu j ) − λ ⎡ ul ,lj − 1 − (ui , j + u j ,i ),i ⎤ + Fj = 0 , j = 1, 2, 3 , ⎢ ⎥⎦ ∂t 2ν ⎣

so that (1 − 2 ν ) Δu j + ui ,ij +

2ν ⎡ ∂ F − ( μu j ) ⎤ = 0 , j = 1, 2, 3 . λ ⎢⎣ j ∂t ⎦⎥

(20.1.97)

We obtain thus the equations of motion of the homogeneous, isotropic linear elastic solid, subjected to infinitesimal deformations; these equations, expressed in displacements, are G. Lamé’s equations. If μ = const , we can write (1 − 2 ν ) Δu j + ui ,ij +

2ν [ Fj − μuj λ

] = 0,

j = 1, 2, 3 .

(20.1.97')

In vectorial form, we have (1 − 2 ν ) Δu + grad div u +

2ν  ) = 0 , (F − u λ

(20.1.97'')

where the unknown function is the displacement vector u = u ( r ; t ) .

20.1.5.4 Motion of Strings Let be a string of length l, extended along the axis Ox 3 by a static tension T0 , without any external given force; in this case, the kinetic energy is given by T =

μ l 1 l 2 u μdx 3 = ∫ u 32 dx 3 , 2 ∫0 3 2 0

(20.1.98)

Variational Principles. Canonical Transformations

263

where μ = const is the linear density of the string (see Sect. 12.2.1.3). The potential energy is written in the form (we assume that both states of stress and strain are one-dimensional) V =

A l EA l 2 EA l 2 σ33 ε33 dx 3 = ε33 dx 3 = u dx . ∫ ∫ 2 0 2 0 2 ∫ 0 3,3 3

(20.1.98')

μ 2 EA 2 u − u . 2 3 2 3,3

(20.1.98'')

In this case,

L =

The Euler–Ostrogradskiĭ equation

d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L + − =0 ⎜ ⎟ dt ⎝ ∂u 3 ⎠ dx 3 ⎜⎝ ∂u3,3 ⎟⎠ ∂u 3 leads to the equation of longitudinal proper vibration of strings

EAu3,33 − μu3 = 0 ,

(20.1.99)

corresponding to the equations (12.2.20') and (20.1.75'). In case of displacements normal to the axis, let be this one along the Ox 1 –axis, the kinetic energy is given by T =

μ l 1 l 2 u μdx 3 = ∫ u12 dx 3 , 2 ∫0 1 2 0

(20.1.100)

the linear density being constant. The potential energy is calculated in the form (ds is along the deformed form of the string; see also Fig. 12.8) V =

l

l

∫0T0 (ds − dx 3 ) = T0 ∫0 (

l 2 1 2 1 + u1,3 dx 3 , − 1)dx 3 = T0 ∫ u1,3 0 2

(20.1.100')

where we have expanded the integrand into a series after Newton’s binomial and we have neglected terms of superior order (corresponding to the linearization hypothesis). We obtain

L =

μ 2 T0 2 u − u . 2 1 2 1,3

The Euler–Ostrogradskiĭ equation

d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L + − =0 ⎜ ⎟ dt ⎝ ∂u1 ⎠ dx 3 ⎜⎝ ∂u1,3 ⎟⎠ ∂u1

(20.1.100'')

MECHANICAL SYSTEMS, CLASSICAL MODELS

264 leads to

μu1 − T0 u1,33 = 0 ,

(20.1.101)

hence to the equation of proper transverse vibrations of strings, corresponding to the equations (12.2.22').

20.1.5.5 Motion of Bars In case of a straight bar one must take into account the transverse contraction, so that the transverse displacements of the bar Ox 3 –axis will be u1 = − νx1u3,3 , u2 = − ν x 2 u3,3 ,

the

corresponding

displacement

velocities

being

given

by

u1 = − νx1u 3,3 , u2 = − ν x 2 u 3,3 . We obtain the kinetic energy T =

μ 2 μ l 1 2 u1 + u 22 + u 32 ) dΩ = ∫ ( u 32 + ν 2 iO2 u 3,3 ( ) dx 3 , ∫ 2 ΩA 2 0

(20.1.102)

where iO is the gyration radius of the cross section with respect to the pole O, given by AiO2 =

∫∫ A ( x1

2

+ x 22 ) dA ,

(20.1.102')

while μ is the linear density, constant on the cross section. The potential energy is given by a formula of the form (20.1.98'), so that

L =

μ 2 1 2 2 u 3 + ν 2iO2 u 3,3 − EAu3,3 . ( ) 2 2

(20.1.102'')

The Euler–Ostrogradskiĭ equation (in the form of the equation (7.2.11')) d2 ⎛ ∂L ⎞ d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L − − + =0 dx 3 dt ⎜⎝ ∂u 3,3 ⎟⎠ dt ⎜⎝ ∂u 3 ⎟⎠ dx 3 ⎜⎝ ∂u 3,3 ⎟⎠ ∂u 3

leads to the A.E.H. Love’s equation EAu3,33 − μ ( u3 − ν 2 iO2 u3,33

) = 0.

(20.1.103)

Neglecting the term which multiplies ν 2 , we find the longitudinal proper vibrations equation of the straight bars in the classical form (20.1.99), in fact the equation (12.2.42). In case of the transverse vibrations of the straight bar, the kinetic energy due to the motion of translation and to the rotation of the cross section is T =

2 l ∂ l 2 1 l 2 1μ 1 l 1 μ ∫ u1 dx 3 + I 2 ∫ ⎡ (u1,3 ) ⎤ dx 3 = μ ∫ u12 dx 3 + μi22 ∫ u1,3 dx 3 , ⎢ ⎥ 0 0 0 0 2 2A 2 2 ⎣ ∂t ⎦ (20.1.104)

Variational Principles. Canonical Transformations

265

where I 2 is the moment of inertia with respect to the neutral axis of the bar ( Ox 2 -axis), i2 being the corresponding gyration radius. Observing that ε33 = − x1u1,33 , σ33 = E ε33 , we can write

V =

l 1 1 2 E u 2 dx x 2 dx dx = EI 2 u1,33 , 2 ∫ 0 1,33 3 ∫A 1 1 2 2

(20.1.104')

so that 1 2

1 2

2 2 L = μ ( u12 + i22 u1,3 . ) − EI 2u1,33

(20.1.104'')

As in the preceding case, the Euler–Ostrogradskiĭ equation d2 ⎛ ∂L ⎞ d2 ⎛ ∂L ⎞ d ⎛ ∂L ⎞ d + − − ⎜ ⎟ dx 3 dt ⎝ ∂u1,3 ⎠ dx 32 ⎜⎝ ∂u1,33 ⎟⎠ dt ⎜⎝ ∂u1 ⎟⎠ dx 3

⎛ ∂L ⎞ ∂L ⎜ ∂u ⎟ + ∂u = 0 1 ⎝ 1,3 ⎠

leads to the proper transverse vibrations equation of the straight bar in the form EI 2 u1,3333 + μ ( u1 − i22 u1,33

) = 0,

(20.1.105)

corresponding to the equation (12.2.47).

20.2 Canonical Transformations In general, applying directly the Hamiltonian formalism (instead of the Lagrangian one), the difficulty to solve a problem of mechanics does not diminish sensibly; the differential equations are practically equivalent. The advantage of using a Hamiltonian formalism consists – particularly – in a profound understanding of the formal structure of mechanics; there appears a greater liberty to choose canonical co-ordinates (generalized co-ordinates and momenta). The corresponding formulations, just because of their abstraction, allow to elaborate new theories concerning matter, nature, Universe; they constitute a starting point for quantum as well as for statistical mechanics. In Sect. 18.2.3.1 we have seen that point transformations let invariant Lagrange’s equations; a study of the transformations which let invariant Hamilton’s equations is thus necessary, so that the use of the Hamiltonian formalism is once more justified.

20.2.1 General Considerations. Conditions of Canonicity In what follows, canonical transformations are introduced; especially, accent is put on the statement of canonicity conditions. We mention also various applications, including those in the perturbation theory. A particular attention is given to infinitesimal canonical transformations.

266

MECHANICAL SYSTEMS, CLASSICAL MODELS

20.2.1.1 Preliminary Considerations. Examples Let be a transformation of canonical variables of the form Q j = Q j ( q , p ; t ) , Pj = Pj ( q , p ; t ) , j = 1, 2,..., s ,

(20.2.1)

where Q j and Pj are functions of class C 2 , having the Jacobian ∂ (Q , P ) ⎤ J = det ⎡ ≠ 0, ⎢⎣ ∂ ( q , p ) ⎥⎦

(20.2.1')

with Q ≡ {Q1 ,Q2 ,...,Qs } , P ≡ {P1 , P2 ,..., Ps } , q ≡ {q1 , q 2 ,..., qs } , p ≡ { p1 , p2 , ..., ps } ; the time t plays the rôle of a parameter in this transformation. Hamilton’s function becomes H ( q , p ;t ) = H ( q (Q , P ;t ) , p (Q , P ; t ) ;t ) = H ( Q , P ; t ) . The transformation (20.2.1), (20.2.1') is called canonical transformation if Hamilton’s canonical system (19.1.14) keeps its form, hence if ∂H  ∂H , Pj = − , j = 1, 2,..., s . Q = ∂ Pj ∂Q j

(20.2.2)

The theorem of implicit functions and the condition (20.2.1') ensure the existence of the inverse transformation qk = qk (Q , P ; t ) , pk = pk ( Q , P ; t ) , k = 1, 2,..., s ,

(20.2.1'')

these functions being also of class C 2 . The canonical co-ordinates q , p and Q , P correspond to the same representative point in the space Γ 2 s . The transformations (20.2.1), (20.2.1'), and (20.2.1'') are called complete canonical transformations if they do not depend explicitly on time. Together with A. Sommerfeld, these transformations are called contact transformations too (Sommerfeld, A., 1962). Other authors give this denomination to the transformations (20.2.1) to which we add the function T = T ( q , p ;t ) ,

(20.2.3)

of the same class C 2 (hence, for which t is no more a parameter), so that ∂ ( Q , P ;T ) ⎤ det ⎡ ≠ 0. ⎢⎣ ∂ ( q , p ; t ) ⎥⎦

(20.2.3')

But it is not necessary to consider such transformations in the following study; they can be useful in a relativistic modelling of mechanical phenomena. Starting from canonical equations of a complicated form, we can obtain simpler equations of motion by means of canonical transformations; practically, the Hamiltonian H can have a simpler structure than Hamilton’s function H . In particular,

Variational Principles. Canonical Transformations

267

if we find the canonical transformation for which H = const (independent of Q and P ) along the trajectory of the representative point, then the solution of the canonical system (20.2.2) will be Q j = Aj , Pj = B j , j = 1, 2,..., s ,

(20.2.4)

where Aj and B j are constants; the functions (20.2.1) are, in this case, first integrals of the canonical system (19.1.14). Let be the transformation Q j = αq j , Pj = δ p j , j = 1, 2,..., s ,

(20.2.5)

where α and δ are non-vanishing constants; the equations (20.2.2) are easily verified, together with the equations (19.1.14), in case of the new Hamiltonian

H = αδH .

(20.2.5')

Hence, this transformation is a complete canonical transformation. As well, the transformation Q j = β p j , Pj = γq j , j = 1, 2,..., s ,

(20.2.6)

where β and γ are non-zero constants, and for which H = − βγ H ,

(20.2.6')

is also a complete canonical transformation. Analogically, we can show that the transformation Q j = p j tan t , Pj = q j cot t , j = 1, 2,..., s ,

(20.2.7)

with H = −H +

2 QP , sin 2t j j

(20.2.7')

is a canonical one. The direct proof of this affirmation is rather complicated. For instance, we have 1 p j + p j tan t , cos2 t ∂H ∂H 2 ∂H 1 = − + tan t + Q = − pj ; sin 2t j ∂ Pj ∂ Pj ∂Q j cos2 t Q j =

taking into account (19.1.14), the first equation (20.2.2) is verified. As well, it is not completely clear how one can obtain such transformations. For this goal, we will give – in what follows – various conditions of canonicity and will put in evidence the possibility to construct canonical transformations.

MECHANICAL SYSTEMS, CLASSICAL MODELS

268

20.2.1.2 Complete Canonical Transformations Let be a complete canonical transformation Q j = Q j (q , p ) , Pj = Pj (q , p ) , j = 1, 2,.., s ,

(20.2.8)

where Q j , Pj are functions of class C 2 which verify the condition (20.2.1'). We consider the differential form of the first degree (the one-form) ψ = pk dqk − Pj dQ j .

(20.2.9)

The canonical form (20.1.41) of Hamilton’s principle leads to t1

t1

t0

t0

δ ∫ [ pk dqk − H ( q , p ; t ) dt ] = δ ∫

[ Pj dQ j

− H ( Q , P ; t ) dt ] .

To obtain the canonical equations (20.2.2), corresponding to Livens’s Theorem 20.1.6, as a consequence of the complete canonical transformation (20.2.1), it is sufficient that the one-form (20.2.9) be an exact differential, hence dψ = 0 ; as a matter of fact, this represents a sufficient condition of complete canonicity for the transformation (20.2.8). On the basis of Poincaré’s lemma, we have d(dψ ) = 0 , the operator d being Cartan’s external differential operator (see App., Subsec. 1.2.2). Corresponding to the reciprocal of this lemma, if dψ = 0 , then there exists a scalar function V so that ψ = dV ,

(20.2.9')

with V = V ( q , p ;Q , P ) , which cannot depend explicitly on time; in fact, the function V can depend only on two from the four sets of co-ordinates q , p ,Q and P . Corresponding to the form (19.1.45), we can write Pfaff’s forms of Hamilton ω = pk dqk − H ( q , p ; t ) dt , Ω = Pj dQ j − H ( Q , P ; t ) dt .

(20.2.10)

We have proved in Sect. 19.1.1.9 that the two forms have the same bilinear covariant, hence they lead both to the same canonical equations (the same associate differential system), if and only if Ω = c ω − dV ,

(20.2.10')

where c is a constant, while V = V ( q , p ;Q , P ) is a function of class C 2 ; we make the same observation as above for the arguments of the function V . The necessity of introducing the constant c will be clearly put in evidence in Sect. 21.1.2.3, in the study of integral invariants. We notice that c ω − Ω = cpk dqk − Pj dQ j − ( cH − H ) dt = dV ;

because V = ∂V / ∂t = 0 , it follows

Variational Principles. Canonical Transformations

269 (20.2.10'')

H = cH .

We introduce thus the differential form ψc = cpk dqk − Pj dQ j = dVc , c ≠ 0 , c = const .

(20.2.11)

The constant c is called the valency of the canonical transformation. If c = 1 , then the transformation is univalent; in this case H = H . Usually, one has to do with univalent canonical transformations. We observe that the condition (20.2.9') (with c = 1 ) is always sufficient; in some particular cases, it is also necessary (the transformations are univalent). In the particular case in which V = 0 , the complete canonical transformation is called, together with E. Mathieu, homogeneous, and the relation pk dqk = Pk dQk

(20.2.9'')

takes place; in this case, if the variable pk is multiplied by a factor, then the variable Pk is multiplied by the same factor. Taking into account (20.2.9), (20.2.9'), we may write ∂Q j ∂V ⎛ ∂Q j ⎞ ∂V dqk + dpk ⎟ = dqk + dpk , pk dqk − Pj ⎜ ∂ pk ∂ pk ⎝ ∂qk ⎠ ∂qk

where we have considered V = V (q , p ) ; there results

pk − Pl

∂Ql ∂Ql ∂V ∂V , − Pl , k = 1, 2,..., s . = = ∂qk ∂qk ∂ pk ∂ pk

Eliminating the function V between these relations or (it is the same thing) writing that the mixed derivatives of second order of this function −

−

∂ Pl ∂Ql ∂ 2Ql ∂ P ∂Ql ∂ 2Ql ∂ 2V ∂ 2V − Pl = , − l − Pl = , ∂q j ∂qk ∂q j ∂qk ∂q j ∂qk ∂qk ∂q j ∂qk ∂q j ∂qk ∂q j

∂ Pl ∂Ql ∂ 2Ql ∂ P ∂Ql ∂ 2Ql ∂ 2V ∂ 2V , − l , − Pl = − Pl = ∂ p j ∂ pk ∂ p j ∂ pk ∂ p j ∂ pk ∂ pk ∂ p j ∂ pk ∂ p j ∂ pk ∂ p j

δjk −

∂ Pl ∂Ql ∂ 2Ql ∂ P ∂Ql ∂ 2Ql ∂ 2V ∂ 2V − Pl = , − l − Pl = ∂ p j ∂qk ∂ p j ∂qk ∂ p j ∂qk ∂qk ∂ p j ∂qk ∂ p j ∂qk ∂ p j

do not depend on the order of differentiation, we obtain the conditions of complete canonicity of the considered transformation in the form

[q j , qk ] = 0 , [ p j , pk ] = 0 , [q j , pk ] = δjk ,

j , k = 1, 2,..., s ,

(20.2.12)

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where δjk is Kronecker’s symbol; we can write 2 [ s (s − 1)/ 2 ] + s 2 = s (2s − 1) such conditions. We have introduced the Lagrange brackets by relations of the form (19.1.50), where q j , p j , j = 1, 2,..., s , play the rôle of integration constants in the integrals (20.2.8) of the canonical system (20.2.2). The conditions (20.2.12) can be written also in the form

∂Ql ⎞ ∂Ql ⎞ ∂ ⎛ ∂ ⎛ = , pk − Pl p j − Pl ⎜ ⎟ ⎜ ∂q j ⎝ ∂qk ⎠ ∂qk ⎝ ∂q j ⎟⎠ ∂ ⎛ ∂Ql ⎞ ∂ ⎛ ∂Ql ⎞ P P = , ∂ p j ⎜⎝ l ∂ pk ⎟⎠ ∂ pk ⎜⎝ l ∂ p j ⎟⎠ ∂Ql ⎞ ∂Ql ⎞ ∂ ⎛ ∂ ⎛ ; = −P p − Pl ∂ p j ⎜⎝ k ∂qk ⎟⎠ ∂qk ⎜⎝ l ∂ p j ⎟⎠ there exists thus a function V so that ∂V / ∂qk and ∂V / ∂ pk be given by the relations from which we started. We obtain then (20.2.9) and (20.2.9'). Hence, the conditions (20.2.12) are equivalent to (20.2.9), (20.2.9'), being thus sufficient conditions of complete canonicity. If c ≠ 1 , then the conditions (20.2.12) take the form

[q j , qk ] = 0 , [ p j , pk ] = 0 , [q j , pk ] = cδjk ,

j , k = 1, 2,..., s ,

(20.2.12')

being necessary and sufficient conditions of complete canonicity. Obviously, we can write also the sufficient conditions of complete canonicity

[Q j ,Qk ] = 0 , [ Pj , Pk ] = 0 , [Q j , Pk ] = δjk ,

j , k = 1, 2,..., s ,

(20.2.12'')

corresponding to the inverse complete canonical transformations. As well, if c ≠ 1 , then we may write the necessary and sufficient conditions of complete canonicity 1

[Q j ,Qk ] = 0 , [ Pj , Pk ] = 0 , [Q j , Pk ] = c δjk ,

j , k = 1, 2,..., s .

(20.2.12''')

Using the Poisson brackets (defined by the relation (19.1.54)), we see that one can write Q j =

∂Q j ∂Q j ∂Q j ∂ H ∂Q j ∂ H − = ( Q j , H ) , Pj = ( Pj , H ) , q + p = ∂qk k ∂ pk k ∂qk ∂ pk ∂ pk ∂qk

along the integral curves of the canonical system (19.1.14); as well, we have dQ j =

∂Q j ∂Q j ∂ Pj ∂ Pj dqk + dp , dPj = dq + dp . ∂qk ∂ pk k ∂qk k ∂ pk k

Taking into account the Lagrange brackets (20.2.12), we obtain

Variational Principles. Canonical Transformations

271

∂Ql ∂ H ⎞ ⎛ ∂ Pl ∂ Pl ⎛ ∂Q ∂ H ⎞ Ql dPl − Pl dQl = ⎜ l − ⎟ ⎜ ∂q dqk + ∂ p dpk ⎟ q p p q ∂ ∂ ∂ ∂ ⎠ j j j ⎠⎝ k k ⎝ j ∂ Pl ∂ H ⎞ ⎛ ∂Ql ∂Ql ⎛ ∂P ∂H ⎞ −⎜ l − ⎟ ⎜ ∂q dqk + ∂ p dpk ⎟ ∂ ∂ ∂ ∂ q p p q ⎠ j j j ⎠⎝ k k ⎝ j ∂H ∂H ⎞ ∂H ∂H ⎛ ⎛ = ⎜ [ qk , p j ] + [ q j , qk ] ⎟ dqk + ⎜ [ q j , pk ] ∂ p + [ pk , p j ] ∂q q p ∂ ∂ j j j j ⎝ ⎠ ⎝ ∂H ∂H = dq + dp = dH − H dt ; ∂qk k ∂ pk k

⎞ ⎟ dpk ⎠

if H = H , then we are led to the canonical system (20.2.2) (the expression dH − H dt is invariant by a complete canonical transformation). Thus, we verify once more that (20.2.9') constitutes a sufficient condition for the considered transformations be complete canonical, Hamilton’s function preserving its form also in the new canonical co-ordinates. Let us represent the Jacobian (20.2.1') as the determinant of a compound matrix ⎡ ⎡ ∂Q j ⎢ ⎢ ∂q ⎣ k J = det ⎢ ⎢ ⎡ ∂ Pj ⎢ ⎢ ∂q ⎣⎣ k

⎤ ⎡ ∂Q j ⎥ ⎢ ∂p ⎦ ⎣ k ⎤ ⎡ ∂ Pj ⎥ ⎢ ⎦ ⎣ ∂ pk

⎤⎤ ⎥⎥ ⎦⎥ , j , k = 1, 2,..., s , ⎤⎥ ⎥⎥ ⎦⎦

the index j indicating the row and the index k the column in the four submatrices with s × s elements. We will interchange the first row of submatrices with the second one, and then the first column of submatrices with the second one; the determinant of the new matrix will be as well J (with the same sign). We multiply then the submatrices of the first row, as well as the submatrices of the first column, successively, by −1 ; transposing this matrix, we obtain the determinant

J∗

T ⎡ ⎡ ∂ Pj ⎤ T ⎡ ⎡ ∂ Pj ⎤ ⎡ ∂ Pj ⎤ ⎤ − ⎢ ⎢ ∂p ⎥ ⎢ ⎢ ∂ p ⎥ ⎢ ∂q ⎥ ⎥ ⎣ k ⎦ ⎣ ⎣ k ⎦ k ⎦ ⎢ ⎥ = det = det ⎢ ⎢ ∂ Q ∂ Q ⎢⎡ ∂P T j ⎤ ⎡ j ⎤ ⎥ ⎢ ⎡⎢ − j ⎤⎥ ⎢ ⎢ − ∂ p ⎥ ⎢ ∂q ⎥ ⎥ ⎣⎣ ⎣ k ⎦ ⎦ k ⎦ ⎣⎢ ⎣ ∂qk ⎦

T ⎡ ∂Q j ⎤ − ⎢ ∂p ⎥ ⎣ k ⎦ T ⎡ ∂Q j ⎤ ⎢ ⎥ ⎣ ∂qk ⎦

⎤ ⎥ ⎥. ⎥ ⎥ ⎦⎥

We notice that J ∗ = J , so that

⎡ [ [ q j , pk ] ] J 2 = JJ ∗ = det ⎢ ⎣⎢ [ [ p j , pk ] ]

[ [ qk , q j ] ] ⎤ ⎥ [ [ qk , p j ] ] ⎦⎥

= 1,

(20.2.13)

where we took into account the conditions of complete canonicity (20.2.12); hence J = ±1 . In case of a transformation of valency c , we have J = ±c s . Because the Jacobian J does not vanish, the condition (20.2.1') is always fulfilled in case of a complete canonical transformation.

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272

Let F = F ( q , p ) be a function of class C 1 . We construct the differential form of first degree ϕ = − ( Pj , F ) dQ j + ( Q j , F ) dPj ;

(20.2.14)

developing the Poisson brackets ( Q j , F ) , ( Pj , F ) , j = 1, 2,..., s , we obtain

∂F ∂F ⎞ ∂F ∂F ⎞ ⎛ ⎛ + [ q j , qk ] ϕ = ⎜ [ qk , p j ] ⎟ dqk + ⎜ [ pk , p j ] ∂q + [ q j , pk ] ∂ p ⎟ dpk . q p ∂ ∂ j j ⎠ j j ⎠ ⎝ ⎝ If the relations (20.2.12) take place, then the transformation ( Q , P ) → ( q , p ) is complete canonical, the differential form being given by ω = dF , where F is an arbitrary function of class C 1 . Taking into account (20.2.14), we have

(Qj , F ) =

∂F ∂F , ( Pj , F ) = − , j = 1, 2,..., s . ∂ Pj ∂Q j

(20.2.14')

In particular, if F = H , then we find Hamilton’s equations, written by means of the Poisson brackets (equations of the form (19.1.60')) Q j = ( Q j , H ) , Pj = ( Pj , H ) , j = 1, 2,..., s .

(20.2.14'')

If we equate F to Qk and Pk , successively, we obtain

(Q j ,Qk ) = 0 , ( Pj , Pk ) = 0 , (Q j , Pk ) = δjk ,

j , k = 1, 2,..., s ;

(20.2.15)

these relations constitute sufficient conditions of complete canonicity, equivalent to the conditions (20.2.12). Analogically, the conditions ( c ≠ 1 )

(Q j ,Qk ) = 0 , ( Pj , Pk ) = 0 , (Q j , Pk ) = cδjk ,

j , k = 1, 2,..., s ,

(20.2.15')

represent necessary and sufficient conditions of complete canonicity. Obviously, we may write also the sufficient conditions of complete canonicity

( q j , qk ) = 0 , ( p j , pk ) = 0 , ( q j , pk ) = δjk ,

j , k = 1, 2,..., s ,

(20.2.15'')

corresponding to the inverse complete canonical transformations. As well, if c ≠ 1 , then we can write the necessary and sufficient conditions of complete canonicity 1

( q j , qk ) = 0 , ( p j , pk ) = 0 , ( q j , pk ) = c δjk , Let be the Pfaffian

j , k = 1, 2,..., s .

(20.2.15''')

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273

∂ pk ∂q ∂ pk ⎛ ∂qk ∂q ∂q ∂ p ∂p dq − k dp = dQl + k dPl ⎞⎟ − k ⎛⎜ k dQl + k dPl ⎞⎟ ∂ Pj k ∂ Pj k ∂ Pj ⎜⎝ ∂Ql ∂ Pl ∂ P ∂ Q ∂ Pl ⎠ ⎠ j ⎝ l ∂Q j ∂Q j dqk + dp , = [Ql , Pj ] dQl + [ Pl , Pj ] dPj = dQ j = ∂qk ∂ pk k

where we took into account the sufficient conditions of complete canonicity (20.2.12''); analogically ∂ Pj ∂ Pj ∂ pk ∂q dqk − k dpk = − dPj = − dqk − dp . ∂Q j ∂Q j ∂qk ∂ pk k

By identification, we obtain the relations ∂Q j ∂ Pj ∂ Pj ∂ pk ∂Q j ∂q ∂p ∂qk , , j , k = 1, 2,..., s . = = − k , = − k , = ∂qk ∂ Pj ∂ pk ∂ Pj ∂qk ∂Q j ∂ pk ∂Q j

(20.2.16) In this case, we may write

(Q j ,Qk ) =

∂Q j ∂Qk ∂Q j ∂Qk ∂ p ∂ql ∂q ∂ pl − = − l + l , ∂ql ∂ pl ∂ pl ∂ql ∂ Pj ∂ Pk ∂ Pj ∂ Pk

as well as other analogous relations; we find thus the relations between Poisson brackets for the inverse complete canonical transformations (20.2.8) and Lagrange brackets for the corresponding direct transformations

(Q j ,Qk ) = [ Pj , Pk ] , ( Pj , Pk ) = [Q j ,Qk ] , (Q j , Pk ) = [Q j , Pk ] .

(20.2.16')

If c ≠ 1 , then the relations (20.2.16) are of the form ∂Q j ∂ Pj ∂ Pj ∂ p ∂Q j ∂q ∂p ∂q =c k , = −c k , = −c k , = c k , j , k = 1, 2,..., s , ∂qk ∂ Pj ∂ pk ∂ Pj ∂qk ∂Q j ∂ pk ∂Q j

(20.2.16'') while the relations (20.2.16') become

(Q j ,Qk ) = c 2 [ Pj , Pk ] , ( Pj , Pk ) = c 2 [Q j ,Qk ] , (Q j , Pk ) = c 2 [Q j , Pk ] , j , k = 1, 2,..., s .

(20.2.16''')

Analogically, one can establish the relations 1 1 [ p j , pk ] , ( p j , pk ) = 2 [ q j , qk ] , c2 c 1 ( q j , pk ) = 2 [ q j , pk ] , j , k = 1, 2,..., s ; c

( q j , qk ) =

(20.2.16iv)

MECHANICAL SYSTEMS, CLASSICAL MODELS

274 if c = 1 , then we have

( q j , qk ) = [ p j , pk ] , ( p j , pk ) = [ q j , qk ] , ( q j , pk ) = [ q j , pk ] ,

j , k = 1, 2,..., s .

(20.2.16v) Thus, starting from (20.2.12'') and (20.2.12), we find again the sufficient conditions of complete canonicity (20.2.15) and (20.2.15''), respectively; as well, the conditions (20.2.12''') and (20.2.12') allow us to write the necessary and sufficient conditions of complete canonicity (20.2.15') and (20.2.15'''), respectively.

20.2.1.3 Canonical Transformations In case of an arbitrary transformation (20.2.1), (20.2.1'), in which the time appears explicitly, we consider the one-form ψ = pk d qk − Pj dQ j − ( H − H ) d t .

(20.2.17)

In this case too, a sufficient condition of canonicity imposes the differential form (20.2.17) to be an exact differential, hence ψ = dW ,

(20.2.17')

where W = W ( q , p ,Q , P ; t ) is a function of class C 2 ; taking into account the canonical form (20.1.41) of Hamilton’s principle, one can easily prove this affirmation. If W = 0 , then the transformation is homogeneous. Introducing Pfaff’s forms of Hamilton (20.2.10), the necessary and sufficient condition of canonicity will be Ω = c ω − dW ,

(20.2.18)

where c is a non-vanishing constant, while W = W ( q , p ,Q , P ; t ) is a function of class C 2 or c ω − Ω = cpk dqk − Pj dQ j − ( cH − H ) dt = dW .

(20.2.18')

As a matter of fact, in this case we introduce the differential form ψc = cpk dqk − Pj dQ j − ( cH − H ) dt = dWc .

(20.2.18'')

For t = t ∗ , arbitrary but fixed, the condition (20.2.18') becomes Pj dQ j = cpk dqk − dW ( q , p ,Q , P ; t ∗ ) ;

(20.2.19)

this is a necessary and sufficient condition for the transformation Q j = Q j ( q , p ; t ∗ ) , Pj = Pj ( q , p ; t ∗ ) , j = 1, 2,..., s , be complete canonical. For the transformation

(20.2.1) be canonical, it is necessary to be canonical for any fixed t (the time is considered as a parameter), with the same valency c .

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275

Supposing that the transformation (20.2.1) is canonical for any t , it follows that we have (20.2.19) for any t . After transformation, let us define the Hamiltonian H by H = cH + W + PjQ j ;

(20.2.19')

we obtain thus

(

)

Pj Q j dt = −W dt − c H − H dt .

Summing member by member with (20.2.19) (where the differentials dQ j and dW are not extended to the time t , which is fixed), we find again the condition (20.2.18'). Hence, the condition (20.2.18) is a necessary and sufficient condition of canonicity. On the basis of the above affirmation, it results that also the conditions (20.2.12) are sufficient conditions of canonicity, while the conditions (20.2.12') are not only sufficient but also necessary for the canonicity of the transformation. One can then make analogous affirmations for all conditions established at the previous subsection by means of Lagrange’s or Poisson’s brackets. As well, the Jacobian (20.2.1') is non-zero in the general case of canonical transformations too. Let qk = qk ( q 0 , p 0 ; t ) , pk = pk ( q 0 , p 0 ; t ) , k = 1, 2,..., s ,

(20.2.20)

be the solution of the canonical equations (19.1.14), with the initial conditions q 0 = q ( t0 ) and p 0 = p ( t0 ) . According to Lagrange’s Theorem 19.1.4, the Lagrange brackets [ q j0 , qk0 ] , [ p j0 , pk0 ] , [ q j0 , pk0 ] do not depend on time; for t = t0 we have

[q j0 , qk0 ] = 0 , [ p j0 , pk0 ] = 0 , [q j0 , pk0 ] = δjk , Hence, the transformation

( q 0 , p0 ) → ( q , p )

j , k = 1, 2,..., s .

given by (20.2.20) is a univalent

canonical transformation. More general, we can state that the values of the canonical co-ordinates at the moment t + τ can be expressed as functions of the values of the same co-ordinates at the moment t in the form q j ( t + τ ) = q j ( q ( t ) , p ( t ) ; τ ) , p j ( t + τ ) = p j ( q ( t ) , p ( t ) ; τ ) , j = 1, 2,..., s ;

,

(20.2.20')

hence, the variation of the canonical co-ordinates q and p during the motion can be considered as a canonical transformation. We can state that the general integral of a dynamical system, on one hand, and the canonical transformations of this system, on the other hand, are two problems covering reciprocally themselves from the point of view of their contents.

20.2.1.4 Point Transformations A transformation of the form Q j = Q j ( q ) , Pj = Pj ( q , p ) , j = 1, 2,..., s ,

(20.2.21)

MECHANICAL SYSTEMS, CLASSICAL MODELS

276 with the functional determinant

∂ (Q , P ) ⎤ ⎡ ∂Q j J = det ⎡ = det ⎢ ⎢⎣ ∂ ( q , p ) ⎥⎦ ⎣ ∂qk

⎤ ⎡ ∂ Pj ⎤ ⎥ det ⎢ ∂ p ⎥ ≠ 0 , ⎦ ⎣ k ⎦

(20.2.21')

is called a point transformation; in this case, we also have ⎡ ∂Q j ⎤ ⎡ ∂ Pj ⎤ det ⎢ ≠ 0 , det ⎢ ⎥ ⎥ ≠ 0. ⎣ ∂qk ⎦ ⎣ ∂ pk ⎦

(20.2.21'')

The condition (20.2.9') of complete canonicity can be written in the form pk dqk = Pj dQ j = Pj

∂Q j dqk , ∂qk

in case of a homogeneous transformation; hence, it is sufficient to have pk = Pj

∂Q j , k = 1, 2,..., s , ∂qk

for the point transformation (20.2.21), (20.2.21') be completely canonical and homogeneous. Multiplying both members by ∂qk / ∂Ql , k , l = 1, 2,..., s , and summing, we may state that the point transformation Q j = Q j ( q ) , Pj = pk

∂qk , j = 1, 2,..., s , ∂Q j

(20.2.22)

linear with respect to the generalized momenta, is complete canonical and homogeneous. We notice that, in the second group of relations, one introduces Q j = Q j ( q ) , j = 1, 2,..., s , to have the transformations written in the form (20.2.21). The condition (20.2.21') is, in this case, automatically fulfilled, as we have seen in Sect. 20.2.1.2. We can write the relations of inverse transformation qk = qk ( Q ) , pk = Pj

∂Q j , k = 1, 2,..., s , ∂qk

(20.2.22')

where we introduce qk = qk ( Q ) in the second group of relations, to have the transformation written in the form pk = pk ( Q , P ) , k = 1, 2,..., s . Let us introduce also a more general transformation, of the form Q j = Q j (q ; t ) , Pj = Pj (q , p ; t ) , j = 1, 2,..., s ,

(20.2.23)

where the time appears explicitly, and which verifies the condition (20.2.21') (implicitly the conditions (20.2.21'')). The canonicity condition is

Variational Principles. Canonical Transformations

277

⎛ ∂Q j ⎞ dqk + Q dt ⎟ + ( H − H ) dt ; pk dqk = Pj dQ j + ( H − H ) dt = Pj ⎜ ∂ q ⎝ k ⎠

hence, it follows pk = Pj

∂Q j , k = 1, 2,..., s . ∂qk

As in the previous case, we can state that the generalized point transformation Q j = Q j ( q ; t ) , Pj = pk

∂qk , j = 1, 2,..., s , ∂Q j

(20.2.24)

linear with respect to the generalized momenta, is canonical and homogeneous; we make Q j = Q j ( q ; t ) in the second group of relations, to have transformations of the form (20.2.23). We notice that the Hamiltonian is transformed in the form H = H + Pj Q j .

(20.2.24')

The relations of inverse transformation are qk = qk ( Q ; t ) , pk = Pj

∂Q j , k = 1, 2,..., s , ∂qk

(20.2.25)

where we introduce qk = qk ( Q ; t ) , k = 1, 2,..., s , in the second group of relations; the Hamiltonian is given by H = H + pk qk .

(20.2.25')

20.2.1.5 Free Canonical Transformations. Generating Functions of Canonical Transformations Let be a transformation (20.2.1), (20.2.1') for which the supplementary condition ⎡ ∂Q j ⎤ det ⎢ ⎥ ≠0 ⎣ ∂ pk ⎦

(20.2.26)

takes place; such a transformation is called a free transformation. In this case, on the basis of the theorem of implicit functions, the first group of relations (20.2.1) allows us to calculate pk = pk ( Q , q ; t ) , and then Pj = Pj ( Q , q ; t ) , j , k = 1, 2,..., s ; as well, the function W can be expressed in the form W ( q , p ; t ) = S ( Q , q ; t ) . Thus, the sufficient condition (20.2.17), (20.2.17') takes the form pk dqk − Pj dQ j − ( H − H ) dt =

∂S ∂S dQ j + dq + Sdt ; ∂Q j ∂qk k

MECHANICAL SYSTEMS, CLASSICAL MODELS

278 we obtain

pk =

∂S ∂S , Pj = − , j , k = 1, 2,..., s , ∂qk ∂Q j H = H + S .

(20.2.27) (20.2.28)

These relations define the canonical transformations (sufficient conditions). Indeed, differentiating the first relation (20.2.27) with respect to time, we may write pk =

∂ 2S ∂ 2S  ∂ 2S ; ql + Ql + ∂ql ∂qk ∂Ql ∂qk ∂t ∂qk

inverting the differentiation order and using the relations (20.2.27), (20.2.28), we obtain pk =

∂ pl ∂P ∂ q − l Q + (H − H ) . ∂qk l ∂qk l ∂qk

We have H ( Q , P ; t ) = H ( Q , P ( Q , q ; t ) ; t ) and H ( q , p ; t ) = H ( q , p ( Q , q ; t ) ; t ) , so that ∂ (H − H ∂qk

)

=

∂ H ∂ Pl ∂ H ∂ H ∂ pl ; − − ∂ Pl ∂qk ∂qk ∂ pl ∂qk

we find the relation pk +

∂ pl ∂H = ∂qk ∂qk

⎛ q − ∂ H ⎞ − ∂ Pl ⎜ l ∂ p ⎟ ∂q ⎝ l ⎠ k

∂H ⎞ ⎛  ⎜ Ql − ∂ P ⎟ . ⎝ l ⎠

Analogically, the second relation (20.2.27) leads to ∂P ∂H Pj + = − l ∂Q j ∂Q j

∂ H ⎞ ∂ pk ⎛  ⎜ Ql − ∂ P ⎟ − ∂Q j ⎝ l ⎠

⎛ q − ∂ H ⎞ . ⎜ k ∂p ⎟ ⎝ k ⎠

Assuming that Hamilton’s equations (20.2.2) take place and taking into account (20.2.26), the second group of relations leads to qk = ∂ H / ∂ pk ; replacing in the first group of relations, we obtain the second group of Hamilton’s equations (19.1.14). Hence, the representation (20.2.27) is sufficient to obtain a free canonical transformation. If the Hessian of the function S is non-zero ⎡ ∂ 2S ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂Q j ∂qk ⎦

(20.2.29)

then we obtain the generalized co-ordinates Q from the first subsystem (20.2.27); by replacing in the second subsystem (20.2.27), we are led to the complete determination

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279

of the canonical co-ordinates Q , P as functions of the co-ordinates q , p and of the time t , hence to the transformation (20.2.1). As well, starting from the condition (20.2.29) we obtain the generalized co-ordinates q from the second subsystem (20.2.27); by replacing in the first subsystem (20.2.27), we get the inverse canonical transformation (20.2.1''). The relation (20.2.28) puts into evidence the connection between Hamilton’s functions corresponding to the two canonical systems. Hence, we may state Theorem 20.2.1 (Jacobi). A function S ( Q , q ; t ) of class C 2 which verifies the condition (20.2.29) generates a free canonical transformation by means of the relations (20.2.27). Indeed, starting from the first subsystem (20.2.27), we obtain Q j = Q j ( q , p ; t ) ,

j = 1, 2,..., s ; this subsystem will be identically satisfied if we replace the generalized co-ordinates Q j thus expressed. Differentiating with respect to pl , we obtain (we observe that the function S depends on pl only by means of the generalized coordinates Q j ) ∂ pk ∂ 2 S ∂Q j = = δk l . ∂Q j ∂qk ∂ pl ∂ pl

The Hessian (20.2.29) does not vanish, hence the corresponding matrix is non-singular; therefore, the matrix [ ∂Q j / ∂ pl ] , which is the inverse of the previous one, is also nonsingular, the condition (20.2.26) being thus fulfilled. The function S is called the generating function of free canonical transformations. If the Jacobian (20.2.26) is identically zero, then there exists at least one relation connecting the generalized co-ordinates q ,Q and the time t . Let us assume that the matrix [ ∂Q j / ∂ pk ] is of rank s − 1 ; in this case, there is only one relation of the above mentioned form, let be Ω ( Q , q ; t ) = 0 . The relations (20.2.27), (20.2.28) are replaced by pk =

∂S ∂Ω ∂S ∂Ω , Pj = − , j , k = 1, 2,..., s , +λ −λ ∂Q j ∂Q j ∂qk ∂qk H = H + S + λ Ω ,

(20.2.27') (20.2.28')

where λ is a non-determinate Lagrange’s multiplier. Analogically, if the matrix [ ∂Q j / ∂ pk ] is of rank s − r , then there exist r relations connecting the generalized co-ordinates q ,Q and the time t , while the formulae of type (20.2.27'), (20.2.28') contain r Lagrange’s multipliers. For example, a point transformation of the form (20.2.21), which is a homogeneous complete canonical transformation, includes s relations linking the generalized coordinates q to the generalized co-ordinates Q ; as a matter of fact, any homogeneous complete canonical transformation includes at least such a relation. Indeed, the condition of complete canonicity pk dqk = Pj dQ j implies s homogeneous equations

MECHANICAL SYSTEMS, CLASSICAL MODELS

280 Pj

∂Q j = 0 , k = 1, 2,..., s ; ∂ pk

hence, to have non-zero solutions Pj , det [ ∂Q j / ∂ pk

]

must vanish, so that

Ω (Q , q ) = 0 . Let us put the problem to determine the generating function S , which leads to a special form of the Hamiltonian H . For instance, if H = 0 , then we have S + H = 0 , where H = H ( q , p ; t ) = H ( q1 , q 2 ,..., qs , ∂ S / ∂q1 , ∂S / ∂q 2 ,..., ∂ S / ∂qs ; t ) ; we find thus again the Hamilton-Jacobi equation (19.2.9), the function S being the corresponding unknown function. We have seen in Sec. 20.2.1.1 that, in this case, 2s first integrals of the initial canonical system are determined; the two subsystems (20.2.27) correspond to the two sequences (19.2.10), which give the solution of Hamilton’s system in the Hamilton–Jacobi theorem. Hence, the problem to determine the canonical transformation for which H = 0 is equivalent to the integration of Hamilton’s canonical system. If we use the necessary and sufficient condition (20.2.18'), then the generating function S = S ( Q , q ; t ) leads to the representation

cpk =

∂S ∂S , Pj = − , j , k = 1, 2,..., s , ∂qk ∂Q j

(20.2.27'')

with

H = cH + S , c ≠ 0 , c = const ,

(20.2.28'')

which, in general, is not univalent. In case of a complete canonical transformation, the generating function S = S ( Q , q ) does not depend explicitly on time ( S = 0 ) and the Hamiltonian is transformed by the relation (20.2.10''). We notice that, in such a transformation, the function H does not change essentially; to can obtain a canonical transformation having a simpler Hamiltonian, we must use a generating function S which contains the time explicitly. For instance, let be the affine transformation Q j = αq j + β p j , Pj = γq j + δ p j , αδ − βγ ≠ 0 ,

(20.2.30)

where α, β , γ , δ are constants; one sees easily that, if c = αδ − βγ , then the necessary and sufficient condition (20.2.11) is fulfilled. In this case, 1 ( αγq j q j + 2 βγq j p j + βδ p j p j ) 2 1 1 1 = − ( αq j + β p j ) ( γq j + δ p j ) + cq j p j = − ( Q j Pj − cq j p j ) . 2 2 2 V = −

We notice that det [ ∂Q j / ∂ pk

] = β s ; hence, if

(20.2.30')

β ≠ 0 , then the transformation is free

(as one can directly see from the first relation (20.2.30)). If β = γ = 0 , then we find

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281

again the transformation (20.2.5), which is not free; as well, if α = δ = 0 , then we obtain the transformation (20.2.6), which is free. We mention that the transformation (20.2.7) is also a free transformation too. For a natural mechanical system, the generalized co-ordinates q1 , q 2 ,..., qs define its position and, together with the generalized momenta p1 , p2 ,..., ps , define its state (positions and velocities of its points). But this specificity of canonical co-ordinates does not remain valid, in general, in case of a canonical transformation; indeed, the quantities Q1 ,Q2 ,...,Qs do no more define the position of the mechanical system, but define its state only together with the quantities P1 , P2 ,...., Ps . The quantities Q1 ,Q2 ,...,Qs define the position of the mechanical system only in the case of a point transformation. However, the non-point canonical transformations are important in the theory of Hamiltonian systems, because they can lead to a Hamiltonian having a simpler structure. Let be the canonical transformation generated by the function S (Q , q ) = Q j q j ;

(20.2.31)

Q j = p j , Pj = −q j , j = 1, 2,..., s .

(20.2.31')

we obtain

The canonical co-ordinates invert their rôles, the above affirmations being thus justified. At the same time, the denomination of conjugate co-ordinates given to qk and pk is also justified. In Sect. 19.2.1.6 we have introduced Hamilton’s principal function S , which satisfies the Hamilton–Jacobi equation; indeed, taking into account the deterministic character of mechanics (we have det [ ∂q j / ∂ pk0 ] ≠ 0 ) and using the final equations of

motion (20.2.20), Hamilton obtained pk0 = pk0 ( q , q 0 ; t ) , k = 1, 2,..., s . In this case, the function

S = S ( q ,q 0 ;t )

will be a generating function of canonical

transformations, so that the transformation (20.2.20) can be considered as a univalent free canonical transformation.

20.2.1.6 Other Functions Generating Canonical Transformations We notice that d (Q j Pj

) = Pj dQ j

+ Q j dPj ; the sufficient condition of canonicity

(20.2.17), (20.2.17') has the form pk dqk + Q j dPj − ( H − H ) dt = dW1 , W1 = W + Q j Pj

in this case. Assuming the supplementary condition ⎡ ∂ Pj ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂ pk ⎦

(20.2.32)

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282

the second group of relations (20.2.1) leads to pk = pk ( P , q ; t ) and then to Q j = Q j ( P , q ; t ) , j , k = 1, 2,..., s ; as well, the function W1 can be expressed in the form W1 ( Q , P , q , p ; t ) = S1 ( P , q ; t ) . The condition of canonicity allows us to write pk =

∂ S1 ∂ S1 , Qj = , j , k = 1, 2,..., s , ∂qk ∂ Pj

(20.2.33)

as well as H = H + S1 .

(20.2.33')

⎡ ∂ 2 S1 ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂ Pj ∂qk ⎦

(20.2.32')

If the Hessian

then we obtain the canonical transformation (20.2.1) from the first subsystem (20.2.33). Hence, a function S1 ( P , q ; t ) of class C 2 , which verifies the condition (20.2.32'), determines a canonical transformation (20.2.33), (20.2.33'). As well, the relation d ( qk pk ) = pk dqk + qk dpk leads to the sufficient condition of canonicity −qk dpk − Pj dQ j − ( H − H ) dt = dW2 , W2 = W − qk pk .

If the supplementary condition ⎡ ∂Q j ⎤ det ⎢ ⎥ ≠0 ⎣ ∂qk ⎦

(20.2.34)

is fulfilled, then the first group of relations (20.2.1) leads to qk = qk ( Q , p ; t ) , so that Pj = Pj ( Q , p ; t ) , j , k = 1, 2,..., s ; analogically, the function W2 can be expressed in the form W2 ( Q , P , q , p ; t ) = S 2 ( Q , p ; t ) . The sufficient condition of canonicity leads to the representation qk = −

∂S2 ∂S , Pj = − 2 , j , k = 1, 2,..., s , ∂ pk ∂q j

(20.2.35)

as well as to H = H + S2 .

(20.2.35')

⎡ ∂ 2S2 ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂Q j ∂ pk ⎦

(20.2.34')

If the Hessian

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283

then we obtain the generalized co-ordinates Q j from the first subsystem (20.2.35); replacing then in the second subsystem (20.2.35), the canonical transformation (20.2.1) is completely specified. By way of consequence, a function S 2 ( Q , p ; t ) of class C 2 , which verifies the condition (20.2.34'), leads to a canonical transformation (20.2.35), (20.2.35'). Analogically, using the relations d (Q j Pj

) = Pj dQ j

+ Q j dPj , d ( qk pk ) = pk dqk + qk dpk ,

the sufficient condition of canonicity −qk dpk + Q j dPj − ( H − H ) dt = dW3 , W3 = W − qk pk + Q j Pj

is resulting. If ⎡ ∂ Pj ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂qk ⎦

(20.2.36)

then the second group of relations (20.2.1) allows us to write qk = qk ( P , p ; t ) and then Q j = Q j ( P , p ; t ) , j , k = 1, 2,..., s , wherefrom W3 ( Q , P , q , p ; t ) = S 3 ( P , p ; t ) . By means of the sufficient canonicity condition, we obtain the representation qk = −

∂S 3 ∂S 3 , Qj = , j , k = 1, 2,..., s , ∂ Pj ∂ pk

(20.2.37)

as well as H = H + S 3 .

(20.2.37')

⎡ ∂ 2S3 ⎤ det ⎢ ⎥ ≠ 0. ⎣ ∂ Pj ∂ pk ⎦

(20.2.36')

We assume that

In this case, from the first subsystem (20.2.37) there result the generalized momenta Pj ; replacing in the second subsystem (20.2.37), we obtain also the generalized coordinates Q j , the canonical transformation (20.2.1) being thus specified. Wherefrom, a function S 3 ( P , p ; t ) of class C 2 , which verifies the condition (20.2.36'), determines the canonical transformation (20.2.37), (20.2.37'). Using the necessary and sufficient condition of canonicity (20.2.18'), we obtain representations corresponding to canonical transformations of non-unitary valency c . If the functions which occur do not depend explicitly on time (for instance, S1 = S1 ( P , q ) , S 2 = S 2 ( Q , p ) , S 3 = S 3 ( P , p ) ), then one obtains representations corresponding to complete canonical transformations.

MECHANICAL SYSTEMS, CLASSICAL MODELS

284

Starting from the free transformation (20.2.27), (20.2.28) with H = 0 , we find again the Hamilton–Jacobi equation (19.2.9); one obtains the same result starting from the transformation (20.2.33), (20.2.33'). As well, starting from the transformation (20.2.35), (20.2.35') or from the transformation (20.2.37), (20.2.37'), we find again the HamiltonJacobi type equation (19.2.68). As an application, let us consider the generating function S1 ( P , q ; t ) = f j ( q ; t ) Pj ,

(20.2.38)

⎡ ∂ 2 S1 ⎤ ⎡ ∂ fj ⎤ det ⎢ = det ⎢ ⎥ ⎥ ≠ 0. ⎣ ∂qk ⎦ ⎣ ∂ Pj ∂qk ⎦

(20.2.38')

for which

The formulae (20.2.33) allow us to write Q j = f j ( q ; t ) , pk =

∂ fj P , j , k = 1, 2,..., s , ∂qk j

(20.2.38'')

obtaining thus a point transformation; if we take into account (20.2.38'), then we may calculate Pj = Pj ( q , p ; t ) from the second relation (20.2.38''). We also notice that H = H + fj Pj ,

(20.2.38''')

finding thus again the results of Sect. 20. 2.1.4. An interesting particular case is S1 = q j Pj ,

(20.2.39)

to which corresponds the identical transformation Q j = q j , Pj = p j , j = 1, 2,..., s .

(20.2.39')

As well, let q j0 , p j0 be a stationary point for an autonomous system (which does not depend explicitly on t ) of Hamiltonian H , so that q j = q j0 , p j = p j0 , j = 1, 2,..., s , is a solution for the canonical equations. The generating function S1 = ( q j − q j0

)( Pj

− p j0

)

(20.2.40)

leads to the transformation Q j = q j0 + q j , Pj = p j0 + p j , j = 1, 2,..., s ,

(20.2.40')

which measures the deviation with respect to the equilibrium solution. More general, may be q j0 = q j0 ( t ) and p j0 = p j0 ( t ) , j = 1, 2,..., s , do represent a known solution

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285

of the canonical equations of an autonomous system; it can be a first application in the perturbation theory, which will be considered in the next subsection.

20.2.1.7 Applications to the Perturbation Theory Let (20.2.20) be the solution of the canonical equations (19.1.14) of a mechanical system of Hamiltonian H and initial conditions q 0 = q ( t0 ) and p 0 = p ( t0 ) ; we try now to obtain the motion of a "perturbed" system of Hamiltonian H + H 1 , hence the solution of the canonical system qk =

∂ ∂ ( H + H 1 ) , pk = − ( H + H 1 ) , k = 1, 2,..., s . ∂ pk ∂qk

(20.2.41)

If the initial conditions are considered as new variables in (20.2.20), then these formulae determine a univalent free canonical transformation (as we have seen in Sects. 20.2.1.3 and 20.2.1.5), which transforms the canonical system (19.1.14) into a canonical system of Hamiltonian H = 0 , so that q j0 = 0 and p j0 = 0 , j = 1, 2,..., s ; as well, (20.2.20) transforms the canonical system (20.2.41) into a canonical system of Hamiltonian H = H 1 (because H − ( H + H1 ) = 0 − H = S , where S is the generating function), from which q j0 =

∂H1 ∂H , p j0 = − 01 , j = 1, 2,..., s . 0 ∂ pj ∂q j

(20.2.42)

So, the variables q 0 and p 0 have the following property: they are constant for the nonperturbed motion (equal to the initial conditions), while for the perturbed motion they are functions of time and initial values; by convention, we represent these variables in the form qk[ 0 ] = qk[ 0 ] ( q 0 , p 0 ; t ) , pk[ 0 ] = pk[ 0 ] ( q 0 , p 0 ; t ) , k = 1, 2,..., s , being – in fact – the general solution of the system (20.2.20), in which the Hamiltonian is the "perturbed energy" H 1 . By replacing this function in formulae (20.2.20), we find the solution corresponding to the perturbed motion.

Fig. 20.5. Perturbed motion of a particle.

Using the theory of the canonical transformations, the integration of the canonical system (20.2.41) was replaced by the integration of the canonical system (19.1.14), (20.2.42); the general solution of the system (20.2.41) will thus be

MECHANICAL SYSTEMS, CLASSICAL MODELS

286

q j = q j ( q [ 0 ] ( q 0 , p 0 ; t ) , p[ 0 ] ( q 0 , p 0 ; t ) ) , p j = p j ( q [ 0 ] ( q 0 , p 0 ; t ) , p[ 0 ] ( q 0 , p 0 ; t ) ) , j = 1, 2,..., s .

(20.2.41')

We notice that the “energy perturbation” of the mechanical system is equivalent to the perturbation of the “initial conditions”. Let us consider a fixed representative point P0 in the hyperplane t = t0 and let us build up in this hyperplane a non-perturbed trajectory P0 P , given by (20.2.20); as well, the trajectory P0Q0 in the same hyperplane corresponds to the displacement of the initial position P0 (at the moment t0 ) till the initial position Q0 (at the moment t ), given by (20.2.41') (Fig. 20.5). From the representative point Q0 , let us set up the nonperturbed corresponding trajectory Q0Q . The representative point Q will correspond to the position of the system in the perturbed motion at the moment t , while P0Q will correspond to the trajectory in the perturbed motion. The perturbation is thus put in evidence by the “displacement” PQ . Thus, the perturbed motion can be considered as a “compound motion” in the phase space; the representative point is in motion along the non-perturbed trajectory, but this trajectory is displaced (in general, deformed) because of the perturbations of the initial conditions.

20.2.1.8 Infinitesimal Canonical Transformations Let be the infinitesimal transformation Q j = q j + δq j = q j + ε f j ( q , p ; t ) , Pj = p j + δ p j = p j + εg j ( q , p ; t ) , j = 1, 2,..., s ,

(20.2.43)

where ε is a small parameter. One can obtain such a transformation starting from the canonical transformation Q j = Q j ( q , p ; t ; λ ) , Pj = Pj ( q , p ; t ; λ ) , where λ is a parameter which for λ = λ0 leads to the identical transformation (20.2.40'); if we put λ = λ0 + ε , and expand into a series after the powers of ε , neglecting

O ( ε2 ) , then

we find the canonical transformation (20.2.43), which is completely determined if we know the functions f j and g j , j = 1, 2,..., s . Hamilton’s function is transformed in the form H = H + δH = H + εH 1 ,

(20.2.43')

while the generating function will be of the form S1 + δS1 = q j Pj + εK ,

(20.2.43'')

where K = K ( q , P ; t ) ; we notice that K ( q , P ; t ) = K ( q , p ; t ) + O ( ε ) . Applying the formulae (20.2.33), one has Qj = q j + ε

∂K ∂K , Pj = p j − ε , j = 1, 2,..., s , ∂ pj ∂q j

(20.2.44)

Variational Principles. Canonical Transformations

287

as well as

H = H + εK ;

(20.2.44')

hence δq j = ε f j = ε

∂K ∂K , δ p j = εg j = − ε , δH = εH 1 = εK . ∂ pj ∂q j

(20.2.44'')

Thus, for each function K = K ( q , p ; t ) of class C 1 one obtains an infinitesimal canonical transformation; as a matter of fact, taking into account the condition (20.2.32'), the function K must be of class C 2 . If K = K ( q , p ) , non-depending explicitly on time, then H = H , and the transformation is an infinitesimal complete canonical transformation. In this case, let F ( q , p ) be a function of class C 1 ; for an infinitesimal complete canonical transformation given by (20.2.44) we have a variation of the form δF =

∂F ∂F ∂F ∂K ⎞ ⎛ ∂F ∂K − = ε (F,K ) , δq j + δ pj = ε ⎜ ∂q j ∂ pj ∂ p j ∂q j ⎟⎠ ⎝ ∂q j ∂ p j

(20.2.45)

where a Poisson bracket has been put in evidence. The operator D=

∂K ∂ ∂K ∂ − ∂ p j ∂q j ∂q j ∂ p j

(20.2.45')

is the operator of the infinitesimal canonical transformation; in this case δF = εDF .

(20.2.45'')

We notice that the infinitesimal canonical transformation (20.2.44) transforms the function F in itself if (F,K ) = 0 .

(20.2.45''')

In particular, for F = qk and F = pk respectively, we get δqk = ε ( qk , K ) = ε Dqk , δ pk = ε ( pk , K ) = ε Dpk , k = 1, 2,..., s ;

(20.2.46)

we can thus express the variations of the canonical co-ordinates, hence the infinitesimal complete canonical transformation corresponding to a generating function K ( q , p ) of class C 1 by means of a Poisson bracket, using the relations (20.2.46). By these relations one can show that the infinitesimal generator of the spatial translations along a fixed direction is the projection of the linear momentum of the mechanical system on the same direction; as well, the infinitesimal generator of the spatial rotations about a fixed axis is the projection of the angular momentum of the mechanical system on the very same axis.

MECHANICAL SYSTEMS, CLASSICAL MODELS

288 If F = H in (20.2.45), then we have

δH = ε ( H , K ) ;

(20.2.46')

comparing with the third relation (20.2.44''), we obtain ( ( H , K ) = − ( K , H ) ) ( K , H ) + K = 0 .

(20.2.47)

Taking into account Theorem 19.1.5, we may state that the generating function K is a first integral of the canonical system (19.1.14). As well, we may state that the first integrals of the motion are generators of the infinitesimal canonical transformations. Particularly, for K = H we see that we must have H = 0 , hence H = H ( q , p ) ; wherefrom, an infinitesimal canonical transformation for which the Hamiltonian is a generating function is an infinitesimal complete canonical transformation. If, in this case, we put ε = δt , introducing a time variation, then we find again Hamilton’s equations written by means of the Poisson brackets in the form δqk = ( qk , H ) δt , δ pk = ( pk , H ) δt , k = 1, 2,..., s .

(20.2.48)

To pass from the co-ordinates q , p at the moment t to the co-ordinates q + δq , p + δp at the moment t + δt , it is sufficient to effect an infinitesimal complete canonical transformation of the co-ordinates q , p , of small parameter δt and of generating function H ( q , p ) . Especially, starting from the initial values of the generalized co-ordinates, we attain their values at a neighbouring moment, obtaining the incipient motion. One may thus state that the motion of the representative point on its trajectory is obtained by a succession of infinitesimal complete canonical transformations. Hence, the motion of a mechanical system corresponds to the evolution or to the continuous development of a canonical transformation; we can say that the Hamiltonian is the generator of this motion in time. If DF = 0 , then F ( q , p ) is an invariant of the canonical transformation, hence a first integral of the equations of motion. We can consider the equations (20.2.48) as being the canonical equations written in finite differences; we are thus led to an iterative method to integrate them (obtaining a finite number of points of the trajectory of the representative point). Let qk ( t ) , pk ( t ) and qk ( t ) + δqk , pk ( t ) + δ pk , k = 1, 2,..., s , be two neighbouring solutions of the system of canonical equations (19.1.14), corresponding to two neighbouring initial states. In this case, the functions δqk (t ) and δ pk (t ) , of class C 1 , verify the equations of the variations d ∂ 2H ∂ 2H δq j = δqk + δp , dt ∂ p j ∂qk ∂ p j ∂ pk k d ∂ 2H ∂ 2H δ pj = − δqk − δ p , j = 1, 2,..., s . dt ∂q j ∂qk ∂q j ∂ pk k

(20.2.49)

The knowledge of a first integral f ( q , p ; t ) = const of the canonical system leads to the obtaining of a particular solution for these equations. We will thus show that

Variational Principles. Canonical Transformations

δq j = ε

289

∂f ∂f , δ pj = − ε , j = 1, 2,..., s , ∂ pj ∂q j

(20.2.49')

are the solutions of the equations (20.2.49). Indeed, introducing in these equations, we have

d ∂ 2H ∂ f ⎞ ∂H ∂ f ⎞ ⎛ ∂ 2H ∂ f ⎡ ∂ ⎛ ∂H ∂ f δq j = ε ⎜ − ⎟ = ε ⎢ ∂ p ⎜ ∂q ∂ p − ∂ p ∂q ⎟ ∂ p ∂ q ∂ p ∂ p ∂ p ∂ q dt j k k k ⎠ k k k ⎠ ⎝ j k ⎣ j ⎝ k ∂f ∂H ∂ 2 f ⎞ ⎤ ∂ ⎛ ∂f ⎛ ∂H ∂ 2 f ⎞ −⎜ − ⎟ ⎥ = − ε ∂ p ⎜ ∂ p pk + ∂q qk ⎟ q p p p p q ∂ ∂ ∂ ∂ ∂ ∂ ⎠ j j j ⎝ k k k ⎠⎦ k k ⎝ k

(

)

∂2 f ∂ df ⎛ ∂2 f ⎞ ⎡ d ⎛ ∂f ⎞ ∂ ⎛ ∂f ⎞⎤ +ε ⎜ − f + ε ⎢ ⎜ p k + qk ⎟ = − ε ⎟− ⎜ ⎟⎥ p p p q p d t ∂ ∂ ∂ ∂ ∂ j j j k k ⎝ ⎠ ⎣ dt ⎝ ∂ p j ⎠ ∂t ⎝ ∂ p j ⎠ ⎦

= −ε

∂ df d ⎛ ∂f ⎞ d ⎛ ∂f ⎞ ε +ε ⎜ = , j = 1, 2,..., s , ⎟ ∂ p j dt dt ⎝ ∂ p j ⎠ dt ⎜⎝ ∂ p j ⎟⎠

where we have supposed that f is a function of class C 2 and have taken into consideration that df / dt = 0 ; analogically, d d ⎛ ∂f ⎞ , j = 1, 2,..., s , δp = − ⎜ ε dt j dt ⎝ ∂q j ⎟⎠

the above affirmation being thus proved. Hence, the infinitesimal transformation (20.2.49') transforms each trajectory in a neighbouring one and the whole family of paths in itself.

20.2.2 Structure of Canonical Transformations. Properties In what follows, we will consider some structure properties of the canonical transformations; we mention especially the group properties and the rôle played by the Symplectic group. The invariance of Lagrange and Poisson brackets and of the volume element to such transformations, as well as the important rôle played by the involution functions are put into evidence.

20.2.2.1 Structure of a Canonical Transformation. The Generating Function In Sects. 20.2.1.5 and 20.2.1.6 we have shown how one can obtain canonical (eventually, complete canonical) transformations, starting from the generating functions S , S1 , S 2 or S 3 . These generating functions characterize the structure of the respective canonical transformations; for instance, if there exists a generating function S , then we have to do with a free transformation. In general, let be 4s quantities qk , pk ,Q j , Pj , j , k = 1, 2,..., s , linked by the canonical transformations (20.2.1), for which the condition (20.2.1') is verified; one can choose 2s independent quantities in the form

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290

q1 , q 2 ,..., qr , pr +1 , pr + 2 ,..., ps ,Q1 ,Q2 ,...,Qm , Pm +1 , Pm + 2 ,..., Ps , 0 ≤ r , m ≤ s . (20.2.50)

Indeed, we may state Theorem 20.2.2 (Carathéodory’s lemma). Let be 2s dependent functions Q1 ,Q2 ,...,Qs , P1 , P2 ,..., Ps , which depend on the independent quantities q1 , q 2 ,..., qs , p1 , p2 ,..., ps ; from the 4s quantities Q , P , q , p one may always choose 2s independent quantities so that between them do not be any pair of conjugate quantities qk , pk or Q j , Pj , j , k = 1, 2,..., s . We must mention that the proof of this lemma has been given by Carathéodory for the particular case in which the relations connecting the dependent variables to the independent ones correspond to a canonical transformation; in fact, the lemma has a general character. If we conveniently index again the generalized co-ordinates q and Q , and the generalized momenta p and P , then the 2s independent quantities can be represented in the form (20.2.50). We use the relations s

∑

k =r +1 s

∑

j =m + 1

pk dqk = d

Pj dQ j = d

s

∑

qk pk −

∑

Q j Pj −

k =r +1 s

j = m +1

s

∑

k =r +1 s

∑

qk dpk ,

j =m + 1

Q j dPj ;

in this case, the necessary and sufficient condition of canonicity (20.2.18') is written in the form s ⎛ r ⎞ ⎛ m c ⎜ ∑ pk dqk − ∑ qk dpk ⎟ − ⎜ ∑ Pj dQ j − ⎝ k =1 ⎠ ⎝ j =1 k =r +1

⎞ Q j dPj ⎟ − ( cH − H ) dt = dU , j = m +1 ⎠ (20.2.51) s

∑

where U =W −c

s

∑

k = r +1

qk pk +

s

∑

j =m + 1

Q j Pj ;

(20.2.51')

we may assume that the function U depends on the variables (20.2.50) and on time. In this case, ∂U ∂U , cql = − , k = 1, 2,..., r , l = r + 1, r + 2,..., s , ∂qk ∂ pl ∂U ∂U , Qj = , i = 1, 2,..., m , j = m + 1, m + 2,..., s , Pi = − ∂Qi ∂ Pj

cpk =

(20.2.52) (20.2.52')

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291

the relations thus obtained corresponding to the considered canonical transformations of valency c . The function U is the generating function of the canonical transformation. Let U be a function of class C 2 for which we suppose that ⎡ ∂ 2U ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂ Rj ∂rk ⎦

(20.2.53)

where rk , Rj , j , k = 1, 2,..., s , represent the variables (20.2.56) in the mentioned order; we can solve the equations (20.2.52) with respect to Pi , i = 1, 2,..., m , and Q j , j = m + 1, m + 2,..., s . Introducing in the relations (20.2.52'), we obtain the searched relations of the canonical transformations (20.2.1).

20.2.2.2 Group Properties. The Symplectic Group We introduce the 2s × 2s matrix associated to the Jacobian (20.2.1') in the form ⎡ ⎡ ∂Q j ∂ ( Q , P ) ⎤ ⎢ ⎢⎣ ∂qk J = ⎡ = ⎢ ⎢⎣ ∂ ( q , p ) ⎥⎦ ⎢ ⎡ ∂ Pj ⎢ ⎢ ∂q ⎣⎣ k

⎤ ⎡ ∂Q j ⎥ ⎢ ∂p ⎦ ⎣ k ⎤ ⎡ ∂ Pj ⎥ ⎢ ⎦ ⎣ ∂ pk

⎤⎤ ⎥⎥ ⎦⎥ , ⎤⎥ ⎥⎥ ⎦⎦

(20.2.54)

where j , k = 1, 2,..., s ; we consider also the matrix ⎡ 0 −E⎤ I= ⎢ ⎥, ⎣E 0 ⎦

(20.2.55)

where E is the s × s unit matrix. The matrix I is called the canonical matrix. We notice that 2 ⎡ 0 −E⎤ ⎡ 0 −E⎤ ⎡ −E I2 = ⎢ = ⎥⎢ ⎥ ⎢ ⎣ E 0 ⎦ ⎣ E 0 ⎦ ⎢⎣ 0

0 ⎤ ⎡ E2 = − ⎥ ⎢ − E2 ⎥⎦ ⎢⎣ 0

0 ⎤ ⎡E 0 ⎤ −1 = − ⎥ ⎢ 0 E ⎥ = − E = − II , E2 ⎥⎦ ⎣ ⎦

the last unit matrix having 2s rows and 2s columns; one obtains I −1 = − I , so that this matrix is non-singular. Introducing the transpose matrix J T , let us calculate ⎡ ⎡ ∂Q j ⎢ ⎢ ∂q ⎣ k JT I = ⎢ ⎢ ⎡ ∂Q j ⎢⎢ ⎣⎢ ⎣ ∂ pk

then

⎤ ⎥ ⎦

T

⎤ ⎥ ⎦

T

⎡ ∂ Pj ⎢ ∂q ⎣ k ⎡ ∂ Pj ⎢ ⎣ ∂ pk

T ⎡ ⎡ ∂ Pj ⎤ ⎤ ⎥ ⎥ ⎡ 0 − E ⎤ ⎢ ⎢ ∂q ⎦ ⎥ ⎢⎣ k ⎥ = ⎢ T ⎥ ⎢E 0 ⎦ ⎢ ⎡ ∂ Pj ⎤ ⎥⎣ ⎥ ⎥ ⎢ ⎦ ⎦ ⎣⎢ ⎣ ∂ pk

⎤ ⎥ ⎦

T

⎤ ⎥ ⎦

T

⎡ ∂Q j −⎢ ⎣ ∂qk ⎡ ∂Q j −⎢ ⎣ ∂ pk

T ⎤ ⎤ ⎥ ⎥ ⎦ ⎥ ; T ⎤ ⎥⎥ ⎥ ⎦ ⎦⎥

MECHANICAL SYSTEMS, CLASSICAL MODELS

292 ⎡ ⎡ ∂ Pj ⎢ ⎢ ∂q ⎣ k J T IJ = ⎢ ⎢ ⎡ ∂ Pj ⎢⎢ ⎢⎣ ⎣ ∂ pk

T ⎤ ⎡ ∂Q j ⎥ ⎢ ∂q ⎦ ⎣ k T ⎤ ⎡ ∂Q j ⎥ ⎢ ∂q ⎦ ⎣ k

⎤ ⎡ ∂Q j ⎥ − ⎢ ∂q ⎦ ⎣ k ⎤ ⎡ ∂Q j ⎥ − ⎢ ∂p ⎦ ⎣ k

⎡ [ [ q j , qk = ⎢ ⎢⎣ [ [ q j , pk

]] ]]

T ⎤ ⎡ ∂ Pj ⎥ ⎢ ∂q ⎦ ⎣ k T ⎤ ⎡ ∂ Pj ⎥ ⎢ ∂q ⎦ ⎣ k

− [ [ q j , pk

⎤ ⎥ ⎦ ⎤ ⎥ ⎦

]] ⎤ ⎥ [ [ p j , pk ] ] ⎥⎦

⎡ ∂ Pj ⎢ ∂q ⎣ k ⎡ ∂ Pj ⎢ ∂p ⎣ k

T ⎤ ⎡ ∂Q j ⎥ ⎢ ∂p ⎦ ⎣ k T ⎤ ⎡ ∂Q j ⎥ ⎢ ∂p ⎦ ⎣ k

⎤ ⎡ ∂Q j ⎥ − ⎢ ∂q ⎦ ⎣ k ⎤ ⎡ ∂Q j ⎥ − ⎢ ∂p ⎦ ⎣ k

T ⎤ ⎡ ∂ Pj ⎥ ⎢ ∂p ⎦ ⎣ k T ⎤ ⎡ ∂ Pj ⎥ ⎢ ∂p ⎦ ⎣ k

⎤⎤ ⎥⎥ ⎦⎥ ⎤ ⎥⎥ ⎥⎥ ⎦⎦

⎡ 0 −c E ⎤ = ⎢ = cI , 0 ⎥⎦ ⎣cE

where we took into account the necessary and sufficient conditions of canonicity (20.2.12'). We obtain thus a necessary and sufficient condition of canonicity, equivalent to the conditions (20.2.12'), and we can state Theorem 20.2.3 The necessary and sufficient condition for the transformation (20.2.1) in the space Γ 2s be canonical consists in the existence of a scalar constant c ≠ 0 , such that J T IJ = c I .

(20.2.56)

This condition is written in the form J T IJ = I

(20.2.57)

in case of a univalent canonical transformation. A matrix J which satisfies the condition (20.2.57) is called a Symplectic matrix. Because detI = ±1 and the determinant of a product of matrices is equal to the product of the determinants of the corresponding matrices, we obtain detJ = ±1 , hence the Symplectic matrices are nonsingular; as a matter of fact, we have given also another proof in Sect. 20.2.1.2. A matrix J which satisfies the relation (20.2.56) will be called a generalized Symplectic matrix of valency c ≠ 0 ; starting from this relation, one can show that detJ = ±c s , obtaining again a known result. Hence, the generalized Symplectic matrices are also non-singular. Let J1 and J 2 be two 2s × 2s generalized Symplectic matrices; the matrix J 2 J1 is a matrix which has the same number of rows and columns. We have J1T IJ1 = c1 I , J 2T IJ 2 = c2 I , wherefrom J1T ( c2−1 J 2T IJ 2 ) J1 = c1 I ; but J1T J 2T = ( J 2 J1 )T , so that

( J 2 J1 )T I ( J 2 J1 ) = c2c1 I , the matrix J 2 J1 being also a Symplectic matrix. The associativity property of the Symplectic matrices follows from the associativity property of the matrix product. Let be the unit matrix E , so that EJ = JE = J , ET = E and ET IE = I ; one notices that this matrix is Symplectic, playing the rôle of neutral element in the set of these matrices (the corresponding canonical transformation is univalent). Multiplying the relation (20.2.56) at left by the matrix I −1 and at right by the matrix J −1 and taking into account that II −1 = JJ −1 = E , we find the remarkable relation I -1 JI = c J −1 ;

(20.2.58)

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293

hence, the inverse matrix J −1 does exist. We notice that detJ ≠ 0 , so that

( J −1 )T

= ( JT

)−1 ; in this case, taking into account the relation (20.2.56) (we assume

that the matrix J is a generalized Symplectic matrix), we may write

( J −1 )T

= IJ −1 = ( J T

)

−1

IJ −1 = c −1 ( J T

−1

) ( J T IJ ) J −1

= c −1 I ,

the matrix J −1 being Symplectic too. We can thus state that the set of Symplectic matrices forms a group: the Symplectic group, defined in a real linear space, of even dimension 2s , and denoted by Sp ( 2s , \ ) . The essential property of this group consists in making invariant certain antisymmetric bilinear form defined on \2s . There results that the set of canonical transformations forms a group too. Analogously, the set of generalized Symplectic matrices forms the generalized Symplectic group. We notice also that, if J is a generalized Symplectic matrix, besides its inverse J −1 , the transpose matrix J T is also a generalized Symplectic matrix; indeed, starting from the relation

( J −1 )T IJ −1

= c −1 I and calculating the inverse at right and at left, we

have ⎡ −1 ⎣( J

)T IJ −1 ⎦⎤

−1

= ( J −1

= − ( JT

)−1 I −1 ⎣⎡ ( J −1 )T ⎦⎤

)T IJ T

−1

= − JI ⎡⎣ ( J −1

)−1 ⎤⎦

T

= −c I −1 = c I .

Hence, we may write the necessary and sufficient condition of canonicity (20.2.56) also in the form JIJ T = c I ;

(20.2.56')

as well, in case of univalent canonical transformations, the condition (20.2.57) becomes JIJ T = I .

(20.2.57')

20.2.2.3 Invariance Properties The univalent canonical transformations admit three basic invariants: the volume element in the phase space, the Poisson bracket and the Lagrange bracket. For instance, let us define the univalent canonical transformation by means of the generating function S1 = S1 ( P , q ; t ) . Before the transformation, the volume element is given by ∂ (q, p ) ⎤ dq1 dq2 ...dqs dp1 dp2 ...dps = det ⎡⎢ dq1 dq2 ...dqs dP1 dP2 ...dPs ⎣ ∂ ( q , P ) ⎥⎦

and after the transformation by ∂ (Q , P ) ⎤ dQ1 dQ2 ...dQs dP1 dP2 ...dPs = det ⎡⎢ dq1 dq2 ...dqs dP1 dP2 ...dPs , ⎣ ∂ ( q , P ) ⎥⎦

MECHANICAL SYSTEMS, CLASSICAL MODELS

294 with

0 ⎤ ⎡ E 2 ∂ (q, p ) ⎤ ⎡ ⎢ ⎥ = det ⎡ ∂ pk ⎤ = det ⎡ ∂ S1 det ⎢ det = ∂ ∂ p p ⎢ ∂P ⎥ ⎢ ∂ P ∂q ⎤ ⎡ ⎤⎥ ⎢⎡ ⎣ ∂ ( q , P ) ⎥⎦ ⎣ j ⎦ ⎣ j k ⎣⎢ ⎣⎢ ∂q ⎦⎥ ⎢⎣ ∂ P ⎥⎦ ⎦⎥ ⎡ ⎡ ∂Q ⎤ ⎡ ∂Q ⎤ ⎤ ∂ (Q , P ) ⎤ ⎡ ∂ 2 S1 ⎡ ∂Q j ⎤ ⎡ ⎢ det det = det ⎢⎣ ∂q ⎥⎦ ⎢⎣ ∂ P ⎥⎦ ⎥ = det ⎢ = ⎢ ∂q ∂ P ⎥ ⎢⎣ ∂ ( q , P ) ⎥⎦ ⎢ ⎥ ⎣ ∂qk ⎦ ⎣ k j E ⎦⎥ ⎣⎢ 0

⎤ ⎥, ⎦ ⎤ ⎥, ⎦

where we took into account the representation (20.2.33). The two determinants at the right are equal because the corresponding matrices are one the transpose of the other (or, using Schwartz’s theorem, if S1 is a function of class C 2 ); it follows that dQ1 dQ2 ...dQs dP1 dP2 ...dPs = dq1 dq2 ...dqs dp1 dp2 ...dps .

(20.2.59)

We can thus state Theorem 20.2.4 The volume element in the phase space is invariant with respect to a univalent canonical transformation. We obtain the same result, observing that ∂ (Q , P ) ⎤ dQ1 dQ2 ...dQs dP1 dP2 ...dPs = det ⎡⎢ dq1 dq2 ...dqs dp1 dp2 ...dps ; ⎣ ∂ ( q , p ) ⎥⎦

in Sect. 20.2.1.2 we have seen that J = ±1 , while for the identical transformation (20.2.39') we have J = 1 , so that we find again the relation (20.2.59). In case of a canonical transformation of valency c we obtain dQ1 dQ2 ...dQs dP1 dP2 ...dPs = cs dq1 dq2 ...dqs dp1 dp2 ...dps .

(20.2.59')

Concerning the Poisson bracket of the functions ϕ = ϕ ( q , p ; t ) , ψ = ψ ( q , p ; t ) we can write

∂ϕ ∂ψ ∂ϕ ∂ψ ⎛ ∂Φ ∂Q j ∂Φ ∂ Pj ⎞ ⎛ ∂Ψ − =⎜ + ∂ql ∂ pl ∂ pl ∂ql ∂ Pj ∂ql ⎟⎠ ⎜⎝ ∂Qk ⎝ ∂Q j ∂ql ∂Φ ∂ Pj ⎞ ⎛ ∂Ψ ∂Qk ∂Ψ ∂ Pk ⎞ ∂Φ ⎛ ∂Φ ∂Q j −⎜ + + = ⎜ ⎟ ⎟ ∂ Pj ∂ pl ⎠ ⎝ ∂Qk ∂ql ∂ Pk ∂ql ⎠ ∂Q j ⎝ ∂Q j ∂ pl

∂Qk ∂Ψ ∂ Pk ⎞ + ∂ pl ∂ Pk ∂ pl ⎟⎠

( ϕ, ψ ) =

+

∂Φ ∂Ψ (Q j , Pk ∂Q j ∂ Pk

∂Φ ∂Ψ ( Pj ,Qk j ∂Qk

) + ∂P

∂Ψ (Q j ,Qk ∂Qk

∂Φ ∂Ψ ( Pj , Pk ) , j ∂ Pk

) + ∂P

where Φ (Q , P ; t ) = ϕ ( q (Q , P ;t ) , p (Q , P ;t ) ;t ) , Ψ (Q , P ;t ) = ψ ( q (Q , P ;t ) , p (Q , P ;t ) ; t ) ;

)

Variational Principles. Canonical Transformations

295

taking into account the sufficient conditions of canonicity (20.2.15), it follows

( ϕ, ψ ) = ( Φ, Ψ ) ,

(20.2.60)

so that we can state Theorem 20.2.5 The Poisson bracket is invariant with respect to a univalent canonical transformation. In case of a canonical transformation of valency c , the necessary and sufficient conditions of canonicity (20.2.15') lead to

( ϕ, ψ ) = c ( Φ, Ψ ) .

(20.2.60')

Let us consider also the Lagrange bracket with respect to the parameters u and v in the form ∂ql ∂ pl ∂q ∂ pl ∂q ∂ Pj ⎞ ⎛ ∂ pl ∂Qk ∂ pl ∂ Pk ⎞ ⎛ ∂q ∂Q j − l =⎜ l + l ⎟ ⎜ ∂Q ∂v + ∂ P ∂v ⎟ ∂u ∂v ∂v ∂ u ∂ ∂ ∂ ∂ Q u P u j j k ⎠ ⎝ ⎠⎝ k ∂q ∂ Pk ⎞ ⎛ ∂ pl ∂Q j ∂ pl ∂ Pj ⎞ ∂Q j ∂Qk ⎛ ∂q ∂Qk −⎜ l + l ⎟ ⎜ ∂Q ∂u + ∂ P ∂u ⎟ = ∂u ∂v [Q j ,Qk ] ∂ ∂ ∂ ∂ Q v P v j j k ⎝ k ⎠⎝ ⎠ ∂Q j ∂ Pk ∂ Pj ∂Qk ∂ Pj ∂ Pk + Q j , Pk ] + Pj ,Qk ] + [ [ [ Pj , Pk ] , ∂u ∂ v ∂u ∂v ∂u ∂v

[ u , v ]q , p =

where Qj ( t ;u , v ) = Qj ( q ( t ;u , v ) , p ( t ;u , v ) ;t ) , Pj ( t ; u , v ) = Pj ( q ( t ; u , v ) , p ( t ; u , v ) ; t ) , j = 1, 2,..., s ;

the sufficient conditions of canonicity (20.2.12'') lead to

[ u , v ]q , p = [ u , v ]Q ,P ,

(20.2.61)

and we may state Theorem 20.2.6 The Lagrange bracket is invariant with respect to a univalent canonical transformation. If the transformation is of valency c , then the necessary and sufficient conditions of canonicity (20.2.12''') allow us to write

[ u , v ]q , p =

1 [ u , v ]Q ,P . c

(20.2.61')

20.2.2.4 Functions in involution. Lie’s Theorem In Sect. 19.1.2.3 we have introduced the canonical conjugate first integrals (the Poisson bracket of which is constant) and the first integrals in involution (the Poisson bracket of which is zero). In general, a set of functions of q , p and t , of class C 1 , for which the Poisson bracket of each pair is equal to zero, is a set of functions in

MECHANICAL SYSTEMS, CLASSICAL MODELS

296

involution; obviously, the generalized co-ordinates Q j = Q j ( q , p ; t )

and the

generalized momenta Pj = Pj ( q , p ; t ) , j = 1, 2,..., s , have, everyone, this property. In general, s arbitrary functions of q , p and t , of class C 1 , cannot be the generalized co-ordinates Q 1 ,Q2 ,...,Qs of a canonical transformation; the following question arises: is it sufficient that these functions be in involution to have this property ? More general, we can formulate the question: Let be given s functions Q j = Q j ( q , p ; t ) in involution; can one find other s functions Pj = Pj ( q , p ; t ) , j = 1, 2,..., s , so that the transformation ( q , p ) → ( Q , P ) be canonical ? Obviously, the answer is affirmative in case of a point transformation. In general, let be the functions in involution ϕi = ϕi ( q , p ; t ) , i = 1, 2,..., s , of class C 2 , so that det [ ∂ϕi / ∂ pk ] ≠ 0 ; because there is not an identical relation connecting the generalized co-ordinates q and Q and the time t , we have to solve the equations Qi = ϕi ( q , p ; t ) with respect to p , obtaining pl = ψl ( q ,Q ; t ) , i , l = 1, 2,..., s . We can write the identity ϕi ( q , ψ ; t ) − Qi = 0 , i = 1, 2,..., s , with respect to the variables q and Q , where we have denoted ψ ≡ { ψ1 , ψ2 ,..., ψs } . Calculating the partial derivatives with respect to ql , there results ∂ϕi ∂ϕ ∂ψl + i = 0 , i , k = 1, 2,..., s ; ∂qk ∂ pl ∂qk

multiplying by ∂ϕ j / ∂ pk and summing for all the values of k , we find ∂ϕi ∂ϕj ∂ϕ ∂ϕj ∂ψl + i = 0 , i , j = 1, 2,..., s . ∂qk ∂ pk ∂ pl ∂ pk ∂qk

Taking the antisymmetric part with respect to the indices i and j and observing that ( ϕi , ϕj ) = 0 , i , j = 1, 2,..., s , we may write (we invert the dummy indices k and l in the first sum) ∂ϕj ∂ϕi ∂ψl ∂ψ ∂ϕi ∂ϕj ∂ψl ∂ϕi ∂ϕj ⎛ ∂ψk − = − l ⎞⎟ = 0 , i , j = 1, 2,..., s ; ⎜ ∂ pl ∂ pk ∂qk ∂ pl ∂ pk ∂qk ∂ pk ∂ pl ⎝ ∂ql ∂qk ⎠

we obtain thus a system of s 2 homogeneous linear equations to determine s 2 unknowns ∂ψk / ∂ql − ∂ψl / ∂qk , the determinant of which is equal to

( det [ ∂ϕi / ∂ pk ] )2 s ≠ 0 . It results ∂ψk ∂ψl = , k , l = 1, 2,..., s ; ∂ql ∂qk

(20.2.62)

hence, there exists a function S = S ( q ,Q ; t ) so that pk = ψk ( q ,Q ; t ) =

∂S , k = 1, 2,..., s , ∂qk

(20.2.62')

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297

[ ∂ 2 S / ∂Q j ∂qk ]

(if not, an identical relation connecting

with a non-singular matrix

q , p and t must exist). If we take also

Pj = −

∂S , j = 1, 2,..., s , ∂q j

(20.2.62'')

we find a free canonical transformation, the answer to the above question being, in general, affirmative. Let us consider the case in which the s functions ϕi , i = 1, 2,..., s , are first integrals of the system of canonical equations. The canonical equations in new canonical co-ordinates derive from Hamilton’s function H = H + S = H ( Q , P ; t ) ; but the canonical co-ordinates Q are constant in time along the integral curves of the canonical system in q , p and t , while the function H cannot depend on the generalized momenta P . Much more, we can choose the canonical transformation so as to have H = 0 , the generalized momenta P being also constant in time along the trajectory of the representative point of the mechanical system. Hence, the knowledge of s first integrals in involution gives the possibility to determine other s first integrals, so that the 2s distinct first integrals do specify, completely, the solution of the problem. Indeed, if we replace Q by ϕ ≡ {ϕ1 , ϕ2 ,..., ϕs } in (20.2.62'), then we obtain an identity in q , p and t ; differentiating successively with respect to q j , p j and t , it follows ∂ψk ∂ψ ∂ϕl ∂ψk ∂ϕl + k = 0, = δ jk , j , k = 1, 2,..., s , ∂Ql ∂ p j ∂q j ∂Ql ∂q j

(20.2.63)

∂ψ ψk + k ϕ l = 0 , k = 1, 2,..., s . ∂Ql

(20.2.63')

Because ϕl is a first integral (it verifies a relation of the form (19.1.61)), we can write ∂ψ ∂ψk ∂ψ ⎛ ∂ϕ ∂ H ∂ H ∂ϕl ⎞ − ; ψk = − k ϕ l = (ϕ ,H ) = k ⎜ l ∂Ql ∂Ql l ∂Ql ⎝ ∂q j ∂ p j ∂q j ∂ p j ⎟⎠

taking into account (20.2.63), we obtain ∂ H ∂ψk ∂H , k = 1, 2,..., s . − ψk = − ∂ p j ∂q j ∂qk

Replacing p by ψ in H ( q , p ; t ) , we obtain G ( Q , q ; t ) , hence ∂G ∂H ∂ H ∂ϕj = + , k = 1, 2,..., s , ∂qk ∂qk ∂ p j ∂qk

MECHANICAL SYSTEMS, CLASSICAL MODELS

298 so that

∂G ψk + = 0 , k = 1, 2,..., s . ∂qk

(20.2.64)

From (20.2.62) and (20.2.64) it follows that

ψk =

∂S , G = − S , ∂qk

(20.2.65)

with S = S ( Q , q ; t ) ; if this function corresponds to the canonical transformation (20.2.62'), (20.2.62''), then the new function of Hamilton is H = G + S , expressed by

Q , P and t . Taking into account (20.2.65), we have H = 0 . We obtain thus Pj = Pj ( q , p ; t ) , j = 1, 2,..., s , that is s new first integrals of the canonical system in q , p and t ; these first integrals are in involution, because the Poisson brackets of all pairs which can be formed vanish. We notice that this result corresponds to Liouville’s Theorem 19.1.9. The above considerations can be presented also in a slightly different form. Starting from ϕk ( q , p ; t ) = ak , ak = const , we may calculate pk = ψk ( q , a ; t ) , k = 1, 2,..., s , with a ≡ {a1 , a2 ,..., as } . Replacing the generalized momenta p by ψ in H ( q , p ; t ) , we obtain the function G ( q , a ; t ) . In this case, ψk dqk − Gdt = dS , with S = S ( q , a ; t ) , is an exact differential; other s relations ∂ S / ∂ak = bk , k = 1, 2,..., s , give the generalized co-ordinates. We build up the function S by means of the s first integrals in involution, and then find again the Hamilton–Jacobi equation (19.2.9). In the particular case in which ϕ k = 0 , k = 1, 2,..., s , and H = 0 , with ϕs = H ( q , p ) , as = h being the energy constant, we have pk = ψk ( q , a ) = ∂ S / ∂qk , k = 1, 2,..., s , with S = S ( q , a ) ; we find again the relations (19.2.15'''), obtaining thus the solution of the problem. A system of functions u1 , u2 ,..., um of class C 2 in a domain D of variables q1 , q 2 ,..., qs , p1 , p2 ,..., ps , t , so that ( u j , uk

) = 0,

j , k = 1, 2,..., s , is called a system

in involution. We may state Theorem 20.2.7 (S. Lie). If v = v (u ) , w = w (u ) are functions of class C 2 , where u ≡ {u1 , u2 ,..., um } is a system in involution defined on the domain D of variables q , p and t , then ( v,w ) = 0 .

(20.2.66)

A direct calculus proves the theorem. We have seen in Sect. 20.2.1.8 that a function F is transformed in itself by an infinitesimal canonical transformation if ( F , K ) = 0 (the formula (20.2.45''')). Let be such a transformation with K = u j ; because we have to do with a system in involution,

Variational Principles. Canonical Transformations

299

any function uk is transformed in itself, hence also the function v . We have thus

( uj ,v ) = 0 ,

j = 1, 2,..., s . Let us consider now an infinitesimal canonical

transformation with K = v ; any of the functions u j is transformed in itself, hence also the function w . The relation (20.2.66) takes place, the Theorem 20.1.7 being once more justified. This theorem can be stated also in a modified form, sometimes useful. Thus, if the equations v = 0 and w = 0 are consequences of the equations u j = 0 , j = 1, 2,..., s , then ( v , w ) = 0 . In fact, this form of the theorem is known as Lie’s theorem. As an application, let us consider the functions ϕ1 , ϕ2 ,..., ϕs in involution and the equations Q j − ϕj ( q , p ; t ) = 0 , j = 1, 2,..., s ; solving with respect to the generalized momenta, we find p j − ψj ( Q , q ; t ) = 0 , j = 1, 2,..., s . Equating the generalized coordinates to constants, the last equations are consequences of the first ones. But the first members of these equations form a system in involution; we can thus say the same thing about the first members of the second system of equations, so that

( pj

∂ ( p j − ψj ) ∂ ( pk − ψk ) ∂ ( p j − ψj ) ∂ ( pk − ψk ) − ∂qi ∂ pi ∂ pi ∂qi ∂ψj ∂ψj ∂ψ ∂ψ δ + k δ ji = − = − + k = 0. ∂qi k i ∂qi ∂qk ∂q j

− ψj , pk − ψk

)=

We find again the relation (20.2.62) as a consequence of Lie’s theorem.

20.2.2.5 First Integrals Linear with Respect to the Generalized Momenta As it is known, in case of a dynamical system described by Lagrangian co-ordinates, one of them (for example, qs ) being cyclic, the corresponding generalized momentum remains constant during the motion (see Chap.19, Subsec. 1.1.8). One can state an inverse property, i.e. Theorem 20.2.8 Any autonomous system which has a first integral linear with respect to the generalized momenta can be described by means of some generalized coordinates conveniently chosen, so that one of them be cyclic. Indeed, let be the first integral ψj p j , where ψj = ψj (q ) , j = 1, 2,..., s . We consider the system of equations dq1 dq dqs = 2 = ... = , ψ1 ψ2 ψs

(20.2.67)

which defines a family of curves in the space Λs ; the solutions are defined as intersections of s − 1 independent integrals of the partial derivatives equation ψj

∂z = 0. ∂q j

(20.2.67')

We denote these integrals by ϕj ( q ) = const , j = 1, 2,..., s − 1 ; let be now the point transformation

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300

Q j = ϕ j , j = 1, 2,..., s ,

(20.2.68)

where ϕ1 , ϕ2 ,..., ϕs −1 are the above obtained functions. We choose then Qs = ϕs , so that we have dq1 dq dqs = 2 = ... = = dQs ψ1 ψ2 ψs

(20.2.69)

for the previously defined displacements. To obtain Qs , we integrate, e.g., dQs =

dq1 , ψ1

where Ψ1 = ψ1 ( q1 ,Q1 ,Q2 ,...,Qs −1 ) . Let us suppose that the transformation (20.2.68) is reversible, so that we may obtain qk = Fk ( Q1 ,Q2 ,...,Qs ) , k = 1, 2,..., s ; for a displacement along one of the paths defined by (20.2.67) (for which Q j = const , j = 1, 2,..., s − 1 ) we obtain ∂ Fk / ∂Qs = ψk , k = 1, 2,..., s . We consider now the point transformation (see Sect. 20.2.1.4) qk = Fk , Pk = p j

∂ Fj , k = 1, 2,..., s ; ∂Qk

we have

Ps = p j

∂ Fj = p j ψj = const , ∂Qs

in the new variables, so that Qs is a cyclic co-ordinate, the Theorem 20.2.8 being thus proved.

20.3 Symmetry Transformations. Noether’s Theorem. Conservation Laws In 1918, Emmy Noether proved her famous theorem relating the physical properties of a mechanical (in general, physical) system and the conservation laws, when the equations of motion emerge from a variational principle. Thus, she established a general method to derive conservation laws in classical mechanics; but this method can be applied equally well in case of relativistic mechanics, quantum mechanics, electromagnetism etc. This study is, obviously, on the line of the Erlangen programme. The importance of the results thus obtained is significant because, in general, the conservation theorems are not deduced systematically, but only in particular cases. By introducing symmetry transformations, certain properties leading to Noether’s theory are put into evidence; one obtains thus integration constants (first integrals). Some considerations concerning Lie groups are made, and then various applications to discrete mechanical systems, resulting conservation laws, are given.

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301

20.3.1 Symmetry Transformations. Noether’s Theorem In what follows, we will present the symmetry transformations which allow to state Noether’s theorem; an important particular case is that of the Lagrangian which does not depend explicitly on time. We mention also the rôle played by the reciprocal of Noether’s theorem. 20.3.1.1 Equations of Motion Let be a physical system (for instance, the mechanical system considered in Sect. 20.1.5.1), the states of which are determined by means of the independent variables x ≡ { x i , i = 1, 2,..., n } and the functions of state (dependent variables) u α , α = 1, 2,...,m ; the derivatives of the functions of state are denoted by uiα = ∂u α / ∂ xi , i = 1, 2,..., n . The time t has not a privileged position and can be anyone of the variables xi . We assume that the equations of motion derive from the functional

∫Ω L ( x i , u

A =

α

, uiα ) dΩ ,

(20.3.1)

where dΩ = dx1 dx 2 ...dxn is the volume element of the domain Ω , and the Lagrangian density L (corresponding to the unit volume) is of class C 2 . To obtain the equations of motion, we introduce the variations of the independent variables xi′ = xi + δxi = xi + εηi ( x ) , i = 1, 2,..., n ,

(20.3.2)

where ε is a small parameter, while ηi and δx i are arbitrary functions and infinitesimal functions of class C 2 , respectively. The relations (20.3.2) define an infinitesimal transformation of the domain Ω into Ω ′ . Consequently, the state functions and their derivatives are transformed according to the relations (analogously, we take δu α = εξ α ( x ) , where ξ α are arbitrary functions of class C 2 ) δu α ( x ) = u ′α ( x ′) − u α ( x ) , δuiα = ui′α ( x ′) − uiα ( x ) , i = 1, 2,..., n , α = 1, 2,...,m ,

(20.3.3)

where we used similar notations, while the variation of the functional (20.3.1) is given by δA =

∫Ω ′ L ( xi′ , u ′

α

, ui′α ) dΩ ′ −

∫Ω L ( x i , u

α

, uiα ) dΩ ,

(20.3.1')

where dΩ ′ = dx1′dx 2′ ...dx n′ . The Jacobian of the transformation (20.3.2) is ⎡ ∂ ( x j + δx j ) ⎤ ∂ ⎡ ∂ x j′ ⎤ ⎡ J = det ⎢ ⎥ = det ⎢ δ jk + ∂ x δx j ⎥ = det ⎢ x x ∂ ∂ ⎣ k k ⎣ k ⎦ ⎣ ⎦

⎤, ⎥⎦

MECHANICAL SYSTEMS, CLASSICAL MODELS

302 hence J =1+

∂ δx + O ( ε 2 ) , ∂x j j

(20.3.4)

where we have used Einstein’s summation convention for dummy Latin indices from 1 to n. The volume elements associated with the domains Ω and Ω ′ are thus related by dΩ ′ = JdΩ , and for the inverse transformation we will have J −1 = 1 −

∂ δx + O ( ε2 ) . ∂xi i

(20.3.4')

With these definitions and using Einstein’s summation convention for dummy Greek indices from 1 to m, we may write the Taylor series expansion

L ( x i′ , u ′α , ui′α ) = L ( x i + δx i , u α + δu α , uiα + δuiα ) ∂L ∂L ∂L α α 2 = L ( x i , u α , uiα ) + δx i + α δu + α δui + O ( ε ) . ∂xi

∂u

∂ui

Hence, taking into account (20.3.4), the variation (20.3.1') of the functional A can be written in the form δA =

⎛

∫Ω ⎜⎝ L

∂ ∂L ∂L ∂L δx + δx + δu α + δu α ∂ x i i ∂ x i i ∂u α ∂uiα i

⎞ dΩ . ⎟ ⎠

(20.3.1'')

As, in general, δuiα ≠ ∂ ( δu α ) / ∂ x i , we introduce the variations δ∗u α ( x ′ ) = u ′α ( x ′ ) − u α ( x ′ ) , δ∗uiα ( x ′ ) = ui′α ( x ′ ) − uiα ( x ′ ) ,

(20.3.5)

to integrate by parts the last term in (20.3.1''). Thus, we have the Taylor series

δ∗u α ( x ′ ) = δ∗u α ( x + δx ) = δ∗u α (x ) +

∂ ( δ∗u α ) δx i + O ( ε2 ) , ∂ xi

where δ∗u α and δx i are variations of order ε; by neglecting O ( ε2 ) , we have δ∗u α ( x ′ ) = δ∗u α (x ) . Therefore, from (20.3.3) we obtain δu α ( x ) = [ u ′α ( x ′ ) − u α ( x ′ ) ] + [ u α ( x ′ ) − u α ( x ) ] , δuiα ( x ) = [ ui′α ( x ′ ) − uiα ( x ′ ) ] + [ uiα ( x ′ ) − uiα ( x ) ] ,

while the notations (20.3.5) and new expansions into Taylor series lead to δu α (x ) = δ∗u α (x ) + uiα δx i , δuiα (x ) = δ∗uiu (x ) + uijα δx j ,

(20.3.5')

Variational Principles. Canonical Transformations

303

where the mixed derivatives of second order uijα = ∂ 2u α / ∂ x i ∂ x j , i , j = 1, 2,..., n , have been also introduced. The second relation (20.3.5) may be written also in the form δ∗uiα ( x ′ ) =

∂ ∂ δ∗u α ( x ′ ) , [ u ′α ( x ′ ) − u α ( x ′ ) ] = ∂ xi′ ∂ xi′

(20.3.5'')

where we took into consideration the first relation (20.3.5); analogically,

δ∗uiα (x ) =

∂ ∂ δ∗u α (x ) , δuiα (x ) = δ∗u α (x ) + uijα δx j . ∂ xi ∂xi

(20.3.5''')

Introducing the operator of total differentiation (L depends on xi explicitly or by the agency of the functions u α and uiα ) d ∂ ∂ ∂ , i = 1, 2,..., n , = + uiα + u jiα dxi ∂ xi ∂u α ∂u jα

(20.3.6)

we observe that

d ∂ ∂L ∂L ∂L δx + δx + uiα δx + uijα δx ; ( L δx i ) = L dx i ∂xi i ∂ xi i ∂u α i ∂uiα j taking into account also the relations (20.3.5'), the variation (20.3.1'') of the functional

A becomes δA =

∂L

⎡ d

∫Ω ⎢⎣ d xi ( L δxi ) + ∂u α δ∗u

α

+

∂L δ∗uiα ∂uiα

⎤ dΩ . ⎥⎦

(20.3.7)

The last term in δA can be written in the form ∂L d ⎛ ∂L d ⎛ ∂L ⎞ α α ⎞ α ⎜ ⎜ α ⎟ δ∗u , α δ∗ui = α δ∗u ⎟ − ∂ui dx i ⎝ ∂ui ⎠ dx i ⎝ ∂ui ⎠

so that δA =

⎡ d ⎛

∫Ω ⎢⎣ dxi ⎝⎜ L δxi

+

∂L δ∗u α ⎞⎟ + [ L ]α δ∗u α ∂uiα ⎠

⎤ dΩ , ⎦⎥

(20.3.7')

where

[ L ]α =

∂L d ⎛ ∂L ⎞ , α = 1, 2,...,m , − ∂u α dx i ⎜⎝ ∂uiα ⎟⎠

(20.3.8)

is the Euler–Lagrange derivative (corresponding to the functional derivative (20.1.79)). Finally, by means of the equations (20.3.5') we obtain the variation of the action A in the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

304 δA =

⎡ d ⎛

∫Ω ⎢⎣ dxi ⎝⎜ L δxi

∂L ∂L α δu α − u δx ⎞ + [ L ]α ( δu α − u αj δx j ) ⎤⎥ dΩ . ∂uiα ∂uiα j j ⎠⎟ ⎦

+

(20.3.7'')

To obtain the equations of motion, according to the conditions of Hamilton’s principle, we assume that the variations δx i vanish identically inside and on the boundary of the domain Ω (i.e. δx i ≡ 0 in Ω and on ∂Ω, and the integration domain remains unchanged), and the variations of the functions of state u α are identically zero on the boundary of Ω ( δu α ∂Ω ≡ 0 ). By way of consequence, applying the fluxdivergence formula (A.2.67'), we show that the first term in (20.3.7'') vanishes and we obtain δA =

∫Ω [ L ]

α

δu α dΩ ;

(20.3.9)

equating to zero the variation δA and using the second basic lemma of the variational calculus, we obtain the equation of motion in the differential form

[ L ]α = 0 , α = 1, 2,...,m .

(20.3.10)

The function L which leads to a certain type of equations of motion is not uniquely determined. Indeed, these equations are invariant if we replace the function

L by λL,

where λ is an arbitrary non-zero constant (scale transformation) or by L + df j / dx j , where f j ( xi , u α ) , j = 1, 2,..., n , are arbitrary functions of class C 2 (divergence transformation). Indeed, observing that df j ∂ fj ∂ fj α ∂ fi ∂ ⎛ df j ⎞ = + , = α uj , dx j ∂x j ∂u ∂uiα ⎜⎝ dx j ⎟⎠ ∂u α ∂ 2 fj ∂ 2 fj ∂ ⎛ df j ⎞ ∂ ⎛ ∂ fj ⎞ ∂ ⎛ ∂ fj uα = = + + ⎜ ⎟ β ⎜ dx ⎟ β β α j β ∂ x j ⎝ ∂u ⎠ ∂u α ⎜⎝ ∂u β ∂u ⎝ j ⎠ ∂u ∂ x j ∂u ∂u

it is easy to show that

[ df j / dx j ]α

d ⎛ ∂ fj ⎞ ⎞ α ⎟ u j = dx ⎜ β ⎟ , j ⎝ ∂u ⎠ ⎠

= 0 . In particular, when the only independent

variable is the time t we obtain the gauge transformations discussed in Sect. 18.2.3.2. 20.3.1.2 Symmetry Transformations. Noether’s Theorem By symmetry transformations we denote the transformations of independent and dependent variables under which the form of the equations of motion remains invariant. Thus, the general form of a transformation of independent variables is xi′ = ϕi ( x ) , i = 1, 2,..., n ,

and it changes the functions of state into

(20.3.11)

Variational Principles. Canonical Transformations

305

u ′α ( x ′) = Φ α ( x , u ( x ) ) , α = 1, 2,...,m ,

(20.3.11')

where u ≡ {u 1 , u 2 ,..., u m } . The functional (20.3.1) is invariant with respect to the transformation (20.3.11) if the relation

L ′ ( x i′ , u ′α , ui′α ) dΩ ′ = L ( x i , u α , uiα ) dΩ

(20.3.12)

holds. On the other hand, the equations of motion (20.3.10) are invariant if

L ′ ( x i′ , u ′α , ui′α ) = L ( x i′ , u ′α , ui′α ) +

df j . dx j′

(20.3.12')

Thus, the transformations (20.3.11) are symmetry transformations for the considered mechanical system if and only if both conditions (20.3.12) and (20.3.12') are satisfied. In the case of the infinitesimal transformations (20.3.2), (20.3.3), these conditions become

L ′ ( x i + δxi , u α + δu α , uiα + δuiα ) dΩ ′ = L ( x i , u α , uiα ) dΩ , L ′ ( x i + δxi , u α + δu α , uiα + δuiα ) dΩ ′ = L ( x i + δx i , u α + δu α , uiα + δuiα ) dΩ +

d ( δ f j ) dΩ ; dx j

in the last term we have introduced the variation δ f j (a quantity of order ε) and we have used the relation δ f j ( x ′ ) = δ f j (x ) (we neglect O ( ε2

)

with respect to

observing that dx j′ and dΩ ′ differ from dxi and dΩ, respectively, by

O ( ε ) ),

O ( ε ) (which

can be neglected, because δ f j is of the order of ε). Eliminating

L ′ ( x i + δx i , u α + δu α , uiα + δuiα ) dΩ ′ , we obtain

L ( xi + δxi , u α + δu α , uiα + δuiα ) = ⎡⎢ L ( xi , u α , uiα ) − ( δ fj dx j ⎣ d

) ⎤⎥ J −1 ; ⎦

finally, substituting (20.3.4') and neglecting O ( ε2 ) , we obtain (expansion in a Taylor series) ⎛ δx ∂ + δu α ∂ + δu α ∂ + ∂ δx ⎞ L = − d δ f ( x , u α ) . ⎜ i ∂x i ∂u α ∂uiα ∂ x i i ⎟⎠ dx j j i ⎝ i

(20.3.13)

Hence, an infinitesimal transformation of the form (20.3.2), (20.3.3) is a symmetry transformation if, for a given L, there exist functions f j ( x i , u α ) of class C 2 , so that the equation (20.3.13) be satisfied. It can be shown that, for a given L, the symmetry transformations form a group, which represents the symmetry group of the considered mechanical system.

MECHANICAL SYSTEMS, CLASSICAL MODELS

306

If, in particular, together with E.L. Hill, the left member of the equation (20.3.13) is identically zero, and if the Jacobian of the transformation is equal to unity, then we say that the density L has an invariant form. From (20.3.1') and (20.3.13) we have (we integrate on the arbitrary domain Ω and on the domain Ω ′ , respectively, δA +

d

∫Ω dxi ( δ fi ) dΩ = 0 ,

(20.3.13')

so that (20.3.7'') can be written in the form ⎡ d ⎛

∫Ω ⎢⎣ dxi ⎝⎜ L δxi

+

∂L ∂L α u δx + δ fi ⎞⎟ + [ L ]α ( δu α − u αj δx j ) ⎤⎥ dΩ = 0 . δu α − ∂uiα ∂uiα j j ⎠ ⎦ (20.3.14)

This integral has to vanish on any domain Ω; if we assume that the integrand is a continuous function, then it must vanish at any point x ∈ Ω . Consequently, when the equations of motion (20.3.10) are satisfied, then we obtain an equation of conservation of the form d dx i

⎛ L δx + ∂L δu α − ∂ L u α δx + δ f ⎞ = 0 . ⎜ i i ⎟ ∂uiα ∂uiα j j ⎝ ⎠

(20.3.15)

It follows that an infinitesimal symmetry transformation of the mechanical system is associated with a certain equation of conservation; as the symmetry transformations form a group, the relation (20.3.15) establishes the connection between the symmetry group of a mechanical (physical) system and a certain conservation law. The general connection between the symmetry properties of a mechanical system and the conservation laws is given by Theorem 20.3.1 (E. Noether). If the Lagrangian of a physical (mechanical) system is invariant with respect to a continuous group of transformations with m parameters, then there exist m quantities which are conserved during the evolution of the system. 20.3.1.3 Case of a Generalized Conservative System Let be a mechanical system described by a Lagrangian L = L ( q , q ) , q ≡ {q1 , q 2 ,..., qs } , q ≡ {q1 , q2 ,..., qs } , which does not depend explicitly on time ( L = 0 ), for instance a mechanical system subjected to catastatic constraints and conservative given forces; see Sect. 18.2.3.4). We also assume that neither the Hamiltonian H = H ( q , p ) , p ≡ { p1 , p2 ,..., ps } (which is a consequence of the considered constraints), nor the integrals of motion of the system do not depend explicitly on time. For an infinitesimal transformation of the generalized co-ordinates

Variational Principles. Canonical Transformations

qk′ = qk + δqk , k = 1, 2,..., s ,

307 (20.3.16)

we have δqk =

d δq . dt k

(20.3.17)

We obtain thus the relation between the Lagrangian L ( q , q ) and the Hamiltonian H ( q , p ) by means of the generalized momenta given by (18.2.80); to pass from H to L we use Hamilton’s first equation (19.1.14). If C ( q , p ) is an integral of motion in the Hamiltonian formalism, then in the Lagrangian formalism this integral of motion will take the form D ( q , q ) = C ( q , p ) ; also, along the trajectory of the representative point, this constant of motion will satisfy the relations d d C (q, p ) = D ( q , q ) = 0 . dt dt

(20.3.18)

The variation of the Lagrangian, induced by the transformation (20.3.16), can be written in the form (we use the summation convention of dummy indices from 1 to s; see Sect. 20.1.1.2) δL =

d ⎛ ∂L ⎞ δq − [ L ]k δqk , dt ⎜⎝ ∂qk k ⎟⎠

(20.3.19)

which, along the trajectory of the representative point in Λs , becomes δL =

d ⎛ ∂L ⎞ δq . dt ⎜⎝ ∂qk k ⎟⎠

(20.3.19')

We consider three cases when the relation (20.3.19') leads to conserved quantities (integrals of motion), i.e.: (i) If δL = 0 , then the corresponding integral of motion is D ( q , q ) δτ =

∂L δq = pk δqk , ∂qk k

(20.3.20)

where pk are the generalized momenta in the Hamiltonian formulation and δτ is a constant factor. (ii) If there exists a function f (q ) such that δL =

d δ f (q ) , dt

(20.3.21)

then we obtain the integral of motion ∂f ⎞ ⎛ ∂L − D ( q , q ) δτ = ⎜ δq .  ∂ q ∂ qk ⎟⎠ k ⎝ k

(20.3.21')

MECHANICAL SYSTEMS, CLASSICAL MODELS

308

In this case, we may assume that the Lagrangian of the system has the form L ′ = L − df / dt , so that δL ′ = 0 . Let us notice that the two Lagrangians (L and L ′ ) are related by a divergence transformation; consequently, they are equivalent in the sense that the form of the equations is preserved. (iii) If δL =

d g ( q , q ) , dt

(20.3.22)

then the associated integral of motion is

D ( q , q ) δτ =

∂L δq − g ( q , q ) . ∂qk k

(20.3.22')

In this case, there is no one Lagrangian L ′ to be equivalent to L and to satisfy the relation δL ′ = 0 . It is evident that the first and the second case are particular cases of the third case. Actually, there is no other case (i.e. δL ≠ dg ( q , q ) / dt ) which could lead to an integral of motion associated with the infinitesimal transformation (20.3.16). Note that not all the three cases considered above correspond to symmetry transformations. Indeed, the only independent variable is the time t, while the Lagrangian L does not explicitly depend on t. Hence, the defining relation (20.3.13) takes the form d ∂ ∂ ⎞ ⎛ δL ≡ ⎜ δqk + δqk L = − δ f ( q ;t ) ; dt ∂qk ∂qk ⎟⎠ ⎝

(20.3.23)

it follows that only the first case and the second one correspond to symmetry transformations. We may state Theorem 20.3.1' (E. Noether). To every infinitesimal transformation of the form (20.3.16), inducing a variation of the Lagrangian of the form (20.3.22), it corresponds a conserved quantity defined by (20.3.22'). If the infinitesimal transformation (20.3.16) is a symmetry transformation (i.e., g ( q , q ) = δ f ( q ) ), then there exists an invariant Lagrangian

L′=L −

d f (q ) . dt

(20.3.24)

20.3.1.4 The Reciprocal of Noether’s Theorem The Hessian of a given Lagrangian with respect to the generalized velocities has the elements

H jk = Hkj =

∂ pj ∂ pk ∂ 2L , j , k = 1, 2,..., s , = = ∂q j ∂qk ∂qk ∂q j

(20.3.25)

Variational Principles. Canonical Transformations

309

and det ⎡⎣ H jk ⎤⎦ ≠ 0 is the necessary and sufficient condition for the integrability of Lagrange’s equations. The inverse of the matrix ⎡⎣ H jk ⎤⎦ has the elements

H jk−1 = Hkj−1 =

∂q j ∂q = k , j , k = 1, 2,..., s . ∂ pk ∂ pj

(20.3.25')

The relation (20.3.22') represents the condition which must be satisfied by the most general integral of motion of a generalized conservative mechanical system whose evolution is described by Lagrange’s equations. An equivalent form of this condition is given by (20.3.18); thus we have d ∂D ∂D D ( q , q ) = q + q = 0 . dt ∂qk k ∂qk k

(20.3.18')

In our case L = L ( q , q ) , H = H ( q , p ) and qk = qk ( q , p ) , k = 1, 2,..., s , so that (we use the relations (19.1.13), (19.1.14)) ∂qk ∂ ⎛ ∂H ⎞ ∂ ⎛ ∂H ⎞ ∂ ⎛ ∂L ⎞ ∂ 2 L ∂ql = = = − =− ; ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂q j ∂q j ⎝ ∂ pk ⎠ ∂ pk ⎝ ∂q j ⎠ ∂ pk ⎝ ∂q j ⎠ ∂ql ∂q j ∂ pk

we obtain the generalized accelerations in the form (we use the relations (20.3.25')) qk =

∂q ∂q ∂q ∂ L d ∂ 2 L ∂ql qk = k q j + k p j = − q + k dt ∂q j ∂ pj ∂ql ∂q j ∂ pk j ∂ p j ∂q j

∂ 2L ⎛ ∂L ⎞ ∂q =⎜ − ql ⎟ k , k = 1, 2,..., s . ∂ q ∂ q  ∂ q j l ⎝ j ⎠ ∂ pj

(20.3.26)

In terms of the elements of the inverse Hessian (20.3.25'), and by means of (20.3.26), the condition (20.3.18') becomes

qk

∂D ∂L ⎞ −1 ∂ D ⎛ ∂ 2L , =⎜ ql − H ∂qk ∂qk ⎟⎠ kj ∂q j ⎝ ∂qk ∂ql

(20.3.27)

where D ( q , q ) is an integral of motion determined by (20.3.22'). The condition (20.3.27) can also be expressed by means of the Poisson bracket (note that ∂C ( q , p ) / ∂t = 0 ) (C ( q , p ) , H ) = 0 ,

(20.3.28)

this being a necessary and sufficient condition for C ( q , p ) be a first integral of the canonical system. For K = C and ε = δτ in (20.2.46) and by means of (20.3.25'), we may rewrite the variations of the generalized co-ordinates in the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

310 δqk = ( qk ,C ) δτ =

∂C ∂ D ∂q j ∂D H −1 δτ , k = 1, 2,..., s ; δτ = δτ = ∂ pk ∂q j ∂ pk ∂q j jk

(20.3.27')

the conditions (20.3.27) and (20.3.27') are equivalent to (20.3.22'). It follows that an integral of motion D ( q , q ) is associated with an infinitesimal transformation defined by (20.3.16) and (20.3.27'), such that the condition (20.3.22') be satisfied, and the function g ( q , q ) is determined by ⎛ ∂L ∂D ⎞ g ( q , q ) = ⎜ H jk−1 − D ⎟ δτ . ⎝ ∂qk ∂q j ⎠

(20.3.27'')

The variation of the Lagrangian, induced by this infinitesimal transformation, is given by (20.3.22). The condition (20.3.27) is a necessary condition which has to be satisfied by the most general integral of motion given by Noether’s theorem. Because this condition is a homogeneous partial differential equation of first order, in 2s independent variables ( q , q ) , it has 2s − 1 functional independent solutions D ( q , q ) . Hence, the equation (20.3.27) yields all the constants of motion D ( q , q ) , which do not explicitly depend on time. As well, the equation (20.3.27) is also a sufficient condition for the integrals of motion; we may rewrite it in the form d ⎡ d ⎛ ∂L ⎞ ∂L ⎤ ∂ D D ( q , q ) = ⎢ ⎜ H jk−1 , ⎟− dt ⎣ dt ⎝ ∂qk ⎠ ∂qk ⎦⎥ ∂q j

(20.3.29)

as one can easily prove. We are thus led to d D ( q , q ) / dt = 0 along the integral curves of the system. In case of integrals of motion corresponding to symmetry transformations, we have g ( q , q ) = δ f ( q ) = ( ∂ f / ∂qk ) δqk , hence g ( q , q ) =

∂ f ∂D H −1 δτ ; ∂qk ∂q j jk

(20.3.30)

by means of the relations (20.3.22), (20.3.27'') we obtain D ( q , q ) = E ( q , p ′ ) = pk′

∂E , ∂ pk′

(20.3.31)

where we denoted p ′ ≡ { p1′ , p2′ ,..., ps′ } , with pk′ =

∂L ∂f ∂L ′ , k = 1, 2,..., s . − = ∂qk ∂qk ∂qk

(20.3.31')

Thus, the integral of motion D ( q , q ) has to be a homogeneous function of the generalized momenta (20.3.31'). This condition is equivalent to the condition which has to be satisfied by the generating function of infinitesimal, homogeneous, canonical

Variational Principles. Canonical Transformations

311

transformations; the relation (20.3.27') shows how to obtain the related infinitesimal canonical transformations, starting from a given integral of motion. Also, the conditions (20.3.31), (20.3.31') or the equivalent condition ∂ f ⎞ ∂D ⎛ ∂L D ( q , q ) = ⎜ − H −1 ∂ q ∂ qk ⎟⎠ ∂q j jk  ⎝ k

(20.3.31'')

represent a sufficient condition for the case (ii) discussed above. All these results can be summarized in the form of Theorem 20.3.1'' (E. Noether; reciprocal). Suppose that D ( q , q ) is an integral of motion. To this integral of motion we associate an infinitesimal variation (20.3.27') of the generalized co-ordinates, which induces a variation

δL =

d ⎡ ⎛ ∂L ∂ D ⎞ ⎤ H −1 − D ⎟ δτ ⎥ dt ⎢⎣ ⎜⎝ ∂qk ∂q j jk ⎠ ⎦

(20.3.32)

of the Lagrangian. If and only if there is a Lagrangian L ′ equivalent to the Lagrangian L, as defined in (20.3.24), and D ( q , q ) is a homogeneous function of the first degree in the generalized momenta (20.3.31'), then the integral of motion D ( q , q ) can be correlated with an invariance property of the Lagrangian ( δL ′ = 0 ).

20.3.2 Lie Groups In what follows we present some notions concerning Lie groups, useful in the study of conservation laws; we consider first the one-parameter Lie groups and then the Lie groups with m parameters. 20.3.2.1 One-Parameter Lie Groups We consider the transformations xi′ = fi ( x ;a ) , x ≡ {x1 , x 2 ,..., x 2 s } , i = 1, 2,..., 2s ,

(20.3.33)

which depend on one parameter a, and the identity transformation corresponding to the zero value of the parameter ( a = 0 ), that is x i = fi ( x ;0 ) , i = 1, 2,..., 2s .

(20.3.33')

We suppose that there exists a value a of the parameter a, so that xi = fi ( x ′;a ) , x ′ ≡ { x1′ , x 2′ ,..., x 2′ s } , i = 1, 2,..., 2s ;

(20.3.33'')

hence there is an inverse transformation of the transformation (20.3.33). Let xi′′ = fi ( x ′;b ) , i = 1, 2,..., 2s ,

(20.3.34)

be some other transformation of the form (20.3.33). These transformations constitute a finite one-parameter topological group if there is a value c of the parameter such that

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312

xi′′ = fi ( x ;c ) , i = 1, 2,..., 2s .

(20.3.34')

We notice that the parameter c is a function of the parameters a and b c = ϕ ( a ,b ) , a = ϕ ( a , 0 ) .

(20.3.35)

The group of transformations (20.3.33) is a Lie group if the functions fi are analytic functions in the parameter a and if the function ϕ is analytic in both arguments. In our case, 1 is the minimum number of independent parameters necessary to characterize the elements of the Lie group; this parameter is called essential, while 1 is the order or the dimension of the group. In terms of the general method for infinitesimal transformations, we have

xi′′ = xi′ + dxi′ = fi ( x ′; δa ) = x i′ + c = a + da = ϕ ( a , δa ) = a +

∂ fi ∂a

a =0

δa = xi′ + ui ( x ′ ) δa ,

∂ϕ (a , b ) δa = a + μ (a )δa , ∂b b = 0

(20.3.36) (20.3.36')

where c is related to the transformation xi′ + dx i′ = fi ( x ;a + da ) .

(20.3.36'')

Indeed, the infinitesimal displacement of the point x can be obtained in two ways: as a result of successive applications of two transformations or directly. As well, the relation (20.3.36') is a consequence of the property of associativity. Noting that δa = λ (a )da , λ (a ) = 1/ μ (a ) , we can write the system of differential equations of the group in the form dxi′ = ui ( x ′)λ (a ) , i = 1, 2,..., 2s , da

(20.3.37)

where the functions ui ( x ) form a velocity field. By introducing the parameter α=

a

∫ 0 λ(a)da ,

(20.3.37')

we obtain dx i′ = ui ( x ′) , i = 1, 2,..., 2s . dα

(20.3.37'')

The new parameter α is an additive quantity with respect to the composition of infinitesimal transformations; for an infinitesimal parameter β we have fi ( f ( x ; α ) ; β ) = fi ( x ; α + β ) , f ≡ { f1 , f2 ,..., f2 s } , i = 1, 2,..., 2s .

(20.3.38)

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313

Let be a function F ( x ) of class C 2 . By an infinitesimal transformation of the point x (corresponding to an infinitesimal displacement), the function F has a variation of the form dF ( x ) =

2s

∂F

∑ ∂xi dxi

i =1

=

∂F

2s

∑ ui (x ) ∂xi δa = XF (x )δa

i =1

=

∂F δa ; ∂a

(20.3.39)

the operator X =

∂

2s

∑ ui (x ) ∂xi

i =1

(20.3.40)

is called the generator of the transformations group. From (20.3.39) it follows that we can express this generator also in the form X =

∂ . ∂a

(20.3.40')

Hence, the solution of the system of differential equations (20.3.37'') can be expanded in a power series xi′ = xi + α Xxi +

α2 2 X xi + ... , i = 1, 2,..., 2s , 2!

(20.3.41)

which, for infinitesimal transformations, is reduced to the simple form xi′ = xi + α Xxi = xi + αui ( x ) , i = 1, 2,..., 2s .

(20.3.42)

From (20.3.41) it follows x i′ = fi ( x ; α ) = e αX x i , i = 1, 2,..., 2s ,

(20.3.43)

and we are led to the equations ∂ fi = Xfi , i = 1, 2,..., 2s . ∂α

(20.3.43')

We are now in a position to establish the connection between the canonical transformations and the one-parameter Lie group, formed by the transformations (20.3.33). This will allow us to make use of Noether’s theorem, to find the corresponding conserved quantities. First of all, it is worthwhile to note that a special form of (20.3.43') is obtained if we set fi ( x ; α ) = xi ( α ) , i = 1, 2,..., 2s , and α = t , such that they become the canonical equations dxi = Xt xi , i = 1, 2,..., 2s , dt

(20.3.44)

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314

where we have attached the label t to the generator to emphasize the associated parameter. Now we assume that the transformations (20.3.33) constitute a Lie group and note that, in the phase space Γ 2s , these transformations can be written in a form similar to (20.2.1), that is qi′ ≡ Qi = fi ( q , p ; α ) , pi′ ≡ Pi = qi ( q , p ; α ) , i = 1, 2,..., s .

(20.3.45)

We sought the conditions that have to be verified by these transformations (which form a Lie group) such that they be also canonical transformations. A sufficient condition consisting in the existence of a function V = V ( Q , P ) , whose exact differential is a differential one-form (20.2.9). Thus, if we denote λi =

∂ fi ∂q , μi = i , i = 1, 2,..., s , ∂α ∂α

(20.3.46)

we obtain dQi dα

α=0

= λi ( Q , P ) ,

dPi dα

α =0

= μi ( Q , P ) , i = 1, 2,..., s ,

from (20.3.45), where the dependence of λi and μi on Q and P was put in evidence. Also, the generator (20.3.40) becomes X = λi

∂ ∂ , + μi ∂qi ∂ pi

(20.3.46')

where we have used the summation convention for dummy indices from 1 to s. By means of the power series (20.3.41) and assuming the series expansion V (Q , P ) =

∞

∑ αkVk (Q , P ) ,

k =0

we obtain the sufficient condition

α2 2 α2 ⎛ ⎞ ⎛ ⎞ pi dqi − ⎜ pi + αμi + X μi + ... ⎟ ⎜ dqi + αdλi + dX 2 λi + ... ⎟ 2! 2! ⎠ ⎝ ⎠ ⎝ =

∞

∑ αk dVk (Q , P ) ,

k =0

which has to be identically satisfied. Hence, equating the coefficients of equal power of α yields the relations dV0 = 0 , dV1 = − ( μi dqi + pi dλi ) ,...

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315

All these relations are simultaneously satisfied if we assume that the function V has the form V = αV1 +

α2 α3 2 XV1 + X V1 + ... . 2! 3!

(20.3.47)

To find the functions λi and μi , we introduce the function U ( q , p ) = −V1 − pi λi ,

(20.3.47')

such that dU =

∂U ∂U dq + dp = μi dqi − λi dpi . ∂qi i ∂pi i

The last equality is identically satisfied in the variables qi and pi if we choose λi = −

∂U ∂U , μi = , i = 1, 2,..., s . ∂ pi ∂qi

Thus, we obtain the condition which has to be satisfied such that the transformations (20.3.45) be canonical transformations, the function U being arbitrary. Consequently, in terms of the Poisson bracket, the generator (20.3.46') takes the form X =

∂U ∂ ∂U ∂ − ≡ (U , ) . ∂qi ∂ pi ∂ pi ∂qi

(20.3.46'')

Finally, by applying this operator to the functions ∂U / ∂qk and ∂U / ∂ pk , we obtain the whole group of canonical transformations, that is Qi = qi − α

∂U α2 ∂U + − ..., X ∂ pi 2! ∂ pi

∂U α2 ∂U + + ..., i = 1, 2,..., s . Pi = pi + α X ∂qi 2! ∂qi

(20.3.48)

Note that we are led to the same result by integrating the system of partial differential equations ∂ fi = (U , fi ) , i = 1, 2,..., s , ∂α

(20.3.49)

or by integrating the system of differential equations (20.3.46), written in the form dQi ∂U dPi ∂U , , i = 1, 2,..., s , = =− dα ∂ Pi dα ∂Qi

(20.3.50)

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316

where U ( Q , P ) = −U ( q , p ) . All these three methods used to determine the Lie group of canonical transformations are equivalent. The function U is the generating function of canonical transformations, while the operator X = (U , ) is the generator of the Lie group. It is also easy to see that, for α = t and U = H ( Q , P ) , the system of differential equations (20.3.50) is identical with Hamilton’s equations. We denote by x the set of canonical variables {q , p } . Then, a function f ( x ) is invariant with respect to the Lie group of canonical transformations if the equation f (x ′) =

∞

αk

∑ k ! Xk f ( x ) = f ( x )

(20.3.51)

k =0

is identically satisfied. This can be accomplished if and only if Xf ( x ) = 0 , ∀x ∈ \2 s . Explicitly, by using the expression (20.3.46'') for the generator X, we may rewrite this condition in terms of canonical variables

∂U ∂ f ∂U ∂ f − ≡ (U , f ) = 0 , ∂qi ∂ pi ∂ pi ∂qi

(20.3.52)

where we have introduced the Poisson bracket. It follows that the conserved mechanical quantities are invariants of the group of canonical transformations and also solutions of the equation (20.3.52). If we find m independent invariants of this kind, then we may choose them as generalized co-ordinates (cyclic co-ordinates in this case), and there remain to integrate only 2s − m equations of motion. 20.3.2.2 Lie Groups with m Parameters Let be xi′ = fi ( x ;a ) , a ≡ {a1 , a2 ,..., am } , i = 1, 2,..., s ,

(20.3.53)

a set of general transformations, which depend on m independent parameters, real and continuous, and are such that for a = 0 (i.e. a1 = a2 = ... = am = 0 ) we obtain the identical transformation. We assume that a are essential parameters, and therefore the transformations (20.3.53) form a Lie group with m parameters. In this case, dxi′ =

∂ fi ( x ;a ) δak = ∂ak k =1 a =0 m

∑

∂ fi ( x ;a ) da j , i = 1, 2,..., s ; ∂a j j =1 m

∑

denoting uik ( x ) =

we may write

∂ fi ( x ;a ) , i = 1, 2,..., s , k = 1, 2,..., m , ∂ak a =0

(20.3.54)

Variational Principles. Canonical Transformations m

∑ uik (x )δak , i

dxi =

317

= 1, 2,..., s .

(20.3.54')

k =1

As in case of one-parameter Lie groups, the infinitesimal displacement of the point x will be a differential displacement, and there are two possibilities to obtain it: as a result of two successive transformations xi′ = fi ( x ;a ) and xi′ + dx i′ = fi ( x ; δa ) or directly, xi′ + dx i′ = fi ( x ;a + da ) ; taking into account the associativity property, the considered parameters are connected by relations of the form a j + da j = ϕ ( a ; δa ) = a j +

m

∑ μkj (a )δak

, j = 1, 2,..., m ,

(20.3.55)

k =1

where μkj (a ) =

∂ϕ j (a , b ) , j , k = 1, 2,..., m , ∂bk b =0

b ≡ {b1 , b2 ,..., bm } being parameters in i = 1, 2,..., s . The relations (20.3.55) lead to δak =

m

∑ λjk (a )da j , k j =1

the

(20.3.55')

transformation

xi′′ = fi ( x ′, b ) ,

= 1, 2,..., m ,

(20.3.56)

where m

∑ λlj μkj j =1

= δkl , k , l = 1, 2,..., m ,

(20.3.56')

δkl being Kronecker’s symbol. From the relations (20.3.54') and (20.3.56) we obtain the differential equations of the group ∂ xi = ∂a j

m

∑ uki (x )λjk (a ) , i

= 1, 2,..., s , j = 1, 2,..., m .

(20.3.57)

k =1

If the functions uki are linear independent (what we assume), then we say that the parameters a j are essential. Let F ( x ) be a function of class C 2 . This function has a variation of the form dF ( x ) =

s

∂F

∑ ∂xi dxi

i =1

=

s

m

∂F

∑ ∑ uij (x ) ∂xi δa j

i = 1 j =1

≡

m

∑ X j F (x )δa j j =1

=

m

∂F

∑ ∂a j δa j ; j =1

(20.3.58) by an infinitesimal transformation of the point x (corresponding to an infinitesimal displacement); the operators

MECHANICAL SYSTEMS, CLASSICAL MODELS

318 Xj =

∂

s

∑ uij (x ) ∂xi

, j = 1, 2,..., m ,

(20.3.59)

i =1

are called the generators of the transformation group. As one can see from (20.3.58), the generators can be expressed also in the form ∂ , j = 1, 2,..., m . ∂a j

Xj =

(20.3.59')

The formalism used in the previous subsection is generalized by introducing m generating functions U k (q , p ) , while from (20.3.47') we obtain ∂U j ∂ ∂U j ∂ − = (U j , ∂qi ∂ pi ∂ pi ∂qi

Xj =

),

j = 1, 2,..., m .

(20.3.60)

corresponding to the operator (19.1.58). Let us consider the product of two generators applied to the function F ( x ) Xp Xr F (x ) = =

∂u r uqp i ∂ xq i =1 q = 1 m m

∑∑

∂F + ∂ xi

∂

s

∂

s

∑ uqp ∂xq ∑ uir ∂xi F (x )

q =1 s

i =1

∂ 2F

s

∑ ∑ uqp uir ∂xq ∂xi

, p , r = 1, 2,..., s ;

i =1 q =1

using Schwarz’s theorem (from which, in case of functions of class C 2 , the mixed derivatives of second order do not depend on the order of differentiation), we obtain s

=

s

m

∂F

∑ ∑ ckpr uik ∂xi

i =1 k =1

=

s

∂u r

⎛

∑ ∑ ⎜⎝ uqp ∂xiq

( Xp Xr − Xr Xp ) F (x ) =

i = 1 q =1 s ∂ c kpr uik xi ∂ k =1 i =1 m

∑

∑

− uqr

F (x ) =

∂uip ⎞ ∂ F ∂ xq ⎟⎠ ∂ x i

m

∑ ckpr Xk F (x ) ;

k =1

hence, the generators of the transformations group verify the relation X p X r − Xr X p ≡ ( X p , Xr ) =

m

∑ ckpr Xk

, p , r = 1, 2,..., m .

k =1

(20.3.61)

The Poisson bracket thus obtained is called the commutator of the operators X p and Xr . The quantities c kpr are called the structure constants of the group; the property of antisymmetry ( X p , Xr ) = − ( Xr , X p ) is reflected in these constants in the form k c kpr = −crp , while from the Poisson-Jacobi identity

( Xi , ( X j , X k ) ) + ( X j , ( Xk , X i ) ) + ( X k , ( Xi , X j ) ) = 0 , i , j , k

= 1, 2,..., m , (20.3.62)

Variational Principles. Canonical Transformations

319

which is easily verified, we obtain the relation m

∑ ( cils cljk

l =1

+ c sjl ckil + ckls cijl

) = 0 , i, j ,k,s

= 1, 2,..., m .

(20.3.62')

The structure constants of every Lie group are determined, univocally, by the structure of the group. The commutator of the generators can be evaluated by applying it to an arbitrary function f (q , p ) ; thus we have

( X j , Xk ) f

= ( X j Xk − Xk X j ) f = (U j , (U k , f ) ) − (U k , (U j , f

));

(20.3.63)

from the above relations of antisymmetry and from the relation (20.3.63) we see that the Poisson brackets are skew-symmetric ( (U j ,U k ) = − (U k ,U j ) ) and verify the Poisson–Jacobi identity

(U i , (U j ,U k ) ) + (U j , (U k ,U i ) ) + (U k , (U i ,U j ) ) = 0 , i , j , k

= 1, 2,..., m . (20.3.64)

If we set one of the generating functions U k = H we are led to the Jacobi–Poisson theorem (see Sect. 19.1.2.3) for the case when the time does not appear explicitly. If U i and U j are two conserved quantities ( (U i , H ) = (U j , H

) = 0 ), then (U i ,U j )

is also

a conserved quantity. Finally, from (20.3.61), (20.3.63) and (20.3.64) we obtain the equation for the structure constants cijk of the group m

(U i ,U j ) = ∑ cijk U k

.

(20.3.65)

k =1

20.3.3 Space-Time Symmetries. Conservation Laws In what follows we will consider some particular groups (the translations group, the rotations group, the time translation group and the Galileo group); various conservation laws (conservation of the generalized linear and angular momenta, conservation of the mechanical energy and the motion of the mass centre) are thus put in evidence. We determine then the general form of the Lagrangian for the invariance with respect to symmetry transformations, introducing thus the Galileo–Newton group. 20.3.3.1 Conservation Laws Let be a discrete mechanical system, formed by n particles P α of masses mα , α = 1, 2,..., n , described by the Lagrangian L ( q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) , where qk (t ) , k = 1, 2,..., s , are the generalized co-ordinates in the configuration space Λs . Note that the only independent variable is the time t, such that the generalized

MECHANICAL SYSTEMS, CLASSICAL MODELS

320

co-ordinates play the rôle of state functions. If s = 3n , then the relation between the generalized co-ordinates qk and the Cartesian co-ordinates x jα , j = 1, 2, 3 , of the particle of index α is given by the relation k = 3( α − 1) + j . In terms of the Cartesian co-ordinates, the condition (20.3.13) which has to be satisfied by a transformation in order to be a symmetry transformation has the form

d ⎞ d ⎛ ∂ α ∂ α ∂ ⎜ δt ∂t + δx j ∂ x α + δx j ∂ x α + dt δt ⎟ L = − dt ( δΩ ) , j j ⎝ ⎠

(20.3.66)

where we have used the summation convention of dummy indices (from 1 to 3 and from 1 to n, for Latin and Greek indices, respectively). Consequently, the left side in (20.3.66) has to be the total derivative, with respect to the time variable, of the function Ω ( x jα ; t ) . Also, the relation of conservation (20.3.15), corresponding to the invariance of the Lagrangian with respect to a symmetry transformation, becomes d dt

∂L α ⎞ ∂L ⎡⎛ ⎤ α ⎢ ⎜ L − ∂ x α x j ⎟ δt + ∂ x α δx j + δΩ ⎥ = 0 , j j ⎣⎝ ⎠ ⎦

(20.3.67)

which is equivalent to ∂L α ⎞ ∂L ⎛ α ⎜ L − ∂ x α x j ⎟ δt + ∂ x α δx j + δΩ = const . j j ⎝ ⎠

(20.3.67')

The equations (20.3.66) and (20.3.67') represent the analytic form of Noether’s theorem (see the Theorem 20.3.1) in classical mechanics. Thus, to every symmetry transformation of a mechanical system (that is an infinitesimal transformation which satisfies the condition (20.3.66)) it corresponds a conservative law expressed by (20.3.67'). 20.3.3.2 Translations Group. Conservation of the Generalized Momentum The simplest symmetry transformations are the translations of the origin of the frame of reference (that is of the Cartesian co-ordinates) by a time-independent vector ( δx1 , δx 2 , δx 3 ). These transformations are defined by the relations (the space translations group T) t ′ = t , δt = 0 , x j′α = x jα + δx j , x j′α = x jα , j = 1, 2, 3 , α = 1, 2,...,n .

If we choose δΩ = 0 , then (20.3.66) takes a simpler form 3

⎛

n

∂L ⎞

∑ ⎜⎝ ∑ ∂x αj ⎟⎠ δx j j =1

α =1

= 0;

(20.3.68)

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321

because the infinitesimal displacements δx1 , δx 2 , δx 3 are independent, it follows that n

∂L

∑ ∂x αj

α =1

= 0 , j = 1, 2, 3 .

(20.3.69)

Now, from (20.3.67) we obtain d n ∂L d n ∂L ∑ α δx j = δx j α = 0; dt α =1 ∂ x j dt α∑ = 1 ∂ x j

consequently, the generalized momentum P of components Pj =

n

∑ p αj

α =1

=

∂L

n

∑ ∂x αj

, j = 1, 2, 3 ,

(20.3.70)

α =1

is conserved if the conditions (20.3.69) are satisfied. The relations (20.3.69) represent not only a sufficient condition, but also a necessary ones, because taking into account Lagrange’s equations of motion, the assumption of conservation of the generalized momentum leads to 0=

dPj d n ∂L = α = dt dt α∑ = 1 ∂ x j

n

d ⎛ ∂L ⎞

n

∂L

∑ dt ⎜⎝ ∂x αj ⎟⎠ = ∑ ∂x αj ,

α =1

α =1

j = 1, 2, 3 .

20.3.3.3 Rotations Group. Conservation of the Generalized Moment of Momentum Let us consider now a rotation of the system of orthogonal co-ordinates, defined by the relations xi′ = aij x j , i = 1, 2, 3 ,

(20.3.71)

where a ∈ SO ( 3, \ ) is an orthogonal matrix. In case of an infinitesimal rotation, this matrix has the form a = E + ak δθk ,

(20.3.72)

where ak are the matrices of the generators of the rotations group SO ( 3, \ ) , δθk , k = 1, 2, 3 , represent the infinitesimal variations of the rotation parameters and E is the identity matrix. Hence, we have aij = δij + ( ak δθk )ij = δij + δaij , δaij = − δa ji , i , j = 1, 2, 3 ,

(20.3.72')

δij being the Kronecker symbol. We assume that the transformation (20.3.71) is of the

form (20.3.2), such that, by substituting (20.3.72') into (20.3.71) we have δxi = x j δaij = − x j δa ji .

MECHANICAL SYSTEMS, CLASSICAL MODELS

322

It follows that the variations of the particle co-ordinates are of the form δxiα = − xkα δaki , δxiα = − xkα δaki , i = 1, 2, 3 , α = 1, 2,...,n ,

(20.3.73)

while the independent variable t remains unchanged, that is t ′ = t , δt = 0 . Thus, from the condition (20.3.66), with δΩ = 0 , we obtain ⎛ α ∂L α ∂L ⎞ ⎜ x j ∂ x α + x j ∂ x α ⎟ δa jk = 0 . ⎝ k k ⎠

The matrix δa is skew-symmetric and the variations δθk are independent, such that we can rewrite this result as a system of equations x αj

∂L α ∂L α ∂L α ∂L = 0 , j , k = 1, 2, 3 . α − xk α + x j α − xk ∂xk ∂x j ∂ xk ∂ x αj

(20.3.74)

If these equations are satisfied, then the transformation (20.3.71) is a symmetry transformation and (20.3.67) yields the equation d ⎛ ∂L α d ⎡ 1 ⎛ ∂L α ∂L α ⎞ ⎤ ⎞ x δa = x − α x k ⎟ ⎥ δa jk = 0 , ∂ x j dt ⎜⎝ ∂ xkα j jk ⎟⎠ dt ⎢⎣ 2 ⎝⎜ ∂ xkα j ⎠⎦

which is equivalent to the system of equations x αj

∂L ∂L − x kα = x αj pkα − x kα p αj , j , k = 1, 2, 3 . ∂ xkα ∂ x αj

Therefore, the conserved quantity is the generalized moment of momentum K, which is a skew-symmetric tensor of rank two, of components

K jk = x jα pkα − xkα p jα , j , k = 1, 2, 3 .

(20.3.75)

Conversely, assuming the conservation of the generalized moment of momentum and taking into account Lagrange’s equations of motion, one can write d ⎛ α ∂L d ⎛ ∂L ⎞ ∂L ⎞ ∂L ∂L − x kα = x αj + x αj − xkα x ∂ x αj ⎟⎠ ∂ xkα ∂ x αj dt ⎜⎝ j ∂ xkα dt ⎜⎝ ∂ xkα ⎟⎠ d ⎛ ∂L ⎞ α ∂L α ∂L α ∂L α ∂L , j , k = 1, 2, 3 . − x kα ⎜ α ⎟ = x j α − xk α + xj α − xk dt ⎝ ∂ x j ⎠ ∂ xk ∂ x j ∂xk ∂ x αj

0=

dK jk dt

=

Therefore, the relations (20.3.74), which represent sufficient conditions for the conservation of the generalized moment of momentum K, are necessary conditions in this sense too.

Variational Principles. Canonical Transformations

323

20.3.3.4 Time Translations Group. Conservation of the Generalized Mechanical Energy The time translations are defined by the equations t ′ = t + δt , xk′α = x kα , δxkα = 0 , xk′α = xkα , δxkα = 0 , k = 1, 2, 3 , α = 1, 2,...,n ,

(20.3.76)

and form a one-parameter Abelian group (the time translations group T ). If we choose δΩ = 0 for δt = const , the condition (20.3.66) is reduced to ∂L = L = 0 , ∂t

(20.3.77)

that is the Lagrangian does not depend explicitly on time. Also, the equation of conservation takes the form d ⎛ ∂L α ⎞ x − L ⎟ = 0 . dt ⎜⎝ ∂ xkα k ⎠

It follows that the generalized mechanical energy

E =

∂L α x − L , ∂ xkα k

(20.3.78)

corresponding to the relation of definition (18.2.68), is a conserved scalar quantity (Jacobi’s first integral or a first integral of Jacobi’s type). Reciprocally, if the conservation relation (20.3.77) holds, then we can write 0=

dE d ⎛ ∂L α d ⎛ ∂L ⎞ = x − L ⎟ = ⎜ α dt dt ⎜⎝ ∂ xkα k ⎠ dt ⎝ ∂ xk

∂ L α dL ⎞ α ⎟ xk + ∂ x α xk − dt = − L . ⎠ k

Here, we have used the relation

dL ∂L α ∂L α x + x = L + dt ∂ x kα k ∂ xkα k and Lagrange’s equations of motion. Hence, the condition (20.3.77) is not only a sufficient condition, but also a necessary condition for the conservation of the mechanical energy.

20.3.3.5 The Galileo Group. Theorem of the Mass Centre Motion Let be two reference frames: one considered to be a fixed frame of reference and the other moving uniformly along a straight line relative to the first one, with the velocity v = (v1 , v2 , v3 ) . The transformation connecting the co-ordinates of a particle in the two reference frames is of the form

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324

xk′ = xk + vk t , k = 1, 2, 3 , t ′ = t , δt = 0 ,

(20.3.79)

where the primed co-ordinates correspond to the fixed frame of reference. The operator associated to this transformation acts in the space of the vectors ( r ; t ) = ( x1 , x 2 , x 3 ; t ) and has the associated matrix ⎡1 ⎢0 t ( v ) = ⎢⎢ 0 ⎢ ⎢⎣ 0

0 0 v1 ⎤ 1 0 v2 ⎥ ⎥. 0 1 v3 ⎥ ⎥ 0 0 1 ⎥⎦

(20.3.80)

The set of transformations (20.3.79) forms an Abelian group with three parameters (the components v1 , v2 , v 3 of the velocity), called the Galileo group (denoted here by Γ ). The matrices (20.3.80) satisfy the relation t ( u + v ) = t ( u )t ( v ) ,

where u and v are two arbitrary velocities. Therefore, the set of matrices (20.3.80) forms a representation of Γ . The infinitesimal transformation corresponding to (20.3.79) has the form t′ = t , xk′ = xk + t δvk , xk′ = xk + δvk , k = 1, 2, 3 ,

(20.3.81)

while the variations of the co-ordinates of the system of particles are given by the relations δt = 0 , δxkα = t δvk , δxkα = δvk , k = 1, 2, 3 , α = 1, 2,...,n .

(20.3.81')

With these notations, the condition (20.3.66) becomes ∂L

∑ ⎛⎜⎝ t ∂xkα n

α =1

+

∂L ⎞ d δv = − ( δΩ ) ; ∂ xkα ⎟⎠ k dt

(20.3.82)

assuming that (20.3.69) is satisfied (i.e., the generalized momentum is conserved), we have n d ∂L ( δΩ ) = − ∑ α δvk = − Pk δvk ∂ dt α =1 xk

(20.3.82')

and the right member has to be the total derivative with respect to time of a function of co-ordinates xk .

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325

The centre of mass of the discrete mechanical system has the co-ordinates ξk =

1 mα xkα , k = 1, 2, 3 , M

(20.3.83)

where M =

n

∑ mα

α =1

is the mass of the mechanical system; therefore, taking into account the definition (20.3.70), we obtain

M ξk = mα xkα = Pk , k = 1, 2, 3 .

(20.3.83')

Hence, up to an additive constant, we may choose δΩ = − M ξk δvk ,

(20.3.82'')

such that the equation of conservation (20.3.67) takes the form d ⎡ ⎛ n ∂L t α − M ξk dt ⎢⎣ ⎝⎜ α∑ =1 ∂ xk

⎞ ⎤ ⎟ δvk ⎥ = 0 ⎠ ⎦

and, by the relation (20.3.70), we may write d [ ( Pk t − M ξk ) δvk ] = 0 . dt

Because the variations δvk are independent, we get d ( P t − M ξk ) = 0 , k = 1, 2, 3 , dt k

or Pk t − M ξk = const ; the constant is determined by the conditions

( Pk t − M ξk ) t = 0 = − M ξk0 , k = 1, 2, 3 . Finally, we obtain ξk0 = ξk −

Pk t , k = 1, 2, 3 . M

(20.3.83'')

Taking into account the conservation of the generalized momentum, it follows that the centre of mass of the considered mechanical system has a uniform rectilinear motion with respect to both frames of reference. It is a characteristic of classical mechanics

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326

that, in case of a Lagrangian invariant with respect to space translations, not only the generalized momentum is conserved but the equation (20.3.83'') holds too (the law of motion of the centre of mass).

20.3.3.6 Lagrangians with Certain Symmetry Properties We will focus now on implications of the reciprocal of Noether’s theorem in classical mechanics. We restrict our considerations to two-particle interactions, thus neglecting higher order interactions, and make use of the superposition principle. Therefore, we assume that the Lagrangian is a sum of Lagrangians, some of them associated with single particle motions, while the others correspond to two-particle interactions, that is

L =

n

n

n

∑ Lα ( xkα , xkα ; t ) + ∑ ∑ Lαβ ( xkα , xkβ , xkα , xkβ ; t ) ,

α =1

(20.3.84)

α =1 β = 1

where x α = ( x1α , x 2α , x 3α ) are the Cartesian co-ordinates of the particle P α , and similarly for x β . On this general form of the Lagrangian we impose the conditions (20.3.69), (20.3.74) and (20.3.77), that is we want the Lagrangian be invariant with respect to space translations, space rotations and time translations. In other words, we consider that the space, in which we study the evolution of our discrete mechanical system, is homogeneous (in both space and time variables) and isotropic. In fact, these conditions represent restrictions with respect to the functional form of the Lagrangian in the variables xkα , xkα , t . Firstly, the condition (20.3.77) leads to

L =

n

n

n

∑ Lα ( xkα , xkα ) + ∑ ∑ Lαβ ( xkα , xkβ , xkα , xkβ ) .

α =1

(20.3.84')

α =1 β =1

Then, for every index k = 1, 2, 3 , the condition (20.3.69) can be regarded as a partial differential equation, the solutions of this equation being the first integrals of the system of differential equations (the Lagrange–Charpit sequence) dx k1 dx 2 dx n = k = ... = k = dλ , 1 1 1

where λ is an arbitrary parameter. Thus, we obtain n (n − 1)/ 2 first integrals of the form ϕkαβ = xkα − xkβ = const , α > β , α, β = 1, 2,...,n , of which only n − 1 are linearly independent. Consequently, if the Lagrangian is of the form (20.3.84'), then Lα must be independent of xkα , while Lαβ has to depend on xkα and xkβ only by the agency of the functions ϕkαβ . From the condition (20.3.74), it follows that the Lagrangian

( xkα

− xkβ

has

)( xkα

to

− x kβ ) ,

depend

( xkα

only

− xkβ

)( xkα

on

the

dot

− xkβ ) ,

( xkα

products

− xkβ

)( xkα

xkα xkα ,

− xkβ ) ,

xkα xkα ,

without

summation with respect to α and β , which are invariant with respect to the space rotations. Hence, this particular Lagrangian has to be of the form

Variational Principles. Canonical Transformations

L =

n

n

n

∑ Tα ( xkα xkα ) − ∑ ∑ Vαβ ( ( xkα

α =1

α = 1 β =1

where we have introduced the notations

− x kβ

327

)( xkα

− x kβ ) , xkα xkα , xkβ xkβ ) ,

(20.3.84'')

Lα = Tα and Lαβ = −Vαβ .

In classical mechanics, the equations of motion are invariant with respect to the Galileo group (Galilean invariance of non-relativistic mechanics), such that, by assuming that Vαβ does not depend on the velocities xkα and xkβ , the condition (20.3.82') becomes d ∂Tα ( δΩ ) = − α δvk . ∂ xk dt

Here, Ω is a function of xkα , k = 1, 2, 3 , α = 1, 2,...,n , and the right side is a total derivative of a function of this kind, only if the functions Tα are proportional to xkα xkα , without summation with respect to α. Finally, the Lagrangian which describes the motion of a system of n particles, invariant with respect to the group of space translations, the group of space rotations, the group of time translations and also conform to the Galilean invariance, has to be of the form

L =

1 n mα xkα xkα − 2 α∑ =1

n

n

∑ ∑ Vαβ ( ( xkα

α =1 β =1

− x kβ

)( xkα

− x kβ

)) ,

(20.3.85)

where we have introduced the proportionality factors mα / 2 , α = 1, 2,...,n , to set up the dependence of the functions Tα on xkα xkα , without summation with respect to α; actually, mα is the mass of the particle P α , α = 1, 2,...,n . In case of such a Lagrangian, the generalized momentum of this particle has the components pkα =

∂L = mα xkα , k = 1, 2, 3 , ∂ xkα

(20.3.86)

while the components of the vector associated to the generalized moment of momentum of the particle are Ki =

1 ∈ K =∈ijk xkα pkα , k = 1, 2, 3 , 2 ijk jk

(20.3.87)

both without summation with respect to α.

20.3.3.7 Application of the Reciprocal of Noether’s Theorem The example in the previous subsection shows, in a very explicit manner, the close relation existing between the motion laws in classical mechanics, inferred directly from the Lagrangian, and the geometrical properties of the space-time, which impose the analytical form of the Lagrangian. Now, we will use the form (20.3.85) of the

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328

Lagrangian to illustrate how to find the symmetry group corresponding to a conservation law, that is how to apply the reciprocal of Noether’s theorem. To simplify the formulation, we will consider the motion of a single particle described by the Lagrangian 1 2

L = m ( x12 + x22 + x 32 ) − V ( x1 , x 2 , x 3 ) ,

(20.3.88)

where V ( x ) is the potential energy of the field of forces acting upon the particle. The Hessian of this Lagrangian with respect to velocities and its inverse are given by the identity matrix E of rank three (abstraction of a multiplicative factor) ⎡1 0 0⎤ ⎡1 0 0⎤ ⎢ ⎥ ⎥ 1 ⎢ −1 H = m ⎢0 1 0⎥ , H = ⎢0 1 0⎥ . m ⎢ ⎥ ⎢ ⎥ ⎢⎣ 0 0 1 ⎥⎦ ⎢⎣ 0 0 1 ⎥⎦

(20.3.88)

Under the assumption that the momentum of the particle of position vector r is conserved, we obtain three first integrals (integrals of motion) Dk ( r , r ) = mxk = const , k = 1, 2, 3 ,

(20.3.89)

and from (20.3.27') it follows δx k =

∂ Dk H jk−1δτk = δτk , ∂ x j

]

(20.3.89')

without summation with respect to k = 1, 2, 3 . Hence, the corresponding infinitesimal transformation with respect to which the Lagrangian remains invariant is of the form xk′ = xk + δτk , k = 1, 2, 3 ,

(20.3.90)

that is an infinitesimal transformation of the form (20.3.68). Consequently, the conservation of the momentum leads to the invariance of the Lagrangian with respect to the translations of the Euclidean space E 3 . The conservation of the angular momentum provides three first integrals Dk ( r , r ) = m ∈ijk xi x j , k = 1, 2, 3 .

The first integral D3 ( r , r ) leads to variations of co-ordinates of the form δx1 = − x 2 δτ 3 , δx 2 = x1 δτ 3 , δx 3 = 0

and to the infinitesimal transformation x1′ = x1 − x 2 δτ 3 , x 2′ = x 2 + x1 δτ 3 , x 3′ = x 3 ,

(20.3.91)

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329

respectively. In matric notation, the transformation takes the form ⎡ x1′ ⎤ ⎡ 1 0 0 ⎤ ⎡ x ⎤ ⎡ 0 −1 0 ⎤ ⎢ ⎥ ⎢ ⎥⎢ 1 ⎥ ⎢ ⎥ ⎢ x 2′ ⎥ = ⎢ 0 1 0 ⎥ ⎢ x 2 ⎥ + ⎢ 1 0 0 ⎥ δτ 3 ⎢ ⎥ ⎢ ⎥ ⎢x ⎥ ⎢ ⎥ 0 0 0 ⎦⎥ ⎣⎢ x 3′ ⎦⎥ ⎣⎢ 0 0 1 ⎦⎥ ⎣ 3 ⎦ ⎣⎢

(20.3.92)

 3 δτ 3 , r ′ = Er + X

(20.3.92')

or

 3 is the generator of the group of transformations where E is the identity matrix, and X

associated to the first integral D3 ; by the same method we obtain for the first integrals D1 ( r, r ) and D2 ( r , r ) the generators ⎡0 0 0 ⎤ ⎡ 0 0 1⎤ ⎢ ⎥ ⎢ ⎥  1 = ⎢ 0 0 −1 ⎥ , X 2 = ⎢ 0 0 0⎥ . X ⎢ ⎥ ⎢ ⎥ ⎣⎢ −1 0 0 ⎦⎥ ⎣⎢ 0 1 0 ⎥⎦

(20.3.93)

The product of the three transformations obtained above leads to an infinitesimal transformation defined by the matrix  δτ , u ( δτ1 , δτ2 , δτ 3 ) = E + X k k

(20.3.94)

i , which is a square, orthogonal non-singular matrix u ∈ SO(3, \ ) . The matrices X i = 1, 2, 3 , satisfy the commutation relations

i X j − X jX  i =∈ X  , i , j = 1, 2, 3 , X ijk k

(20.3.95)

and can be identified with the matrices of the generators of the SO(3, \ ) group in exponential parameterization. Consequently, the transformation defined by matrices of the form (20.3.93) form a group isomorphic to SO(3, \ ) , and the relations expressing the conservation of the angular momentum lead to the invariance of the Lagrangian with respect to the proper rotations of the Euclidean space E 3 . At the same time, the property of invariance imposes a restriction on the form of the potential energy V ( x ) in (20.3.88). This function has to depend on the variables x1 , x 2 and x 3 in the form V ( x12 + x 22 + x 32 ) , the norm of the vector r (i.e., ( r ⋅ r )1/ 2 = ( x12 + x 22 + x 32

)1/ 2 )

being invariant with respect to the proper rotations of E 3 . Indeed, from the condition (20.3.27) it follows that V ( r ) is a solution of the partial differential equations

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330

∂ Dk ∂V ∂ Dk ∂V ∂ Dk ∂V + + = 0 , k = 1, 2, 3 , ∂ x1 ∂ x1 ∂ x 2 ∂ x 2 ∂ x 3 ∂ x 3

or ∈ijk x j

∂V = 0 , i = 1, 2, 3 , ∂ xk

which admits the general solution V = V (ϕ ) , where ϕ = xk xk . The method described above can be generalized to systems of n particles because for ∂L / ∂t = 0 and ∂V / ∂ xk = 0 , in the case L = T − V , the function T is a homogeneous function of second degree, respectively a positive definite quadratic form, which can be reduced to its canonical form (i.e., a sum of squares) by means of a point transformation. Hence, the Hessian of the transformed Lagrangian with respect to the generalized velocities is a diagonal matrix, and the relations of conservation of the generalized angular momentum lead to the generators of the SO(3, \ ) group. We have shown in Sect. 20.3.3.4 that whenever the Lagrangian does not explicitly depend on time ( ∂L / ∂t = 0 ) along a system trajectory (i.e., when Lagrange’s equations of motion are satisfied), the generalized mechanical energy D ( r , r ) = E =

∂L x − L ∂x j j

is conserved ( ∂E / ∂t = 0 ). In our case, the only independent variable is the time t, and we consider an infinitesimal transformation of the form t ′ = t + δt , where δt is an arbitrary infinitesimal function of class C 2 . If we choose δΩ = 0 , then the equation of conservation (20.3.67) becomes

−E

d( δt ) d ⎛ ∂L ⎞ δx + = 0, dt dt ⎜⎝ ∂x j j ⎟⎠

which is identically satisfied for δt = const , that is the considered transformation is a time translation. Note that, for time translations δx k (t ) = x k′ (t ′) − x k (t ) = 0 , k = 1, 2, 3 ,

(20.3.96)

because xk′ (t ′) and xk (t ) have to represent the same point in the space of coordinates. Consequently, we also have δxk = 0 ; it follows that the symmetry transformation corresponding to the conservation of energy is the time translation for a one-particle system t ′ = t + δt , xk′ = xk , δx k = 0 , xk′ = xk , δxk = 0 , k = 1, 2, 3 .

(20.3.96')

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331

20.3.3.8 The Galileo–Newton Group Let E 4 be the Euclidean space spanned by the vectors x = ( x1 , x 2 , x 3 ; t ) ≡ ( r ; t ) . The Galileo–Newton group G is the group of endomorphisms of E 4 , of the form r ′ = Rr + vt + a , t ′ = t + τ

(20.3.97)

and it consists of a time translation, defined by τ, a translation in E 3 ⊂ E 4 , defined by a ∈ E 3 , a transformation of the form (20.3.79) from the Galileo group Γ and a rotation R ∈ SO(3, \ ) of the space E 3 . The Galileo-Newton group is a subgroup of the affine group in four dimensions or, in homogeneous co-ordinates, a subgroup of the general linear group GL ( 5, \ ) . The time interval between two events in the space E 4 , Δt = t2 − t1 , as well as the spatial distance between two simultaneous events Δr = r2 − r1 , t2 = t1 , are invariant with respect to the transformations of G. Reciprocally, the Galileo–Newton group is the most general group of linear transformations in E 4 , which preserves the intervals Δt and Δr . The elements of the Galileo-Newton group G depend on ten parameters, and are denoted by g ( τ , a, v , R ) . The induced composition law is defined by g ( τ , a ′, v ′, R ′ ) g ( τ , a, v , R ) = g ( τ ′ + τ , a ′ + R ′a + τ v ′, v ′ + R ′v , R ′R ) .

(20.3.98)

The identity element of the group is g ( 0, 0, 0, 1 ) , where 1 is the identity element in SO(3, \ ) , while the inverse of a given element is g −1 = g ( τ , a, v , R )−1 = = g ( − τ , R −1 ( a − τ v ), − R −1 v, R −1 ) . The Galileo–Newton group is isomorphic to the

group of matrices v1 ⎡ ⎢ R v2 ⎢ v 3 g → t( g ) = ⎢ ⎢ ⎢0 0 0 1 ⎢0 0 0 0 ⎣⎢

a1 ⎤ a2 ⎥ ⎥ a3 ⎥ ⎥ τ ⎥ 1 ⎥⎦⎥

(20.3.99)

of order five, which depend on ten parameters and satisfy the relation t ( g1 g 2 ) = t ( g1 ) t ( g2 ) ,

(20.3.99')

identical with the internal composition law (20.3.98). We will outline now some results from the theory of the Galileo–Newton group. Thus, it can be shown that the group G is a non-compact Lie group. The most important subgroups of the Galileo–Newton group are:

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332

T = { g ( τ , 0, 0, 1 ) }

(a) (b) (c) (d)

T = { g ( 0, a, 0, 1 ) } Γ = { g ( 0, 0, v , 1 ) } SO(3, \ ) = { g ( 0, 0, 0, R ) }

– the group of time translations, – the group of space translations in E 3 , – the Galileo group, – the group of proper rotations in E 3 .

The direct product of the groups T × T forms an invariant Abelian subgroup of G, and the corresponding factor group is the semi-direct product Γ ∧ SO(3, \ ) , that leads to the decomposition

G = (T × T ) ∧ (Γ ∧ SO(3, \ )) .

(20.3.100)

Note that there are also other decompositions of G, like the following one

G = (T × Γ ) ∧ (T ∧ SO(3, \ )) ,

(20.3.100')

where T × Γ is the maximal invariant Abelian subgroup of G. The basis of the Lie algebra associated to G comprises ten elements corresponding to the subgroups T, T, Γ , and SO(3, \ ) . Consequently, the first integrals obtained from the invariance of the Lagrangian with respect to these subgroups, determine the generators of the Lie algebra, corresponding to the relation (20.3.60). Thus, we obtain: (α ) (β ) (γ ) (δ )

H Pk , k = 1, 2, 3 , ξk0 , k = 1, 2, 3 , Ki = (1/ 2) ∈ijk K jk , i = 1, 2, 3 ,

– the generator of T, – the generators of T, – the generators of Γ , – the generators of SO(3, \ ) .

Hence, the generator of an infinitesimal transformation of G has the form X=

1 K δε + Pi δai + ξi0 δvi + Hδτ = K i δθi + Pi δai + ξi0 δvi + Hδτ . 2 ij ij

(20.3.101)

The matric form of these generators follows from the representation (20.3.99). We can also obtain these generators in the form of differential operators. Thus, we consider the space of square integrable functions f ( r ; t ) , defined on E 4 . Then, we obtain the regular representation of G as the set of differential operators u acting in the space of the functions f, defined by the equation u ( τ , a, v , R ) f ( r ; t ) = f ( R −1 ( r − vt − a + vτ ), t − τ ) .

(20.3.102)

Finally, we obtain the basis of the Lie algebra associated to G, as the set of generators H=

∂ , P = ∇ , ξ0 = t ∇ , K = r × ∇ , ∂t

(20.3.103)

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333

that satisfy the commutation relations

) =∈ijk ξk0 , ( Ki , Pj ) =∈ijk Pk , ( ξi0 , H ) = Pi , ( Ki , H ) = ( ξi0 , ξ j0 ) = ( ξi0 , Pj ) = ( Pi , Pj ) = ( Pi , H ) = ( H, H ) = 0 . ( Ki , K j ) =∈ijk

Kk , ( Ki , ξ0j

(20.3.103')

The subgroup structure of G can be determined by means of the Lie subalgebras which may be identified on the basis of these commutation relations. From (20.3.103') we can also obtain the two invariants of the Galileo–Newton group

P 2 = P2 , N 2 = ( ξ 0 × P ) . 2

(20.3.104)

The invariance of the Lagrangian of a mechanical system with respect to the Galileo–Newton group is the necessary and sufficient condition for the conservation of the generalized linear and angular momenta, of the generalized mechanical energy and of the uniform rectilinear motion of the centre of mass. The corresponding Lagrangian has the form (20.3.85). It is easy to show that, under the transformations of the GalileoNewton group, the mechanical energy and the momentum of a free particle are transformed according to the relations 1 E ′ = E + v ⋅ ( Rp ) + m v2 , p ′ = Rp + m v . 2

(20.3.105)

It follows that V = E′ −

1 1 2 p ′2 = E − p , 2m 2m

(20.3.106)

i.e. V is invariant with respect to the transformation of G. To prove (20.3.106) we used the invariance of the dot product with respect to SO(3, \ ) , that is ( Rp ) ⋅ ( Rp ) = p ⋅ p . From a mechanical point of view, V represents the difference between the mechanical energy and the kinetic energy of the particle, that is the energy of the particle at rest ( p = 0 ); hence, we may say that V represents a sort of “intrinsic energy” (of the nature of a potential energy) of the particle. It is worthwhile to mention that, while in the relativistic mechanics the existence of the rest energy of a particle is an obvious result, a similar energy appears here in the context of classical mechanics, from considerations on the symmetry properties of the Lagrangian in a four-dimensional Euclidean space-time.

Chapter 21 Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems Closely connected to the canonical transformations, one can introduce the notions of integral invariant and of ergodicity. The action-angle variables, useful in the study of periodic motions, are then considered. The methods of exterior differential calculus prove their importance to mathematical modelling of the phenomena of nature, in particular to mathematical modelling of mechanics; in this order of ideas, we will present some elements of invariantive mechanics. Besides the formalisms in the spaces Λs and Γ 2s , other formalisms can be used in various cases; we mention thus the Birkhoffian formalism, applicable to a larger circle of problems than of the Lagrangian and Hamiltonian formalisms. The theory of control will allow then a profound insight into optimal trajectories.

21.1. Integral Invariants. Ergodic Theorems After some results concerning the integral invariants of order 2s , one passes to a study of the integral invariants of first order. As well, one makes some general considerations concerning ergodic theorems (Arnold, V.I., 1976).

21.1.1 Integral Invariants of Order 2s In what follows one introduces the notion of integral invariant and one gives some corresponding general results; one passes then to the case of the absolute integral invariant of order 2s . 21.1.1.1 Notion of Integral Invariant. General Considerations We return to the system of differential equations (15.1.22), written in the form x j = X j ( x1 , x 2 ,..., x 2 s ; t ) , j = 1,2,..., 2s

(21.1.1)

for n = 2s + 1 ; if the functions X j and ∂X j / ∂xk , j , k = 1, 2,..., 2s , are definite and continuous in a neighbourhood U of the point ( x10 , x 20 ,..., x 20s ; t0 ), then there exists a neighbourhood U ⊂ U in which the solutions of this system of differential equations is obtained by means of the first integrals (15.1.23). The initial conditions of Cauchy type

P.P. Teodorescu, Mechanical Systems, Classical Models, © Springer Science+Business Media B.V. 2009

335

MECHANICAL SYSTEMS, CLASSICAL MODELS

336

allow to determine the integration constants so that through any point P 0 (x10 , x 20 ,..., x 20s ) ∈ D0 does pass an integral curve (according to the theorem of existence and uniqueness) x k = x k (t ; x10 , x 20 ,..., x 20s ), k = 1, 2,..., 2s ,

(21.1.2)

and only one; vectorially, we can write (we consider 2s -dimensional vectors) r = r (t ; r0 ) .

(21.1.2')

⎡ ∂ ( x1 , x 2 ,..., x 2 s ) ⎤ J = det ⎢ ⎥ ≠ 0, 0 0 0 ⎣ ∂ ( x1 , x 2 ,..., x 2 s ) ⎦

(21.1.2'')

As well,

so that we also may calculate

x k0 = x k0 (t ; x1 , x 2 ,..., x 2 s ), k = 1,2,...,2s ,

(21.1.2''')

r0 = r0 (t , r ) ,

(21.1.2IV)

or

Fig. 21.1 Transformation of the domain

D0 into the domain D

A representative point P 0 corresponds to the state of the mechanical system for t = t0 , in the phase space Γ 2s , so that D0 represents a set of possible states of the system. If we fix r0 , then the trajectory of the representative point which passes through P 0 is given by (21.1.2'). If we fix t , while r0 travels through the domain D0 , then (21.1.2'') corresponds to a continuous and twice differentiable transformation of the domain D0 into a domain D ⊂ Γ 2 s (Fig. 21.1); thus, the domain D represents the set of the states of the mechanical system at the moment t , every state corresponding to a state at a point of the domain D0 . In a physical (mechanical) modelling one can imagine that the representative points are the particles of a fictitious

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337

fluid which occupies the domain D 0 at the moment t0 and the domain D at the moment t . The transformation (21.1.2) is one-to-one and bicontinuous (hence topological), so that a point of D0 passes at a point of D , while any point of D is the image of only one point of D0 ; hence, the transformation (21.1.1) preserves the individuality of the points, as well as the domain D as a material variety. Indeed, the domain D is formed only of the points which have been in D0 at the initial moment. In general, the transformation (21.1.1) conserves any material variety: curves (linear varieties), surfaces (two-dimensional varieties) etc. Let us consider an integrable function F = F ( r ; t ) of class C 1 definite on the set of possible states D of the system at the moment t . We say that the quantity I2s =

∫D F (r ; t ) δV2 s ,

(21.1.3)

where the element of volume δV2 s = δx1 δx 2 ...δx 2 s (we use the operator δ because we calculate for a fixed t ) is an integral invariant of the system (21.1.1) if (δV20s = δx10 δx 20 ...δx 20s )

∫D F (r ; t ) δV2 s

=

∫D

F ( r0 ; t ) δV20s 0

(21.1.3')

for any moment t ; in this case, I 2s is an invariant of the transformation (21.1.2). In general, the integral I 2s depends on t (directly, through the agency of r and of D ); hence, it is necessary that dI 2 s / dt = 0 . Let be a curve of possible states of the system at the moment t and the form of first degree ω1 =

2s

∑ Fi δxi ,

i =1

definite on Γ ; one obtains an integral invariant

∫Γ ω1 = ∫Γ

∫Γ ω1

if

ω1 . 0

(21.1.4)

One can make an analogous statement for a surface of simultaneous states. Let be a m -dimensional manifold Vm and let be Im =

N

k ∫V k∑=1 Fk ( r ; t )δVm , m

(21.1.5)

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338

where δVmk = δx i1 δx i2 ...δx im , the indices i1 , i2 ,..., im taking all the values from 1 to 2s , while the index k (which specifies the volume element) takes all the values

from 1 to N = C 2ms (the number of volume elements); I m is an integral invariant if N

k ∫V k∑=1 Fk ( r ; t )δVm m

=

N

0k ∫ k∑=1 Fk ( r0 ; t0 )δVm Vm0

(21.1.5')

at any moment t . In general, let be a m -chain Vm , 1 ≤ m ≤ 2s , of possible states at the moment t ; N

∑ Fk (r ; t ) δxi

if one gives the form ωm =

1

k =1

∧ δx i2 ∧ ... ∧ δx im , defined on Vm ,

where the indices i1 , i2 ,..., im take all the values from 1 to 2s , the sum having C 2ms terms and where we have introduce the exterior product (see App., Subsec. 1.2.1), then we say that the quantity ∫ ωm is an integral invariant if (Vm0 is the m -chain of states Vm

at the initial moment, from which result, by trajectories, the states of Vm )

∫V

m

ωm =

∫V

0 m

ωm .

(21.1.4')

The order of the invariant is specified by the degree of the form. The integral invariant is called absolute (denoted by I m ) if Vm is an open set or relative (denoted by Im ) if Vm is a closed set (with frontier). Using Stokes’s theorems, we notice that

Im = I m +1 ;

(21.1.6)

one can state Theorem 21.1.1 A relative invariant of order m is equivalent to an absolute invariant of order m + 1 . Thus, in case of a relative invariant of first order we have (see. App. Subsec.2.3.2) 2s

∫∂V k∑=1 Fk δxk 2

=

2s

⎛ ∂F

k ⎜ ∫V j ∑ ,k = 1 ⎝ δx j 2

j
−

∂Fj ⎞ δx δx , δx k ⎟⎠ j k

(21.1.6')

where V2 is a surface which leans upon a closed curve ∂V2 . If we replace the initial values by (21.1.2IV) in (21.1.3'), then we get an invariant of the form

∫D Φ(r ; t ) δV , the integrand of which depends on the system of the chosen initial values; observing that these values can vary arbitrarily, we can state

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339

Theorem 21.1.2 A differential system of the form (21.1.1) has an arbitrary number of integral invariants. 21.1.1.2 Complete Absolute Integral Invariants Let be an infinitesimal domain D , neighbouring the domain D 0 , both arbitrary; in this case, N

∑ Fk ( r ; t )δVmk

k =1

=

N

∑ Fk ( r0 ; t0 )δVm0 k .

k =1

(21.1.7)

Let us denote the differential form in (21.1.7), called “the element of the invariant”, by F ( r ; t ; δr ) . Making the substitution (21.1.2IV) in the second member of this relation, that one becomes a functions of r and t and δr , let be Φ ; we have F ( r0 ; t0 ; δr ) = Φ ( r ; t ; δr ; dt ) .

(21.1.8)

Taking into account how the function Φ has been built up, we notice that its numerical value depends only on the initial values r0 and δr0 and does not depend on the chosen point ( r; t ) on the actual trajectory and nor on the point ( r + δr; t + δt ) chosen on the neighbouring infinitesimal trajectory. Hence, Φ is an invariant form, more general that the invariant from which we started, because we are no more obliged to consider only the simultaneous points. If we assume that t = const in the relations (21.1.2'''), then we must have Φ ( r0 ; t0 ; δr ; 0) = F ( r ; t ; δr ) ;

(21.1.9)

if t is variable, then it will intervene by δt in the calculation of δx k0 . The last differential is the total differential of a first integral, hence it can be expressed by a linear combination of the forms δx k = x k δt ; the quantity (21.1.9) becomes Φ (x i ; t ; δx i ; δt ) = F (x i ; t ; δx i − Xi δt ) .

(21.1.9')

In this case, the absolute integral invariant (21.1.5), which is obtained by replacing δx i by δx i − x i δt , is called complete absolute integral invariant. But we must notice that the methodology to obtain complete invariants can be applied only to absolute invariants; indeed, in case of relative invariants, the invariance of the differential form (21.1.7) under the integral does not result from the invariance of the integral (21.1.5). 21.1.1.3 The Absolute Invariant of Order 2s . Liouville’s Theorem We notice that (21.1.3) is an integral invariant if the condition d F ( r ; t ) δV2 s = 0 dt ∫D

(21.1.10)

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340

is fulfilled. To reach a domain D0 which does not depend on time we effect the transformation (21.1.2'), obtaining thus

(

)

d d dF dJ F ( r ; t ) δV2 s = F ( r ( r0 ; t ); t ) J δV20s = ∫ J +F δV20s ∫ ∫ D D D dt dt 0 dt dt 0

.

We observe that the derivative dJ / dt of the functional determinant can be written as a sum of 2s determinants I i , i = 1, 2,..., 2s , where I i is obtained from J by differentiating the line i ; it results (we take into account (21.1.1)) ⎡ ∂ ( x1 , x 2 ,..., x i −1 , Xi , Xi + 1 ,..., X 2 s ) ⎤ I i = det ⎢ ⎥. ∂ ( x10 , x 20 ,..., x 20s ) ⎣ ⎦

Because

∂Xi = ∂x j0

∂X ∂x k , ∂x j0 k =1 2s

∑ ∂xki

we see that I i can be calculated as a sum of 2s determinants, each one having two identical lines, excepting that determinant which corresponds to k = i ; this determinant is equal to J δXi / δx i . Finally, 2 s ∂X dJ = J ∑ i = J divX , dt i =1 ∂x i

(21.1.11)

where X is the 2s -dimensional vector of components X1 , X 2 ,..., X 2 s . Returning to the actual variables, it results 2s dI 2 s ∂X ⎛ dF = ∫ ⎜ + F∑ i D0 ⎝ dt dt i =1 ∂x i

2s ∂Xi ⎞ ⎞ ⎛ dF 0 ⎟J δV2 s = ∫D ⎜ dt + F ∑ ∂x ⎟ δV2 s i ⎠ ⎠ ⎝ i =1 2s 2s ∂Xi ⎞ ⎤ ∂F dxi ∂ ⎡ ∂F ⎡ ⎤ ( FXi ) ⎥ δV2 s . = ∫ ⎢ + ∑ ⎛⎜ +F ⎟ ⎥ δV2 s = ∫D ⎢ F + ∑ D ⎣ ∂t x d t x x ∂ ∂ ∂ i i ⎠⎦ i ⎣ ⎦ i =1 ⎝ i =1

Because F is a function of class C 1 and the domain D is arbitrary, we can state Theorem 21.1.3 I 2s is an invariant integral if the necessary and sufficient condition F +

2s

∂

∑ ∂xi ( FXi ) = 0

i =1

(21.1.12)

is fulfilled. Taking into account the results in Sect. 15.1.1.4 (formula 15.1.29)), it results that F must be a Jacobi multiplier of the system of equations (21.1.1).

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341

If F = 1 , then the condition (21.1.12) becomes a condition of null divergence ∂X

2s

∑ ∂xii

i =1

= 0,

(21.1.13)

verified by non-divergent systems; in this case

I2 s =

∫D δx1 δx 2 ...δx 2 s

(21.1.13')

is an absolute integral invariant of this system. As well, dJ = 0. dt

(21.1.13'')

As we have seen in Sect. 19.1.1.6, the condition (21.1.13) is verified in case of Hamilton’s system of canonical equations, so that the quantity

I2 s =

∫D δq1 δq2 ...δqs δp1 δp2 ...δps ,

(21.1.14)

which represents the volume of the domain D in the phase space (the phase space is not a metric space, because we cannot establish distances; however, we consider volume elements which are obtained by multiplying a volume element of the space of configurations by the corresponding volume element of the space of generalized momenta), is an integral invariant; we can state Theorem 21.1.4 (Liouville). The volume of a domain of possible (simultaneous) states of a Hamiltonian system remains constants during the motion. Let us imagine a very great number of identical actual states of a system, which differ one from the other only by their initial states qi0 , pi0 , i = 1, 2,..., s ; all these states form a statistical set. Such a set may be the set of molecules of gas in a given volume. The volume, represented by I2s in the space Γ 2s , is called extension of phase; Liouville’s theorem is thus known under the denomination of conservation theorem of extension of phase too. Thus, Liouville’s theorem plays a fundamental rôle in the deductions of Boltzmann’s equation in the kinetic theory of gas. If we remember the mechanical image in Sect. 21.1.1.1 (the fictitious fluid), then we can say that “the flow in the phase space is incompressible”. In case of a canonical system we have J = 1 at the initial moment; from (21.1.13'') it results that J = const , so that J = 1 at any moment t .

21.1.2 Invariants of First Order After some results of general character, one presents the Poincaré and the Poincaré-Cartan invariants; we mention Hwa-Chung Lee’s theorem too. A special attention is given to the connection between the integral invariants and the differential forms.

MECHANICAL SYSTEMS, CLASSICAL MODELS

342 21.1.2.1 General Results The condition that the integral 2s

∫Γ k∑=1 F (r ; t ) δxk ,

I1 =

(21.1.15)

where Γ is a curve of simultaneous positions, be an integral invariant of first order is written in the form dI 1 / dt = 0 . If the curve Γ 0 (the image of the curve Γ by the transformation (21.1.2''')) is given parametrically in the form r0 = r0 (λ ) , then the curve Γ will have an equation of the form r = r (λ ) ; in this case (we take into account (21.1.1)) dx d d ⎛ ∂x i ( δx i ) = δλ ⎞⎟ = δ i = δXi = ⎜ dt dt ⎝ ∂λ dt ⎠

2s

∂X

∑ ∂xki

k =1

δx k ,

so that dI 1 = dt

∫Γ

⎡ 2 s dFk ⎢ ∑ dt δx k + ⎣ k =1

0

2s

⎤

d

⎛ dFk + dt k =1 2s

∑ Fi dt ( δxi ) ⎥⎦ = ∫ Γ ∑ ⎝⎜

i =1

2s

∑ Fi

i =1

∂Xi ⎞ δx = 0. ∂x k ⎠⎟ k

(21.1.16) The integrand is a continuous function and Γ is an arbitrary curve; hence, I 1 is an absolute invariant if and only if the conditions dFk + dt

2s

∑ Fi

k =1

∂Xi = Fk + ∂xk

∂F

2s

∑ ⎛⎜⎝ ∂xki Xi

i =1

+ Fi

∂Xi ⎞ = 0, k = 1, 2,..., 2s , ∂xk ⎟⎠

(21.1.16')

hold. In the particular case in which Γ 0 is a closed curve (let be the curve C 0 ), the curve Γ is, as well, a closed one (the curve C ). Integrating by parts the term (for a fixed t ) 2s

∑ Fi

k =1

∂Xi δx = Fi δXi , ∂x k k

the condition for I i be a relative invariant reads (we take into account (21.1.16) too) ⎡

∂F

⎛ k v∫ C k∑=1 ⎢⎣ Fk + i∑=1 ⎜⎝ ∂xi 2s

2s

−

∂Fi ⎞ ⎤ X δx = 0 . ∂x k ⎟⎠ i ⎥⎦ k

(21.1.17)

Let be Fk = ∂F / ∂xk , k = 1, 2,..., 2s , F = F ( x1 , x 2 ,...x 2 s ; t ) ; the conditions (21.1.16') become ∂ ⎛ ∂F ⎞ + ∂t ⎜⎝ ∂x k ⎟⎠ ∂ = ∂x k

⎛  ⎜F + ⎝

∂

∂F ⎞ ∂F ∂X i ⎤ ⎟ X i + ∂x ∂x ⎥ ⎠ i k ⎦

∑ ⎡⎣⎢ ∂xi ⎛⎝⎜ ∂xk 2s

i =1

∂F ∂ dF ⎞ ∑ ∂xi Xi ⎟⎠ = ∂xk ⎛⎜⎝ dt ⎞⎟⎠ = 0, k = 1, 2,..., 2s , i =1 2s

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343

being verified if dF / dt = 0 , hence if F (x1 , x 2 ,...x 2 s ; t ) = const . Hence, if F is a first integral of the system of equations (21.1.1), then 2s

∂F

∫Γ ∑ ∂xk k =1

δx k

(21.1.18)

is an absolute integral invariant of this system. The inverse implication takes place too. Indeed, let (21.1.18) be an absolute integral invariant of the system (21.1.1); the condition (21.1.16') leads to ∂ (dF / dt )/ ∂xk = 0, k = 1, 2,..., 2s , so that

F − ∫ f (t )dt is a first integral of the considered system of equations. If dF / dt = 0 then F is a dF / dt = f (t ) , f (t ) being an arbitrary function. It results that

determined first integral. 21.1.2.2 Poincaré’s Integral Invariant Passing to Hamilton’s system of canonical equations, we consider the integral (we use the convention of summation of dummy indices, from 1 to s ) I1 =

v∫ C p j δq j .

(21.1.19)

Fig. 21.2 Tube of trajectories of a fictitious fluid (a); projection of a contour on a phase plane

Considering the condition (21.1.16), we can write dI 1 = dt

⎛

v∫ C ⎜⎝ p j

+ pk

∂qk ⎞ δq = ∂q j ⎟⎠ j =

v∫ C δ ( pk qk

v∫ C [ p j δq j

+ δ ( pk qk ) − qk δpk

]

− H ) = 0,

along the integral curves of the canonical equations; the conditions (21.1.17) leads, obviously, to the same result. Hence, I 1 is a relative integral invariant of the canonical system (H. Poincaré’s relative integral invariant).

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344

This invariant integral does not change its value if the closed curve C travels along the tube of the trajectories of the fictitious fluid, model of the trajectories of the representative points, starting from the contour C 0 , corresponding to the initial moment t0 (Fig. 21.2a); these contours consist of simultaneous states. Hence,

v∫ C p j δq j

v∫ C

=

p 0j δq 0j .

(21.1.19')

0

Let be s separated phase planes (qi , pi ), i = 1, 2,..., s . Projecting a contour C (from those considered above) on one of these planes (Fig. 21.2b), we obtain the contours C i , i = 1, 2,..., s . Projecting for each i , it results

v∫ C pi δqi

=

v∫ C

p j δq j = ± Ai , (!), i

where Ai is the area of the domain

(21.1.20)

Di bounded by the contour C i in the plane. The

sense of circulation on C i is specified by the sense of circulation on C ; one takes the sign + if the sense of circulation is clockwise and the sign – otherwise. Hence,

I1 =

v∫ C pi δqi

=

s

∑ ( ± Ai ) .

(21.1.21)

i =1

The contours C and C i vary during the motion (as well, the areas Ai ), the algebraic sum (21.1.21) remaining constant; this is the geometric interpretation of Poincaré’s integral invariance. Let be the system of differential equations q j = Q j (q , p ; t ), p j = Pj (q , p ; t ), j = 1, 2,..., s ,

(21.1.22)

where q ≡ {q1 , q2 ,..., qs }, p ≡ { p1 , p2 ,..., ps } ; we calculate dI 1 = dt =

v∫ C ( p i δqi

v∫ C [ p i δqi

+ pi

d δq dt i

) = v∫

C

( p i δqi + pi δqi )

+ δ ( pi qi ) − qi δpi ] = v∫ ( Pi δqi − Qi δpi ). C

Equating to zero, it follows that the integral must be an exact differential −δH (q , p ; t ) (because t = const ); hence

Pi = −

∂H ∂H , Qi = , i = 1, 2,...s . ∂qi ∂pi

(21.1.22')

We can state Theorem 21.1.5 If the system of differential equations (21.1.22) admits the relative invariant (21.1.19), then it is a canonical system.

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345

21.1.2.3 Universal Integral Invariants. Hwa-Chung Lee’s Theorem Applying Stokes’s formula (see App., Subsec. 2.3.2), the integral I 1 can be written in the form ( I2 = I 1 )

I2 =

∫∫ δp j δq j .

(21.1.23)

Let Qi = Qi (q , p ; t ), Pi = Pi (q , p ; t ), i = 1, 2,..., s , be new canonical variables and H = H (Q , P ; t ), Q ≡ {Q1 ,Q2 ,...,Qs }, P ≡ {P1 , P2 ,..., Ps } , the corresponding function of Hamilton; if q and p are integration constants (e.g., initial values of the variables Q and P ) in the new system, then these relations can be seen as relations of canonical transformations. Let use consider, in the phase space (q , p ) , a two-dimensional variety V2 , which is transformed in the two-dimensional manifold V2′ . By means of two parameters ( u and v ), we can thus express the canonical co-ordinates in the form qk = qk (u , v ) , pk = pk (u , v ) , j , k = 1, 2,..., s In this case, we can write

∫V

2

⎡ ∂ ( qk , pk ) ⎤ δu δv ∂ ( u , v ) ⎥⎦

∫Ω det ⎢⎣

δqk δpk =

∂q ∂p ∂q ∂p = ∫ ⎛⎜ k k − k k ⎞⎟ δu δv = Ω ⎝ ∂u ∂v ∂v ∂u ⎠

∫Ω [ u , v ]

δu δv ,

q ,p

where [ u , v ] is Lagrange’s bracket and Ω is the domain on which one integrates in the plane of variables u and v ; as well, we have

∫V ′ δQj δPj 2

=

⎡ ∂ ( Qi , Pj ) ⎤ δu δv = ∂ ( u , v ) ⎥⎦

∫Ω det ⎢⎣

∫Ω [ u , v ]

Q ,P

δu δv ,

where we have used the invariance of Lagrange’s bracket to a canonical transformation (see Sect. 20.2.2.3). We obtain thus the identity

∫V

2

δqk δpk =

∫V ′ δQ j δPj ; 2

(21.1.23')

we can state that I2 is an absolute integral invariant of second order of the canonical system or of the canonical transformation. Analogously, we can define integral invariants of higher order, that is the relative integral invariant I 2 l −1 =

δpi δqi δpi δqi v∫V ∑ i 2 l −1

1

1

2

...δpil δqil

(21.1.24)

...δpil δqil ,

(21.1.24')

2

j

and the absolute integral invariant

I2 l =

δpi δqi δpi δqi ∫V ∑ i 2l

1

j

1

2

2

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346

where the indices i1 , i2 ,..., il take all the values from 1 to s , so that the sums contain C sl terms; these invariants are defined on manifolds of odd or even number of dimensions, respectively, Using theorems of Stokes type, one can show that I 2 l −1 =

I2 l .

(21.1.24'')

On the same way as in the case l = 2 , using a relation of the form ⎡ ∂ ( X1 , X 2 ,..., X 2 s ) ⎤ = det ⎢ ⎣ ∂ ( u1 , u2 ,..., ul ) ⎥⎦

∑ det ⎡⎢⎣

∂ ( X1 , X 2 ,..., X 2 s ) ⎤ ⎡ ∂ ( x1 , x 2 ,..., x 2 s ) ⎤ det ⎢ ∂ ( x1 , x 2 ,..., x 2 s ) ⎥⎦ ⎣ ∂ ( u1 , u2 ,..., ul ) ⎥⎦

one can show that dI 2 l / dt = 0 . In the particular case l = s , one obtains again the absolute integral invariant I 2s of Liouville. We notice that both I 2 l −1 and I 2 l , l = 1, 2,..., s , do not depend on the structure of Hamilton’s function H , hence do not depend on the considered mechanical system, but only on the canonical form of the equations; thus, these integral invariants are called universal integral invariants. The uniqueness of these invariants has been put in evidence, in 1947, by Hwa-Chung Lee, who made a profound study of them; in this order of ideas, we state Theorem 21.1.6 (Hwa-Chung Lee). If I 1′ =

v∫ C [ Aj (q , p ; t )δq j

+ B j (q , p ; t )δp j

]

(21.1.25)

is a universal relative integral invariant, then I 1′ = cI 1 , c = const ,

(21.1.25')

where I 1 is the relative integral invariant of Poincaré. We prove the theorem in the particular case s = 1 for the universal relative integral invariant I 1′ =

v∫ C [ A(q , p ; t )δq + B (q , p ; t )δp ] ,

in the phase plane (q , p ) , the closed curve C being in this plane. The solution of the Hamiltonian system q = ∂H / ∂p , p = −∂H / ∂q will be of the form q = q (t ; q 0 , p0 ) , p = p (t ; q 0 , p0 ) , with q 0 = q (t0 ), p0 = p (t0 ) . Let be q 0 = q 0 (λ ) and p0 = p0 (λ ) , 0 ≤ λ ≤ l , q 0 (0) = q 0 (l ), p0 (0) = p0 (l ) , the parametric equations of a closed contour C 0 in the phase space (Fig. 21.3); the points of this contour reach at the moment t the contour C , the equations of which (after replacing the parametric representation of q 0 and p0 ) are given by q = q (t ; λ ), p = p (t ; λ ), 0 ≤ λ ≤ l .

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347

If we replace in the integral I 1′ , we obtain a function of parameter t , so that the condition of invariance becomes dI 1′ / dt = 0 . Differentiating under the integral sign and integrating by parts ( v∫ u δv = − v∫ v δu ), we get dI 1′ = dt

v∫ C ⎛⎜⎝ dt δq + A dt δq +

dB d δp + B δp ⎞⎟ dt dt ⎠ d A d B d A dB = v∫ ⎛⎜ δq + Aδq + δp + B δp ⎞⎟ = v∫ ⎛⎜ δq − qδA + δp + p δB ⎞⎟ C ⎝ dt C dt d t dt ⎠ ⎝ ⎠ ∂A ∂B ⎞ ∂ ∂ B A ⎡ ⎛ ⎤ ⎡ ⎛ ⎞ ⎤ = v∫ ⎢ ⎜ − − ⎟ p + A ⎥ δq + ⎢ ⎜ ⎟ q + B ⎥ δp C ⎣ ⎝ ∂p ∂q ⎠ ∂p ⎠ ⎦ ⎣ ⎝ ∂q ⎦ ∂H ∂ H ⎡ ⎞ ⎤ ⎛ ⎞ ⎛ = v∫ ⎢ ⎜ − Z + A ⎟ δq + ⎜ − Z + B ⎟ δp ⎥ = 0, C ⎣⎝ ∂q ∂p ⎠ ⎦ ⎠ ⎝ dA

d

{

}

Fig. 21.3 Representations of the closed contours C 0 and C

where we took into account the canonical equations and we have denoted Z =

∂A ∂B . − ∂p ∂q

The integral thus obtained must vanish for any t (considered as a parameter) and for an arbitrary integration path; hence, the integrand must be a total differential with respect to the variables q and p , so that ∂ ⎛ ∂H ∂ ⎛ ∂H + A ⎞⎟ = + B ⎞⎟ ⎜ −Z ⎜ −Z ∂p ⎝ ∂q ∂p ⎠ ∂q ⎝ ⎠

or (taking into account the notation made) −

∂Z ∂H ∂Z ∂H + + Z = 0 . ∂p ∂q ∂q ∂p

Because H may be chosen arbitrarily, it results that ∂Z / ∂p = ∂Z / ∂q = Z = 0 or Z = c , c = const . We can also write

MECHANICAL SYSTEMS, CLASSICAL MODELS

348

∂ (A − cp ) ∂B = ; ∂p ∂q

hence, there exists a function ϕ (q , p ; t ) ( t is a parameter), so that (A − cp ) δq + B δp =

∂ϕ ∂ϕ δq + δp = δϕ , ∂q ∂p

wherefrom

v∫ C (Aδq + B δp ) = v∫ C (ϕ + cpδq ) = c v∫ C pδq

= cI 1 ,

the theorem being thus proved. In the case s > 1 one uses the same ideas; but the proof is much more complicated. One can state an analogous theorem for l > 1 . 21.1.2.4 The Poincaré-Cartan Integral Invariant We make a study of the variation of the Lagrangian action δAL in the general case in which both the generalized co-ordinates and the time depend at the initial and at the final moments on a parameter λ , so that

t0 = t0 (λ ), t1 = t1 (λ ), qi0 = qi0 (t ; λ ), qi1 = qi1 (t ; λ ), 0 ≤ λ ≤ l , l = 1,2,..., s .

(21.1.26)

The Lagrangian action will be, in this case, of the form

AL (λ ) =

t1 ( λ )

∫t ( λ ) L ( qi (t ; λ ), qi (t ; λ ); t ) dt ;

(21.1.27)

0

differentiating (the limits depend on a parameter) and integrating by parts, we obtain δAL (λ ) = =

∂L ∂L δqi + δqi ⎞⎟ dt ∂ ∂ q q  ⎝ i ⎠ i

L1 δt1 − L0 δt0 + ∫t 1 ⎛⎜

L1 δt1 − L0 δt0 +

t

0

∂L δq ∂qi i

t1 t0

+

t1

∫t

0

d ⎛ ∂L ⎞ ⎤ ⎡ ∂L ⎢⎣ ∂qi − dt ⎝⎜ ∂qi ⎠⎟ ⎦⎥ δqi dt ,

where Lα = L (qi (tα , λ ), qi (tα , λ ); tα ), α = 0,1 . Along the integral curves of the canonical system, we remain with δAL = L1 δt1 − L0 δt0 + pi1 ( δqi )t =t1 − pi0 ( δqi )t =t0 ;

(21.1.27')

Let us consider also the variations of the co-ordinates qiα = qiα (t (λ ); λ ), α = 0,1 , in the form δqiα = qiα δtα + [ ∂qiα (t ; λ )/ ∂λ ]t =tα , i = 1, 2,..., s , α = 1,2 , so that

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

[ δqi ]t =tα = δqiα − qiα δtα , i = 1, 2,..., s , α = 1,2 ;

349

(21.1.28)

replacing in δAL , expressing the generalized velocities by means of the generalized   momenta and taking into account pi qi − L = H (after introducing the generalized momenta), we get (along the integral curves of the canonical system) δAL = [ pi δqi − H δt ]10 = pi1 δqi1 − H 1 δt1 − pi0 δqi0 + H 0 δt0 .

(21.1.27'')

Being situated in the space Γ 2 s + 1 (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) , let be a closed curve C 0 of equations

qi = qi0 (λ ), pi = pi0 (λ ), t = t0 (λ ), 0 ≤ λ ≤ l , i = 1,2,..., s ;

(21.1.29)

Fig. 21.4 Closed tube of trajectories

we put the condition to obtain the same point of the curve C 0 for λ = 0 and λ = l . From any point of this curve starts a unique trajectory, determined by the canonical equations and by the initial conditions, obtaining thus a closed tube of trajectories (Fig. 21.4) qi = qi (t ; λ ), pi = pi (t ; λ ), qi (t ; 0) = qi (t ; l ), pi (t ; 0) = pi (t ; l ),

(21.1.30)

0 ≤ λ ≤ l , i = 1, 2,..., s .

We choose on this tube another closed curve C 1 which has only one common point with every trajectory; thus, to each λ corresponds a trajectory and only a single point on C 1 , so that we can write the equations of the curve C 1 in the form qi = qi1 (λ ), pi = pi1 (λ ), t = t1 (λ ) .

Integrating the relation (21.1.27'') with respect to λ , from 0 to l , it results

(21.1.30')

MECHANICAL SYSTEMS, CLASSICAL MODELS

350

1

AL (1) − AL (0) = ∫ 0 [ pi δqi − H δt ]10 = =

∫0 ( pi δqi 1

v∫ C

1

1

− H 1 δt1 ) −

( pi δqi − H δt ) − 1

∫0 ( pi δqi

v∫ C

1

0

0

+ H 0 δt0

)

( pi δqi − H δt ) = 0, 0

so that

v∫ C

( pi δqi − H δt ) = 0

v∫ C

( pi δqi − H δt ) ; 1

hence, the integral I =

v∫ C ( pi δqi

− H δt ) ,

(21.1.31)

taken on a curve C , obtained by displacement and deformation from C 0 , along the tube of trajectories, remains constant in time. We obtain thus an integral invariant called the Poincaré-Cartan integral invariant, corresponding to the system of canonical equations. E. Cartan extended Poincaré’s integral invariant to contours of non-simultaneous states, by introducing the supplementary term − H δt ; but if C 0 represents a set of simultaneous states (we make δt = 0 ), we find again Poincaré’s integral invariant. To can prove the inverse implication, we consider a tube of trajectories which verify the system of differential equations (21.1.22), so that – on a closed curve surrounding it – the integral (21.1.31) is invariant; such a curve can be represented in parametric form qi = qi (λ , μ ), pi = pi (λ , μ ), i = 1, 2,..., s , t = t (λ , μ ) ,

(21.1.32)

where the parameter λ is fixing the trajectory which starts from a point of the curve C 0 , while the parameter μ corresponds to the point on this trajectory. The system of equations (21.1.22) reads d q1 dq dp dp d qs dps dt = 2 = ... = = 1 = 2 ... = = = χ (q , p ; t )dμ , Q1 Q2 Qs P1 P2 Ps 1

where χ (q , p ; t ) is an arbitrary function in the extended phase space Γ 2 s + 1 (thus, the family of curves μ = const is, at the actual state, an arbitrary family of closed curves, which are not intersecting and which surround the given tube of trajectories). To integrate this system of equations, we can express its solution by means of functions of μ and of the initial values qi0 , pi0 , i = 1, 2,..., s and t0 (for μ = 0 ) in the form

qi = fi (t0 , q 0 , p 0 ; μ ), pi = gi (t0 , q 0 , p 0 ; μ ), i = 1,2,..., s , t = t (t0 , q 0 , p 0 ; μ ) .

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

351

One obtains thus a parametric representation of the family of integral curves of the system (21.1.22). The initial positions of the corresponding representative point will be on the curve C 0 (to obtain the considered tube), given by (21.1.29); we obtain thus the equations (21.1.32). For any μ = const we find a point on each generating trajectory and a corresponding closed curve (with 0 ≤ λ ≤ l ). The integral I becomes a function of μ ; imposing the condition that I be an invariant by passing from a curve C to another closed curve along the tube, we write dI = 0 (we differentiate with respect to μ ). Differentiating with respect to μ under the integral sign, it results

v∫ C

[ dpi δqi + pi dδqi − (dH δt + Hdδt ) ] = 0 ; 0

by permutation of the operators d and δ (the first one represents the differentiation with respect to μ , while the second one represents the differentiation with respect to λ ) and integrating by parts, we obtain (the integrated term vanishes on a closed contour)

v∫ C ( dpi δqi =

⎡⎛

v∫ C ⎢⎣ ⎝⎜ pi

+

− δpi dqi − dH δt + δHdt )

(

)

∂H ⎞ ∂H ⎞ dH δq + ⎛ −q + δp + − + H δt ⎤⎥ χ dμ = 0, ∂qi ⎠⎟ i ⎝⎜ i ∂pi ⎠⎟ i dt ⎦

because δH =

∂H ∂H δq + δp + H δt . ∂qi i ∂pi i

Taking into account (21.1.22), we obtain ⎡⎛

v∫ C ⎢⎣ ⎝⎜ Pi

+

(

)

∂H ⎞ ∂H ⎞ dH δq + ⎛ −Q + δp + − + H δt ⎤⎥ χ dμ = 0 ; ∂qi ⎠⎟ i ⎝⎜ i ∂pi ⎠⎟ i dt ⎦

the integrand must be an exact differential with the arbitrary factor χ , which is possible only if the brackets vanish, being thus led to the equations (21.1.22') (we obtain also dH / dt = H , result which is – in fact – a consequence of the canonical equations). We can thus state Theorem 21.1.7 If the system of differential equations (21.1.22) admits the relative invariant (21.1.31), then this system is canonical. Hence, the Poincaré-Cartan integral invariant can be stated at the basis of Hamiltonian mechanics, because it leads us to the canonical equations. We notice that in this integral invariant the time plays the rôle of a co-ordinate and − H (the mechanical energy with changed sign) plays the rôle of a generalized momentum; the mentioned analogy is particularly useful. Indeed, introducing the new function z = − H (q , p ; t ) , we can calculate p1 = − K (q1 , q2 ,..., qs , z , p2 , p3 ,..., ps ; t ) , so that the integral (21.1.31) becomes

MECHANICAL SYSTEMS, CLASSICAL MODELS

352 I =

v∫ C ( z δt + p2 δq2

+ p3 δq 3 + ... + ps δqs − K δq1 ) ,

(21.1.33)

preserving the aspect of the Poincaré-Cartan integral. Thus, the motion of the system will be described by Hamilton’s differential equations dt ∂K = , d q1 ∂z dq j ∂K dp j = , = d q1 ∂p j dq1

∂z = − K , ∂q1 ∂K , j = 2, 3,..., s , ∂q j

(21.1.33')

where q1 is the independent variable. For instance, in case of the linear oscillator, for which H =

1 2 k 2 p + q = −z , 2m 2

(21.1.34)

we obtain K = −p =

m −2 z − kq 2 ;

(21.1.34')

the canonical equations will be

dt = dq

m k

1 dz , = 0, z 2 dq −2 − q k

being thus led to z = − h , h = const , and to ωt =

∫

k dq q +ϕ, + ϕ = arcsin 2h 2h − q2 k

wherefrom q = a sin ( ωt − ϕ ) , a =

2h ,ω = k

k , m

(21.1.34'')

ϕ being an arbitrary constant.

21.1.2.5 Integral Invariants and Differential Forms We have introduced in Sect. 19.1.1.9 the Pfaff form of Hamilton (19.1.45), which is a form of first degree definite on the space of states S = Γ 2 s × E1 ( E1 is the space of time). More general, we introduce on S the form of second degree, independent on the local system of canonical co-ordinates,

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

dω = dpi ∧ dqi − dH ∧ dt ,

353

(21.1.35)

where we use the exterior product (see. App., Sec. 1.2). Let be a family of solutions of the canonical equations qi = qi (t ; λ ), pi = pi (t ; λ ), λ ≡ {λ1 , λ2 ,..., λr }, i = 1, 2,..., s , which depends on the parameters λj ∈ [a j , bj ], j =,2,..., r ; thus, it is

defined a differentiable and one-to-one transformation T : Λ → S of a domain Λ of the space (t , λ ) into S . Let us consider a domain X ⊂ Em , a domain Y ⊂ En and a differentiable application T : X → Y . As well, let be the function (form of zero degree) f : Y → \ by composition with T one obtains the function (a form of zero degree too) T ∗ f = f D T , defining the application T ∗ : F 0 (Y ) → F 0 ( X ) . In general, one can define the application T ∗ : F p (Y ) → F p ( X ) , which maps forms of degree p , defined on Y , into forms of degree p , defined on X . We mention the basic properties of the application T * : (i) T * ( ω1 + ω2 ) = T * ω1 + T * ω2 , (ii) T * ( ω1 ∧ ω2 ) = (T * ω1 ) ∧ (T * ω2 ) , (iii) d(T * ω ) = T * (dω ) , (iv) T1 : X → Y ,T2 : Y → Z ⇒ (T2 D T1 )* = T1* D T2* . Observing that the form (21.1.35) can be written in the form ∂H ∂H dω = ⎛⎜ dpi + dt ⎞⎟ ∧ ⎛⎜ dqi − dt ⎞⎟ ∂qi ∂pi ⎝ ⎠ ⎝ ⎠

(21.1.35')

and taking into account the canonical equations, we get T * (dω ) =

∂qi ∂pi dλj ∧ dλk = ∂ j ,k = 1 λj ∂λk r

∑

r

∑

j ,k = 1

Ajk (t ; λ ) dλj ∧ dλk .

Poincaré’s Lemma ( d(dω ) = 0 ; see App. , Subsec. 1.2.2) leads to d(T * (dω )) = 0 ; it results, further, A jk = 0 , so that T * (dω ) does not depend on time. We say that a differential form Ω of degree p , defined on S , is an absolute integral invariant if, for any transformation T , defined on a family of trajectories which depend on r parameters λj , j = 1,2,..., r , T *Ω is an independent p -form of t and dt , on the space λ , and if dΩ = 0 . The forms dω ,( ∧ dω )2 ,...,( ∧ dω )s are integral invariants, in this case, because p

d( ∧ dω )p = 0, T * ( ∧ dω )p = ⎡⎣ ∧T * ( ∧ dω ) ⎤⎦ , p = 1,2,..., s .

MECHANICAL SYSTEMS, CLASSICAL MODELS

354

Starting from the above definition, one can show that integrating Ω on any chain transverse to a family of trajectories one obtains a constant; hence, any invariant in the sense of this definition is invariant in the sense of the classical definition. We say that a p -form Ω is a relative integral invariant if dΩ is an integral invariant. Thus, because dω is an absolute integral invariant, it results that ω is a relative integral invariant (the Poincaré-Cartan invariant). We can state that the forms ω ∧ dω , ω ∧ ( ∧ dω )2 ,..., ω ∧ ( ∧ dω )s are, as well, relative invariants, because d [ ω ∧ ( ∧ dω )p ] = ( ∧ dω )p + 1 .

If the points which correspond to transverse chains represent simultaneous chains (for t = t0 = const and t = t1 = const ), then dt = 0 . Poincaré’s invariant will correspond to ω . As well, starting from dω = dqi ∧ dpi and observing that ( ∧ dω )s = (s !)dq1 ∧ dp1 ∧ dq2 ∧ dp2 ∧ ... ∧ dqs ∧ dps = ( −1)s ( s −1)/ 2 (s !)dq1 ∧ dq2 ∧ ... ∧ dqs ∧ dp1 ∧ dp2 ∧ ... ∧ dps ,

we get Liouville’s invariant.

21.1.3 Ergodic Theorems In what follows we present the recurrence theorem of Poincaré; starting from it, we put in evidence the so-called ergodic theorems. 21.1.3.1 Poincaré’s Recurrence Theorem Let be an autonomous system (where the time does not appear explicitly) x j = X j ( x1 , x 2 ,..., x s ), j = 1, 2,..., s ,

(21.1.36)

which has the properties: (i)