Lecture Notes in Physics Edited by J. Ehlers, Miinchen, K. Hepp, Ztirich R. Kippenhahn, Miinchen, H. A. Weidenmiiller, and J. Zittat-tz, Kijln Managing Editor: W. Beiglbijck, Heidelberg
Heidelberg
107 Jerzy Kijowski Wlodzimierz M. Tulczyjew
A Symplectic Framework fol Field Theories
Springer-Verlag Berlin Heidelberg
New York 1979
Editors Jerzy Kijowski Department of Mathematical Methods in Physics University of Warsaw ul. Hoza 74 00-682 Warszawa Poland Wlodzimierz M. Tulczyjew Department of Mathematics and Statistics University of Calgary 2920 - 24th Av. N.W. Calgary, Alberta, T2N lN4 Canada
ISBN 3-540-09538-l ISBN O-387-09538-1
Springer-Verlag Springer-Verlag
Berlin Heidelberg New York New York Heidelberg Berlin
Library of Congress Cataloging in Publication Data Kijowski, J 1943A symplectic framework for field theories. (Lecture notes in physics; 107) Bibliography: p. Includes index. 1. Symplectic manifolds. 2. Field theory (Physics) I, Tulczyjew, II. Title. III. Series. QC174.52.894K54 530.1’4 79-20519 ISBN 0-387-09538-l
W. M., 1931-joint
author.
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Printing and binding: 2153/3140-543210
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CONTENTS Introduction I. An intuitive derivation of symplectic concepts in mechanics and field theory
7
I. Potentiality and reciprocity
7
2. Elastic string
2o
3. Eiastostatics
51
4. Electrostatics
35
II. Nonrelativistic
particle dynamics 4-I
5. Preliminaries 6. Special symplectic structures.
Generating
functions
~2
7. Finite time interval formulation of dynamics 8. Infinitesimal description of dynamics
58
9. Hamiltonian description of dynamics
69
~0. The Legendre transformation
75
11. The Caftan form
75
12. The Peisson algebra
79
III. Field theory
80
15. The configuration bundle and the phase bundle
8o
14. The symplectic structure of Cauchy data on a boundary
85
15. Finite domain description of dynamics
91
16. Infinitesimal description of dynamics
tO0
17. Hamiltonian description of dynamics
116
18. The Legendre transformation
124
19. Partial Legendre transformations. -momentum density
The energy125
20. The Cartan form
143
21. Conservation laws
151
22. The Poisson algebra
157
IV
23. The field infinite 24. Virtual
as a mechanical number
of degrees
tensors
of different
160
of freedom
action and the Hamilton-Jacobi
25. Energy-momentum review
system with an
and stress
approaches
theorem
tensors.
16Zl-
A
168 184
IV. Examples 26. Vector
18zl-
field
191
27. The Proca field 28. The electromagnetic 29. The gravitational
198
field
209
field
231
30. The hydrodynamics Appendices A. Sections B. Tangent
242
mapping
C. Pull-back
of differential
of vertical
F. Tensor product
vectors
of bundles
G. The Lie derivative List of more References
forms
24"3 2#5
D. Jets E. Bundle
24-0
of fibre bundles
important
symbols
2q-8 250 250 252 25zt-
Introduction
~hese notes
contain the formulation
work for classical field theories. on fairly advanced concepts
of a new conceptual frame-
Although the formulation
of symplectic
is based
geometry these notes can not
be viewed as a reformulation
of known structures
elegant terms.
is to communicate to theoretical physi-
Our intention
cists a set of new physical
in more rigorous
and
ideas. We have chosen for this purpose
language of local coordinates
the
which although
involved
is more elemen-
tary and more widely known than the abstract
language
of modern diffe-
rential geometry.
We have given more emphasis to physical
than to mathematical
rigour.
Since the new framework unifies variational nonical formulations same symplectic
intuitions
of field theories
structure
formulations
as different
it is of potential
with ca-
expressions
of the
interest to a wide audien-
ce of physicists. Physicists
have been interested
viding a method of theories. variational
in variational principles
as pro-
of generating first integrals from symmetry properties
Powerful methods formulations.
v i n g variational
of solving field equations
are based on
We develop a systematic procedure
formulations
of physical
theories.
for deri-
Using this proce-
dure we have succeeded in formulating a number of new variational principles such as the formulation tes. ~he usefulness is in progress
of the procedure
on a variational
thermal processes.
of hydrodynamics
is far from being exhausted.
formulation
of hydrodynamics
A new variational principle
vity is also included in these notes.
included in these noWork
including
for ~h~ theory of ~ra-
~his new formulation
suggests
solution of the energy localization problem~
provides
lysing asymptotic
fields at spatial infi-
behaviour
of gravitational
a basis for ana-
nity and throws new light on the Cauchy problem for Einstein's tions and on unified field theories.
a
equa-
Quantum physics lectic
formulations
nonical
is one of the main sources of physical
quantization
tructing
symplectic
cial importance
theories.
and the associated
symplectic
structures.
quantum field theory such as the method fit from the more precise tions to gauge theories
theory.
lagrangian
Physicists
interested
will find the extensive submanifolds
in classical
of classical
We consider
theories
Each theory has an underlying time manifold
interest.
limits
theories
is included
mechanics,
with the boundary
/or statics/
allowed by the physical
The lagrangian
laws governing
problem
to the boundary
they are described
generated
of the new fra-
case.
by variational
the field.
domain in
consists
of
This space
of the boundary value
functions,
principles.
"sta-
by a lagran-
subspace
of the domain.
by generating
ction is the action functional.
A symplectic
of each /compact/
as the set of solutions
are usually
expansions.
and the four-dimen-
fields.
can also be described
spaces
theory for cor-
of the field is described
of the state space.
corresponding
Lagrangian
it is the physical
in the case of dynamical
states
in field
as a special
sional
gian subspace
Applica-
which we call field
space for static field theories
M and the dynamics
may pro-
of quantum theories
submanifolds.
three-dimensional
te space" is associated
in
manifold M which is the one-dimensional
in the case of particle
space-time
Of spe-
The new inter-
systems
of the main features
mechanics
methods
systems.
by modern W.K.B.
a class of physical
although particle
algebras.
Lagrangian
limits and asymptotic
We give a brief description
of cons-
local field theories
and hamiltonian
required
in ca-
throws new light on the re-
use of lagrangian
are the objects
rect formulation
mework.
Poisson
of lagrangian
transformation systems
interested
of Feynman integrals
may be of particular
of the Legendre
lation between
formulation
in symp-
general methods
may be the method of describing
by finite-dimensional
pretation
Physicists
will find in these notes spaces
of interest
Lagrangian
sub-
in other words~
Here the generating
If the boundary
is divided
fun-
into sere-
ral components
then the associated
ce and the lagrangian boundary
consists
symplectic
consist
of end points
out to be mappings. can be considered nonrelativistic tonian fields extend
mechanics
as the result
particle
This
relations
mappings
field
formulated
expense
equations
of introducing
an element
situation
terms
relations
gauge invariance
dynamics
mechanics
systems
for
lead to
governed
problems
by
can be
only at the
into the theory and imsituations.
Gauge invariance
in spaceby im-
of the theo-
gauge conditions. can be retained systems
Without
by using
similar to
compactifying
can still be discussed data,
Such
or compactified
hamiltonian
mechanics.
between initial
the
in terms of sym-
final data and the asymptotic
Boundary problems
other than Cauchy
within the same framework.
are obtained
and does not
is obtained
interesting
suitable
and generalized
be discussed
nal formulations
boundary
elements
conditions.
or outgoing radiation.
problem~can
in terms of hamii-
hamiltonian
must be compact
in relativistic
Cauchy hypersurfaces
incoming
Consequently
family of Cauchy hypersurfaces
by imposing
Within our framework
relations
turn
in the case of a field theory governed
non-intrinsic
asymptotic
ry must be destroyed
plectic
state.
No boundary problems
and this formulation
is a one-parameter
those appearing
boundaries
Already relativistic
Only very special
The Cauohy hypersurfaces
symplectic
In non-
relations
is exceptional
and generalized
excluding physically
posing restrictive
are time-intervals,
in the case of static field theory
equations.
posing conditions
If the
two-term
mapping.
can be formulated
/see ref.[37]/.
in hamiltonian
by hyperbolic
/canonical/
of the initial
mechanics
and flows.
its proper formulation
elliptic
corresponding
spa-
In this way the state at the end of an interval
symplectic
symplectic
relation.
and the corresponding
easily to other field theories.
requires
-time.
domains
space is a product
a symplectic
the~the
may be a symplectic
particle
of pairs
becomes
of two components
relation
relativistic
subspace
symplectic
by considering
limits
Finite-dimensioof domains
contrae-
ting them to points. We attach ~an
special
subspaces.
nerating
We mentioned
function,
many ~enerabinc mations.
importance
furcb~ons
related function
plastic
structure"
ionian
of particle
mechanics
of
sa e ! gr n
an s, hm
rived
/such
describing
level.
can be described
to eao ~ ~th~r by Le~endre is associated
as components
l
ee r e f .
functions
3]I. Most p h y s i c a l
of the enercy-momentum
as generating
sym-
and the Hamil-
can be shown to be generatinz
ifol
by
transfor-
with a "special
~he Lagrangian
functions
tensor/
of lagrangian
are de-
subspaces
fie]@ dynamics.
Chapter I is devoted to the symplectic discrete
of lagran-
of the action as a ge-
subspace
or a "contral mode".
in our approach
functions
the interpretation
mhe same ]agrangian
Each generating
quantities
to generating
and continuous
The notion
systems
analysis
are considered
of reciprocity
of statics.
on a largely
and potentiality
Both
intuitive
of the theory is
discussed. Chapter Ii is a presentation more rigorous manifolds
definitions
is studied principle
of particle
functions
dynamics
structure.
are defined
together with Lagrangian
states within a finite time interval
can be derived from composition
of histories
is stated
of the particle
grangian description
properties
in an infinitesimal system.
of dynamics.
ved in Section 9. Section
sub-
in Section 6.
in Section 7. It is shown that the Hamiltonian
Section 8 dynamics
terms
of the geometric
and their generating
The time evolution
of particle
11 contains
of dynamics.
In
form in terms of jets
Section 8 contains
The hamiltonian
variational
also the la-
description
a formulation
is deri-
of dynamics
in
of the Caftan form. The Caftan form is an object used in the
geometric
formulation
Caratheodory,
of the calculus
de Ponder,
Lepage,
of variations
Dedecker
developed
by Weyl,
and others /see ref. [58],
[~, D2], ~ ] , ~]/" Chapter
ili is the main part of these notes.
The construction
of
canonical momenta of a field is given in Section 13. Field dynamics is first discussed contained
of infinitesimal
dynamics
sense but a natural a definition
generalization
level.
structure
of the concept.
Section 19 contains
associated with a family of control modes.
ly the most complex part of this volume.
is followed by a discussion is formulated
Section 19 is technical-
Results of this section are
section establishes
language.
This
content and proofs.
Dy-
of the Caftan form in Section 20. This
a relation between our symplectic
the geometric formulation
Sections
of the intrinsic
in terms
of the
We consider this
first stated without proofs and in purely coordinate
integrals.
in the strict
of the energy-momentum density as the potential
definition one of the most important results.
namics
A rigorous
starts in Section 16. The sym-
structure used here is not a symplectic
dynamics
This discussion
in Section 14 and 15 stays on a heuristic
formulation plectic
in finite domains of space-time.
of the calculus
The time evolution formulation
22 and 23. An infinitesimal
framework and
of variations of dynamics
of multiple
is derived in
version of the Hamilton-Jacobi
theorem is proved in Section 24. The last section of the chapter contains a detailed discussion
of objects associated with the energy-mo-
mentum of the field. Different definitions and stress-tensors results
of energy-momentum tensors
are compared and new definitions
are proposed.The
of this section are used in a new formulation
of General Re-
lativity given in Chapter IV. Chapter IV contains strate various
features
examples
of field theories
of the new approach.
selected to illu-
The simplest
example
of
a tensor field /the covariant tensor field/ is given in Section 26. The appearance
of constraints
in the h a m i l t o n i a n d e s c r i p t i o n is illu-
strated by the example of Proca field in Section 27. An example of a gauge field /the electromagnetic A new formulation
field/ is discussed
in Section 28.
of the theory of gravity is given in Section 29.
The new formulation
consists
in using the affine connection
~
in spa-
ce-time
as the field configuration.
on the connection vature
~
and its first derivatives
tensor R. The metric
mentum canonically objects
tensor g appears
conjugate
conjugate
standard Einstein
to P
together
by the cur-
as a component
of the mo-
between these two equations.
components.
instead
depending
stein theory of gravity is a very special theory based on an affine
connection
Ricci tensor
mework one of the versions Other possibilities
are equa-
only on the symmetric
of the full Riemann tensor.
r . Using the Lagrangian
of Einstein's of formulating
unified
part
Thus the Ein-
case of the geometric
one can easily reproduce
The
To obtain the
theory of gravity most of these components
of the Ricci tensor
~4]/.
represented
with Einstein's
has 80 independent
ted to zero by using a Lagrangian
on the complete
of the theory depends
to P , and the relation
is a part of dynamics
momentum
The Lagrangian
field
depending
within this fra-
field theories
/see
unified field theories
are
being investigated. The last Section analysis
contains
of hydrodynamics
logy with electrodynamics. ciple can be formulated directly
of a variational
Appendices frequently
In both theories
reveals
equations
a formal
ana-
a simple variational
or Maxwell's
prin-
which do not appear
equations.
and the subsequent
is an example
An
illustrating
The dis-
formulation
the fruitfulness
to field theory.
contain
a short review of several
used throughout
references [29]
our framework
for hydrodynamics
principle
of the new approach
within
of hydrodynamics.
only in terms of potentials
in either Euler's
covery of potentials
the formulation
the notes /further
geometric
details
concepts
may be found in
I. An intuitive
derivation
of symplectic
concepts
in mechanics
and
field theory.
I. Potentiality
and reciprocity.
In the present physical
systems.
Static
ple well understood to introduce
chapter we consider
conceptual
symplectic
sical characteristics
concepts
we use more complicated
cepts derived theories.
examples
described vilinear
to gradually
belonging
mechanism
sition of the point then an infinitesimal (q~
to
~i
+
~ q~
I. I
A
The Einstein notes.
requires
summation
The coeficients
called the force.
=
con-
dynamic
is much cle-
concepts
philosophy
suspended
space Q. The position
If an external
con-
derived
of trea,
[34 .
point
(q~ , i=1,2,3,
sections
The geometric
theories
exam-
symplectic
of these concepts
with the Minkowskian
a single material
by coordinates system.
and the exis-
for use in primarily
to dynamic
in space-time
physical
develope
is
of such phy-
In subsequent
systems.
meaning
Applying
agrees
as statics
three-dimensional
as reciprocity
continuous
the intuitive
from static theories
We consider
expression
of freedom.
in this way are intended
arer in static theories.
ting dynamics
of degrees
for describing
However
of having a sim-
The aim of this chapter
as a natural systems
of static
In this section we begin with a very simple
ple with a finite number
suitable
have the advantage
structure.
of static
tence of potentials.
cepts
theories
a series of examples
the mechanism
elastically
in the
of the point will be in general to a cur-
is used to control the podisplacement to perform
from a position a virtual work
fi ~ qi.
convention
is used here and throughout
fi in the above expression
these
form a covector
f
The force f is actually the force that the control-
ling mechanism
has to exert to maintain
ment shows that for each configuration cessary to maintain ce are functions
fj
1.2
=
?
If the form
~
then the result
is evaluated <~q,~
:
as defining
~j(qi) dqJ
on a virtual
process
{~ along a path
~
A(D
It is an experimentally
.
displacement
~q =
( ~ qi)
1.1.
from a configuration
. The total work performed
I
=
?
well verified
the work A is the same for all paths q. Consequently configuration
I. 5
if a reference
~ to
in this
}
=
? J (qi) dqJ
fact that for elastic
suspensions o) joining the configurations q and
configuration
~
e.g.
an equilibrium
is chosen then the formula
U (q)
=
I ?
a function U on Q. We call
of the system.
The function
~
bhis function the internal
U and the form
mula
.6
O-form
is the integral
.4
defines
a differential
is exactly the virtual work
We consider now a finite displacement a configuration
fj of this for-
~j(qi)
can be interpreted
~.3
~he components
(qi)
=
q. Experi-
q there is a unique force f ne-
this configuration.
of the coordinates
~he above formula
the configuration
=
dU
?
energy
are related by the for-
equivalent to U
I. 7
~j
=
~qJ
If the internal energy is known then form
~
characterizing the sus-
pension system is found from d.6. The internal energy contains complete information of the behaviour of the system. The internal energy is a particular example of an object called the potential.
~he property
of the system leading to the existence of the internal energy is thus called potentiality.
In the language of differential
perty is referred to as exactness of the form An exact form is closed
q.8
d~
~
geometry this pro-
.
:
=
0
or, equivalently,
1.9
? qJ
_
}-%.
~ q~
=
o.
For a potential system the work performed in moving the system around a closed path is zero. Formula 1.9 expresses this property for infinitesimal parallelograms.
We give an alternative physical interpreta-
tion of this formula. Let the system be displaced in such a way as to increment the coordinate qJ by an infinitesimal value ging remaining coordinates.
E
then
without chan-
The increment of the component f
force caused by this displacement is is incremented by
~
~ g ~ql
~6-
1
of the
. If the coordinate qi
is the corresponding increment
of fj. Formula q.9 implies that the two increments are equal. This important physical property is called reciprocity.
A general state-
ment of reciprocity is that the response of one degree of freedom to variations in the control parameter of a second degree of freedom is
10
the Same as the response
of the second degree
in the control parameter
of the first.
If Q is simply connected plies potentiality.
Hence both properties
With no external an equilibrium
forces
position.
dinates
point.
f
J
im-
are equivalent.
~qi) be Cartesian
point will assume
coordinates
If the suspension
the case for sufficiently
then the components
lemma reciprocity
exerted the suspended
Let
origin at the equilibrium is slways
then by Poincar@
of freedom to variations
with the
is linear,
small deflections
which
from equilibrium,
of the force are linear functions
of the eoor-
:
I .10
fj
The linear system
=
is reciprocal
kjiq
i
if and only if the matrix
kij
is sym-
metric
1.41
kij
• he internal
=
energy of a reciprocal
4.12
U(q)
~his function
is normalized
=
We may be interested
behaviour
4 kijqiqj
at the equilibrium
systems whose
configuration
was controlled.
a system will assume
any combination
4.2 is too special to be useful
of the system in such situation.
tric description
configuration.
to it. In general we may want to find
of the system to controlling
Equation
system is
in finding the configuration
when a known force is applied
and force.
•
linear
to vanish
Up to now we considered
the response
kji
of the elastic properties
cription which does not distinguish
of position
for studying the
We introduce
a more symme-
of the suspension,
any particular
method
a des-
of control.
11 The components
of the force together with the coordinates J of the configuration at which the force acts can be used as coor-
qi
dinates
f
(qi fj) of a space F called the phase space of the system.
Points of P represent states of the system. In terms of differential geometry the space F is the cotangent bundle T~Q of the configuration space Q. The cotangent bundle ~ Q
1. q 3
~
If the form (qi fj) to
@
carries a canonical q-form
=
fidq i •
is evaluated on a virtual change of state
(qi+~qi
f~+j Sfj)
then the result
<~f,@>
6f from
is exactly the
virtual work q.1. We denote by D the submanifold of F composed of states with coordinates
(ql fj) satisfying q.2. The construction of the
phase space :is the same for all suspension systems of a point. Each suspension system is characterized by a submanifold D m F. Points of D are the states compatible with the elastic properties
of the suspen-
sion. In the case of a linear suspension the space F is a vector space and D is a linear subspace of P. Let the system be moved from a state ~ to a state f along a path . The whole process is compatible with the elastic properties of the suspension.
This means that the path
~
is contained in D. Pre-
viously we considered processes represented by paths in the configuration space Q,, Since each path projection
~
in D is completely determined by its
]I onto q the two representations of processes are equi-
valent. The work performed in the process previously calculated by integrating the form
~
of the canonical l-form
along
~
@
along
can now be expressed as the integral ~
:
If the system is potential then this integral is the same for all paths
12
belonging
to D joining states
sen
an e q u i l i b r i u m
e.g.
f and f. If a r e f e r e n c e
state
state f is c h o -
then the formula f
_~(f)
.15
defines
a function
U on D. The d i f f e r e n t i a l
of U is equal to the r e s -
m
triction
~/D
of the form
~.16
@
to the s u b m a n i f o l d
d~
The above f o r m u l a /D is thus
means
the canonical
that
a criterion
To find the phase
=
@ID
D
:
.
@/D is exact.
The e x a c t n e s s
of the form
for p o t e n t i a l i t y .
space
interpretation
of r e c i p r o c i t y
we d e f i n e
2-form
1.17
~
in F. The c o o r d i n a t e
expression
.18
gO
=
d{9
of this
=
form follows
df
A
from I.~3
:
dq m
i
The space F t o g e t h e r led in d i f f e r e n t i a l ciprocity
means
with the 2-form geometry
This
I .2O
a symplectic
that the form
1.19
d
is e q u i v a l e n t
gO
is an example
=
o.
cO/D
=
0
to
is cal-
m a n i f o l d f[I],[8], [57]). Re-
@/D is closed
(e/D)
of what
:
13
since
d
(Oi
i
=
(de)J
--
A submanifold D of the symplectic msnifold called an isotropic dimension
submanifold.
(F,~)
The dimension of D is equal to the
of q and thus a half of the dimension
of an isotropic
submanifold
then the submanifold We conclude
of P. If the dimension
is equal to a half of the dimension
is called a lagrangian
that an elastic
satisfying ~.20 is
suspension
submanifold
(cf.[8],[9~).
system is reciprocal
D of states compatible with this suspension
of F
is a lagrangian
if the set submani-
fold of the phase space. The function U defined on D is related ternal
in a simple way to the in-
energy U defined on q. To each configuration
q there corresponds
a unique state f belonging to D. The value of U at q is equal to the value of U at f provided that the reference to the reference
state ~ for U. The function U is called the proper
function of the lagrangian
submanifold D. The function U is called the
generating function of D (of.I%5]).
The term "generating function"
justified by the fact that U contains The differential
state f for U corresponds
complete
is
information about D.
dU can be considered as a mapping from Q to F = T~Q.
The submanifold D is the image of this mapping
.22
D
=
{(qif
As an introduction
)d
f J -
"
to the analysis
of the behaviour
of a suspen-
ded point when the external force instead of the configuration
is con-
trolled we consider a simple special example with one degree of freedom. The system is a spring balance. vitational
A scale is suspended in the gra-
field on a spring with spring constant k. The force applied
to the system is controlled by placing weights
on the scale.
The weights
14
are stored at the level of the e q u i l i b r i u m
p o s i t i o n with no weight.
///////
//////// 0 0
0
The force and the p o s i t i o n elastic
characteristic
are in the r e l a t i o n
of the spring.
f to f+ g f by t r a n s f e r i n g then the w o r k p e r f o r m e d
a weight
B
If the weight
is changed
=
If the force
-
q
U
of the spring
--
= - Z
£f
1
I fdf
-
=
and on c h a n g i n g
~
=
- q£
=
--2-~ I f~
"
1
2--k
energy
f~
the g r a v i t a t i o n a l
by
1.26
.
0
on the internal
1~ kq~
to the scale
£
0
I .25
from
from 0 to f then the total w o r k is
H = - q d If
This work is spent
the
and equal to
f 1.2%
is i n c r e a s e d
~f from the storage
is n e g a t i v e
1.25
f = kq r e f l e c t i n g
=
_ I f~ k
•
energy
of the w e i g h t s
Returning
to the general
example we define
1.2?
0 ~
=
-
in F a l-form
q±df i
.
The integral
£
1.28
B(%)
along a path
~
is
~
ilustrated
l a n c e . Since the difference
1.29
~
~
~
c o n t a i n e d i n D i s the work p e r f o r m e d i n t h e p r o c e s s
of moving the system a l o n g o f such c o n t r o l
:
_
~H
by c o n t r o l l i n g by t h e s p e c i a l
between
=
~
the f o r c e ,
A mechanism
example o f a s p r i n g b a -
and
~H
f i d q i + qidg i
:
is exact
d~
:
,
where
I. 30
~
the work B(~)
depends
:
fi q
i
,
only on the end points
the s~me is t~uo for A ( y )
. She
of the path because
~ntogral f
defines
a function H on D. Obviously
1.32
dR
If the reference internal
=
@~/D
.
state ~ is the same as the one used to define
energy then
the
16
f
-~.33
_HCf)
=
I(e-
d,F)
=
Usually f is an equilibrium state,
1.3~
_~
=
2
-
te it is a form of energy.
in this case
"F/D
Since H is the work performed
__u(f) -
which we interprete
LF(~' )
~(})) = 0 and
.
in the transition from the reference
sta-
Formula 1.34 shows that this energy is compo-
sed of the internal energy ~ of the suspension - ~
up(f) +
system and the term
as the energy stored in the force controlling
mechanism following the example of a spring balance.
Assuming that to
each of the controlling forces(fj)
a unique configu-
there corresponds
ration
~.35
qi
~oi(fj)
~
we define a function H by
1.36
~(fj)
:
~ (~oi(fp,fk)
.
Since
'
qidf /D --
i
-
~i(%)d~
i
it follows from q.32 that
1.3s
~6 i C f j )
By analogy with thermostatics
--
-
~H
~i
we call H the enthalpy of the suspension
system. The function H is a generating function of D since D is com-
17
pletely determined
J.39
by H :
D
The enthalpy energy.
-- {
is an example
The existence
the other.
mode.
I
i
~H
of a potential
of one potential
Hence potentiality
of the control responding
i '%)
from the internal
is implied by the existence
is a property
The formula
to the two control
different
of the system independent
1.35 relating the two potentials
modes
of
is called a Legendre
cot-
transformation
< [53]). Let the controlling
force be changed by incrementing
nent fj by an infinitesimal coordinate
i 6' ~~f
qi is
amount
. The resulting
. If the ~ncrement
J the change
g
~
the compo-
change
of the
is applied to f.m then
~J
of qJ is equal
to
6'2fi"
Since
fj) dfiA dfj 1.4-0 =
it follows
(dq 'j/'~ d f j ) / D
that the response We conclude
independent
of the control mode.
is the isotropy
forces
is
of the system
in one control mode,it
manifestation
of the suspension
of this property
~
system we used geometric
One level is the level of symple-
space F. The sympleotic
of being lagrangian.
reciprocity
geometry
. To this level belong the submanifold The physical
characteristics
tem which belong to this level are the abstract abstract
0
is a property
Established
The geometric
of the phase
sented by 2-form property
that reciprocity
which belong to two levels.
ctic geometry
=
of D.
In above discussion concepts
- dO/D
of the system to controlling
reciprocal.
will hold in any other.
=
independent
is repreD and its of the sys-
potentiality
of the control mode.
and the
The second
le-
18
vel is the so called level of special with different sitions
specific
of the phase
and response
control
of a special
structure
of the system are described
there is asso-
is ~
(the complete
are
~
and
@H.
The
on this level by potentials. reciprocity
of the control-
relation.
The distinction mical theories. mulation
control parameters
will be given in Section
these q-forms
In each control mode the system exhibits -response
:
exterior differential
examples
connected
To this level belong decompo-
into two sets
symplectic
6 3 . In the two considered properties
geometries
With each such decomposition
ciated a q-form on F whose definition
modes.
coordinates
parameters.
symplectic
between
tensor)
descriptions
can be treated
approach
can be used to describe
q mole of an ideal gas ~cf. ~3]). ropy S, the pressure
as different
of the same symplectic
in dyna-
and hamiltonian
(in field theory also the formulation
of the energy-momentum
A similar
is very useful
We shall see that the lagrangian
of dynamics
symplectic
these two levels
by means special
object. the thermostatics
Let the volume V, the metrical
p and the absolute
for-
temperature
of ent-
T be used as coor-
d i n a t e s ~V,S,p,T ) of a manifold F called the phase space.
Together
with the 2-form
q .41
dO
the phase space
=
dV ^ dp + dT A d S
is a symplectic
is governed by the two equations
pV
=
manifold. of state
RT
q .42 pV ~
where R, ~
and k are constants
S k exp C v
and
The behaviour :
of the gas
19
R 1. ~3
It
cV
is
fold its by
easy D of
check
(F,co)
thermostatic one
of the
enero~y, ven
to
the
b y the
that
these
. Points
Gibbs
~-1 equations
of D are
properties.
four
=
The
thermostatic function
and
the
define states
submanifold potentials
the
a lagrangian
of t h e
gas
allowed
D is u s u a l l y :
enthalpy.
internal These
submaniby
described
energy,
functions
free
are
gi-
formulae
Cv,s)
k
V (1-~')
S
=
~" -
=
CvT(1
- in T + in k - in R)
Q('p,T)
=
CpTfl
- in T - in R) + C v T I n
~(s,p)
--
~-
I
I
e x p T.. V
P
e~T
- RT
in V
k + RTIn
,
p,
' P
where rol
Cp = R + c V.
modes
formulae
and for
Reciprocity
are
These
functions
virtual
finds
functions
work
correspond
of c o n t r o l in t h e
four
parameters modes
@U
=
-pdV
+ TdS
,
F
=
-pdV
- SdT
,
~G
=
Vdp
- SdT
,
@H
=
Vdp
+ TdS
.
an
expression
in t h e
to f o u r
are
Maxwell's
different
in e a c h
mode.
:
identities
:
contThe
20
aT
=
p
2. Elastic
string
The conceptual framework derived present
S
section to a more complicated
result is a set of concepts
in Section ~ is applied in the example of a static system.
The
formally identical with those used in Chap-
ter II to describe particle mechanics. We consider an elastic
string aligned with a £oordinate
axis.
Each point of the string is labelled with the value of the coordinate t of its equilibrium position.
The configuration
ched.by forces
applied in the direction
function
R ~.,
q :
2.4
The value q(t)
t
of the string stret-
of t will be described by a
~ R ~ i.e.
~
q(t).
of the function denotes the displacement
corresponding to t from its position in the unstretched configuration configuration.
of the string to the actual position
equilibrium
in the described
External forces applied to the string are described by
a function f
2.2
of the point
t
~
f(t)
21
The force ac~ing on an infinitesimal segment of the string corresponding to the interval
[t,t+~t]
is equal to f(t)'at.
In addition to
external forces we introduce the tension p(t) defined as the internal force which the portion of the string corresponding to the values of the parameter smaller than t applies to the remaining portion of the string. The tension p(t) can be measured by cutting the string at t and replacing the internal forces by measurable external forces which are necessary to maintain the configuration of the string. Let us consider a segment of the string corresponding to the interval p(tl)
~j,t2].
Forces acting on this segment are :
at the ends and f(t) in the interior.
2.3
[%,t2_7
~
t
,
represents a virtual displacement
A
=
If
~q(t)
of the segment then the virtual
work A corresponding to this displacement
2.4
-Pft2) and
is
-p(t2) ~q(t2) + p(t¢ ~q(t~)
+
f(t) ~q(t)dt . t1
If the finite interval val
~,t+m~
2.5
is replaced by the infinitesimal
inter-
then the virtual work is
A
--
[-h¢pct q¢t>l + let>
The virtual displacement by the value
~1,t~
~qCt)
of the infinitesimal segment is represented
and its derivative
~(t)
d =-~
~qCt).
Thus
2.6
Relations between the forces applied to a segment of the string and the configuration of the segment depend on the elastic properties of
22 the string. segment.
Specially
simple are these relations
for an infinitesimal
The equation
2.7
- ~(t)
is the infinitesimal
+ f(t)
version
:
o
of the equation t2 f
D(t 2) + p C t ¢
2.8
+ I fCtl~t
0
tI expressing
the balance
of forces
2.9
p t(t)
is the Hook's
law expressing
to the stretching
=
applied to the segment.
- k (t) ~ (t)
the fact tha@ the tension
of the string.
Equations
segment
Etl,t2~
o
we obtain thus the system of equations
fort
2.41
These
equations
figuration
p(tl)
=
- k(t~)~(t¢
p(t2)
--
_ k(t~)~(t2)
since they give the response tion.
~
[tl,t2]
,
,
give the forces necessary
q of the segment.
is proportional
2.7 and 2.9 lead to
o
dt
For a finite
The equation
for maintaining
a given con-
They are thus the analog of equations of the system to the control
In order to verify the potentiality
of this response
1.2
of configurawe calcula-
23 te the virtual work 2.4 substituting the values 2.11 parameters
of the response
:
A
=
k(t2)4(t2)
gq(t2)
- k(tl)¢(tl)
g q(t I)
-
t2 2.12 v
t1
=
.
tI We consider the space q([t2,tl] ) of all configurations
of the Segment.
This space is the space of smooth functions q defined on the interval [tl,t ~
o
The virtual work 2.12 is the value of the differential
functional U : q([t2,tl] )
of the
, R 1 given by the formula t2
2.fl5
U(q)
=
2
It 1
calculated on the virtual displacement
2 dt ~q. This displacement
treated as a vector tangent to the space qC[tf,tl])
2.14
<~q,dU>
=
k (t) {(t)
can be
at q. We may write
~ {(tdt
t1 It
follows
gurations
that
t h e work i s
independent
o f t h e p a t h between two c o n f i -
and t h e r e s p o n s e o f t h e s y s t e m i s p o t e n t i a l .
of the potential
i s t h e work n e c e s s a r y t o b r i n g
from its unstretched
The v a l u e U(q)
the string
equilibrium configuration
q = 0
segment
to the given
configuration q. A similar analysis applies to an infinitesimal segment. Due to equations 2.? and 2.9 the
2.15
A
--
[k(t){(t)
.
24
i We consider the space qt of all configurations of the infinitesimal segment. As coordinates
i in Qt we use q(t) and ~(t). The formula 2.75
gives the amount of work performed in the virtual displacement from the
configuration
~(t),~(t))
to the configuration
(q(t)+ ~ q ( t ) , ~ ( t ) +
+ $~(t)) . It turns out that this work is the differential of the function
ut(q(t),a(t)).~ t
2.~6
:
,
evaluated on the virtual displacement is the work
per unit lenght
(q(t), 6q(t)) . The value u t
necessary to bring the infinitesimal
string segment from its unstretched equilibrium configuration (q(t) = O, q(t) = 0) to given configuration
=
(q (t) ,q (t)) . We see that
t2 F
u(q)
2.17
=
I ut(q(t)'~(t)) dt t~
We call U the internal energy of the segment and u t the internal energy density. These functions are generating functions of the control-response relation. For a finite segment we prove this using formula
- p(t2) S q ( t 2 ) + p ( t q ) g q ( t n )
+
f(t)~q(t)dt
=
tI t2 r
2.1s
=
$ u(q)
=
$1 ut(q(t) ,~(t))dt t~
Leaving aside all mathematical problems connected with the infinite dimension of the configuration space Q([t2,t~])we can formally write t2 r
~Iut(qCt) ,~Ct))
dt
ti
25
i
2[ ~ u t
2.19
:
-
6q(t)
-
+
]
~q
t. 1 t 2
~%2
Utl
gq(t2 )
~q(tl) + I(~%
d ~ut~ ;q(t)
dr.
tI
The
2.20
expression
~ut
=
Sq
but
d
~q
dt 2{
But
is the so called Lagrange derivative of the energy density u t. Comparing 2.']8 with 2.19 we obtain equations
ut
f(t) 2.21
sq (q(t),a(~))
p(t¢
--
p(t2)
--
9ut 1
ut2 (q(t2), {(t2))
equivalent to 2.1Q which proves that the internal energy is indeed the generating function of the control-response relation. A similar procedure applies to the infinitesimal segment. The virtual work 2.6 is equal to the increment of internal energy. ~hus
This implies equations
~(t) + ~(t)
_l
~u t Z9
2.23 -
p(t)
-6 u t =
~
28
equivalent to 2.7 and 2.9. So far we treated the string as having its configuration programmed by an external mechanism providing forces necessary to maintain a given configuration.
~he work in changing the configuration
formed by the mechanism and accumulated
was per-
in the string in the form of
internal energy. Now we consider a segment of the string whose e n d points only are attached to a position programming mechanism.
~he in-
terior of the segment is left free and assumes an equilibrium
configu-
ration compatible with the position of the end points.
As control pa-
rameters we take therefore the boundary configurations
q(tq)
~hey are coordinates configuration
in the boundary configuration
and q ( t ~ .
space Q (t~'tl)- . ~he
of the interior of the segment is determined by the boun-
dary configurations.
THe response of the system on the control of q(tq]
and q(t2) consists of the forces P(tl)
and -P(t2)
which the position
controlling mechanism has to supply in order to maintain a given configuration. dinates
We take these forces together with configurations
(q(tq),q(t2] ,P(tl],P(t2) ) in the space B (t~'tl)
as coor-
which we call
the boundary phase space. }he virtual work performed by the position controlling mechanism in the virtual displacement
(gq(t2) , ~q(tq])
is equal to
2.24
A
=
- p(t2)
q(t2) + p(tq
Sq(t
We consider the submanifold D(t~'tl ) c P ( ~ ,t~) which consists dary values of solutions
of equations
equal zero. Points of D (t~'~) elastic properties q(t2~
2.1q where the force f(t) is set
are boundary states compatible with the
of the string. The virtual displacement
of the boundary position induces the corresponding
boundary forces
of boun-
(~q(tq), change of
(gp(t~), ~p(t2) ) such that the state of the segment
stays on the submanifold D (t~'~) ring this displacement
~he virtual work 2.24 performed du-
is equal to the evaluation
of the q-form
27
@(t~,t~)
2.25
on the v e c t o r
v = ($q(tl)
=
_ P(t2]dq(t2
) + p(tl) dq(tl )
, ~q(t2) , ~P(tl]
, ~ P ( t 2 ) ) tangent t o D (t~'t~).
Using equations 2.11 with f = 0 we may calculate this work in the following way :
6~(t~'t~)~
_-
k(t 2)~][t2) ~q(t 2) - k(t 1) q(t 1) 6 q ( t I) :
t2
tI 2.26 t2
I k(t){(t) 6 4(t)a~
=
tI t2 =
6
~(t)
4(t)
dt
=
~u(q)
tI Solving the boundary value problem for equations 2.~1 with f = 0 we can find the configuration of the entire segment as a function of (q(tl) ,q(t2)) and thus express the internal energy U as a function of control parameters. She result is the function W(q(tl),q(t2) )
equal
to the value of U(q) evaluated on the solution q of 2.71 with the boundary condibions q(t~) and q(t2). Similarily we can define the function W on D (t~,t~
2.2?
The
by setting
-~(q(h) ,q(t2) ,P(%) ,P(t2)) equation 2.26 shows that
=
w(~(hl ,q(t2)l
•
28 @(t~ ,tO ID(t~ ,tO
2.28
Potentiality
of the control-response
:
dW
relation implies abstract reci-
procity which can be expressed by means of the 2-form
2.29
oD(t~. ,t~)
=
d 0 (l;~ ,t~)
=
-dp(t 2) adq(t 2) +
The equation 2.28 implies
2.30
_-
i.e. the submanifold D (t~'t~)
)
is isotropic.
o
o
The dimension of D (t~ 't4)
is 2 and thus half of the dimension of p(t~ ,t~) . We conclude that D (t~ ,t~) is lagrangiam submanifold
of the symplectic manifold (P (t~ ,t~
co(t~'t~)) . Equation 2.28 means that W is the proper function of D (t~ ,t~! It follows from 2.26 that D (t~'t~)
2.34
Thus,
-P(t2)dq(~2)
is described by the equation
+ n(tl) dqCtl)
--
dW(q(tl),q(t2) )
similarily as in ~.22 we have
2.32
~W - P(t2)
=
~q{t2),
3W p(t¢
which means that W is generating function for D (t~'t~) .
Example Let k(t)
~
k be constant.
Then
=
aq(tl)
I ,
29
p(t~)
Substituting
p(t2)
=
description
-
q (t~)
t2
-
t~
applies to the infinitesimal
The boundary configuration
the infinitesimal
space Q ( ~ '~)
segment I t ,
is now replaced by
space qti with coordinates
configuration
The role of the coordinate
q(t+at)
The role of the tension p ( t + ~ t ) response
q (t2) -
this to equation 2.3~ we obtain
A similar t+~t].
=
(q(t),~(t~ .
is now played by q(t) + ~Ct)-At.
is now played by p(t) + ~(t)'~t.
The
of the system to the control of q(t) and ~(t) consists of the
tension p(t) and its derivative
p(t). We take (q(t)
,~(t)
,p(t)~(tO
as a coordinate
system in the space pi which we call the infinitesit mal phase space. We consider the submanifold D ti c Pti of states which are compatible with the elastic properties from 2.9 and 2.9 that D ti is a 2-dimensional equations
of the string.
It follows
submanifold described by
:
~(t)
=
o
p(t)
=
-
2.~3 kCt)GCt)
According to 2.24 the virtual work per unit length performed by the configuration v = ( ~ q(t~
2.3#
controlling mechanism during the virtual displacement , g~(t)
A
=
, ~p(t)
-~
d
, ~(t))
p(t) gqCt)
tangent to D ti is
=
- ~Ct) gqCt)
This work is equal to the evaluation of the form
- p(t) gG(t)
30
@~
=
- ~(t)dqCt)
A
:
- pCt)d~iCt)
on the vector v :
2.35
U s i n g equations
2.33 we may calculate
2.36
=
A
The f u n c t i o n
2.3?
(q(t)
We can also d e f i n e
~(t)
,~(t))
i of D t
-~(t)=
,-p(t)
=
= ut(q(t) ,~(t))
e q u a t i o n 2.35 shows that
w h i c h means the finite
2.~o
.
the f u n c t i o n ~t on D ti by s e t t i n g
i
the
~ut(q(t),~(t))
function
ut(q(t) ,~(t),p(t),~(t))
2.3~
The
,~(t)
this work in the f o l l o w i n g way
=
~(t)~(t)~i(t)
u t is thus a g e n e r a t i n g
Di
@ ti>
=
d__ut
that u t is the p r o p e r segment
sympleetic
co ti
the s u b m a n i f o l d
manifold
=
function
d @ ti
(P~,co~)
=
of D t" i As in the case of
D ti is a l a g r a n g i a n
submanifold
where
-d~(t)~
dq(t)
- ~p(t) ^ d~(t)
of
31
3. Elastostatics
We consider
in this section an elastic
ce of fixed external of symplectic
forces.
medium under the influen-
The description
of this medium in terms
geometry will serve as a model for symplectic
formula-
tion of field theory. Coordinates
(x ~)
will be used in the physical
three-dimensional
space M. The space M is endowed with a riemannian
metric
whose
denoted by
components
be calculated coordinates
are g ~
. Covariant
with respect
(x~)
brium position
derivatives
to the connection
will also label points
with no external
dium is the displacement whose components
forces.
=
of the medium
described
~ ( x ~)
medium
surface
in the equiliof the me-
by a vector field
. Internal
element then n~ ~ . ~ $
on the negative
tive side. Similarily
forces
can be measured by equivalent
We consider equilibrium
of a string in Section 2 stresses
a domain V ~ M and a piece
deformation
~V
= ~I 8V
of a fixed external
space Q 3V. Elements
tangent
to M and defined
field.
device
The medium responds
of the
of Q ~ V are vec-
on the boundary
is given by
the medium
is placed
to deformations
of the
on the boundary by the defor-
in order to maintain
of these forces
by apply-
of the piece is under the influence
boundary by forces which must be applied controlling
forces
~V. Deformations
force field f. For example
in the gravitational
face density
internal
of the medium which in
to its boundary
V of the domain V. The interior
mation
applies to the posi-
forces.
form the configuration
tor fields
represents
fills the domain V. This piece will be deformed
ing a programmed boundary
element
by cutting the medium and replacing
external
are des-
is the force which the
side of the surface as a tension
will . The
cribed by the stress tensor density p ~ ( x ~) . If n~ . ~ s an oriented
~
~
The configuration
from equilibrium
are functions
~.
tensor
a deformation.
The sur-
32
3.1
_ p~ ~v
If a virtual change
~ V
- p#~
~
(x)n~
is applied to the boundary configuration
SV then the virtual work performed is
3.2
~V
V
We describe with more rigour an infinitesimal piece
~V
of the medium
at a point x ~M. ~he configuration of the piece is described by the values
~x)
of the field
Infinitesimal
~
at x and its derivatives at x :
configurations form the infinitesimal
i If a virtual change ce denoted by Qx" to the infinitesimal configuration
(g~Cx),
f~{x),
configuration spa-
$ ~(x))
~(x))
is applied
then according to
3.2 the virtual work performed is
3 ° 4-
As response parameters we choose the coefficients
3.5
and p ~ f x ) .
=
Together with the infinitesimal configuration they form
a coordinate system
(~(x)
~(x),p~(x)
~(x)) ~
in the space pi i
called the infinitesimal phase space. Not all states of pi x are allowed by the elastic properties of the medium. The allowed states sa-
33
tisfy the equation
~(x)
3.6
=
f#(~)
expressing the balance of forces acting on the infinitesimal element of the medium. They also satisfy the strain-stress relation
3.7
p~Cx)
=
-k~
~ ~
C~)
analogous to the Hook's law 2.9. States satisfying the equations 3.6 and 3.7 w~th given external forces f~Cx) form a submanifold D i C pi -
X
X"
A virtual change of configuration induces a corresponding change of the response parameters
such that the state remains in D i
X °
~he vir-
tua! work A corresponding to this change is the value of the ~-form
evaluated on the vector tangent to D i describing this change. The X
potentiality of the control-response relation means that the form @ ~ I Dix is exact i.e. that
@ xi IDix
3-9
=
dU --
This happens when the matrix k is symmetric
:
3.1o
The function U is the internal energy of the infinitesimal volume of the medium. The corresponding generating function /potential/ U on
qxi
is defined by
84
3.1~ Substituting
3.6 and 3.7 into
3.11 we obtain
1
~ y~.
The f u n c t i o n
2 is the i n t e r n a l
energy density.
cribed by the f u n c t i o n u x
~
3.15
dO x
=
is a s y m p l e c t i c
symmetric
parts
3.17
-
~V.
~B ,ux
bu x I • -p/_ ~?~
~ + dp#~A
- Z~V
and D i is a l a g r a n g i a n x
submanifold
d~Ad~
components
of
~
can be d e c o m p o s e d
into the symmetric
and anti-
:
~
The a n t i s y m m e t r i c element
=
manifold
of the d e r i v a t i v e
des-
with the form
d@ x
The c o v a r i a n t
D i is c o m p l e t e l y x
:
,p~,~) I - ~
~ ( (~,~
i The space Px t o g e t h e r
The s u b m a n i f o l d
=
~{~)
part d e s c r i b e s
The symmetric
+ ~9[r_~1
the r o t a t i o n
part is r e l a t e d
of the
infinitesimal
to the Lie d e r i v a t i v e
of 9:
$5
5.18
2
~C~)
=
~
g~
W and describe
the change of the shape and the volume
Usually the internal ~e~.
energy depends
of the element.
only on the symmetric
part of
This means that the tensor
3.19
k~
gP~ g ~ k ~ ~
satisfies
3.20
k~
It follows
F
=
that the stress
5.24
p ~
k (#~)~ ~
=
tensor density
=
gV~p
~
=
k (~)(# ~)
is symmetric
p~)
4. Electrostatics
The geometric
concepts
developed
in earlier sections
in this section to a true field - the electrostatic In the three-dimensional we consider ~(x).
a dieletric
The configuration
by the electrostatic
physical
are applied
field.
space M with coordinates
(x~)
medium charged with a fixed charge density of the electrostatic
potential
~(x).
field is represented
The electrostatic
field E~ is
defined by
#.I
In addition
E
to the
Cx)
"external
=
-
charges"
•
represented
by
~ there are
36
"internal charges" represented by the electrostatic induction field p~(x) which is a vector density.
If a piece of the dielectric occupy-
ing a domain V is singled out and the rest of the medium is disregarded then the influence of the surroundings
on the singled out piece
has to be replaced by a surface charge -n~p~(x) - ~ s element n ~ . ~ s
of the boundary
ry value
=
~BV
9V
on each surface
of V. We will control the bounda-
~ I ~ V of the potential.
The space of boundary va-
lues will be the boundary configuration space Q 8v. The response of the field is the surface charge density _p~V = n~p~ I 8 V which the potential programming mechanism has to supply. If a virtual change ~ ~V of the boundary configuration is made then the virtual work is
~g
V
Passing to an infinitesimal piece
~V
of the medium at a point x ~ M
we obtain for the virtual work the expression
4.3
6 The infinitesimal configuration is represented by ~ ( x ] =
~(x)
.
and
= -E~(x) and the response is described by ~ ( x )
and p~(x)
The infinitesimal
~(x) =
=
8~p~(x)
configuration space is denoted by @i and
•
"
"X
the infJnfitesimal phase space by P~. States compatible with the dielectric pronerties
of the medium form a subspace D ix of Pxi described
by the field equations•
These equations are
37 stating that the total charge
of the infinitesimal
element
is zero
These
equations
and the relation
reflecting
the dielectric
imply the Poisson
properties
of the medium.
equation
4.6
A virtual ohange virtual
change
of infinitesimal
of response
induces
a correspoding
so that the state of the field remains
work A corresponding
evaluated
configuration
to this change
on the vector tangent
is the value
to D i representing x
on D i. The X
of the l-form
the virtual
chan-
ge. The space pi with the 2-form X
=
is a sympleetic metric
manifold.
d~
- d p ~ A dE~]- z~V
If the dielectric
constant
£*~(X) is a sym-
tensor density
4.9
then the response
4.10
-Idea
£~x~
=
g~(x)
is reciprocal
~ x i ID xi
where U is the internal
=
,
and
dU --
'
energy of the infinitesimal
element
of the
38
electrostatic
field.
corresponding
~.11
The function U is the proper function
generating
- [~d~
function
U is defined
- p~dE~]'~V
=
of D i. The X
i
on Qx by
dU(~E~)
~.& and %.5 into g . 1 1 we obtain
Substituting
=
1_
.
2 The function
! 2 is the internal
energy density.
Complete
information
on D i is contained X
in the function Ux :
ux
The space D i is a lagrangian X
submanifold
p~
~Ux
of the symplectic
manifold
i
~X) " Let X be a vector field on M whose the electrostatic state.
field along X produces
If the electrostatic
can calculate internal
virtual
as a continuous
~V
represented
and not to the charge density
~
the boundary potential the charged dielectric. displacement
programming
plied by $ / of the internal
~V
the
with respect
to
energy due to the
is equal to the Lie derivative
energy treated
medium
~ . This means that
is displaced
The change of the internal
of the domain
medium we
but not to the dielectric
device
of its
~ X ~ to the electrostatic
field and the domain by
Dragging
work by differentiating
We apply the displacement it occupies
are X~(x).
a virtual displacement
field is treated
the corresponding
energy.
components
as a function
/multi-
on M. Since
39
the internal
",ITJ
energy u x is a ~calar density
--
~'X %
~. ~,.(~"ux).
:
• ,', v
this change
ffs equal to
,,,v
X
~.~5
+ #~)]~v The change
of the electrostatic
minus the Lie derivative the internal
field due
to displacement
of the field times g . Hence
energy due to the change
is equal to
the change
of electrostatic
.
of
field is equal
to
[ ~u x
~2 U
The total chan~e static
~u x
of the internal
energy of the piece
field due to the displacement
A. I U +
of the electro-
e.Xm is equal to
-
/',2U
]
t. ~,~,.- Q . u ) -
4.17 - ~x~C~) where
~m is the Kronecker's
system this chan~e performed
by infinitesimal
~.18
of the internal
in the displacement.
of the proceeding
and p ~
symbol.
section.
A
Due to the potentiality
the result with formula
In this formula we replace g X m and
g,X~
to avoid confusion
=
-
g-
of the
energy is equal to the virtual
We compare
displacements
to T~ and T ~
] ~v
[2 X "~
+
"
and ~
and also change
of notation
T,~X#~
~
] ~ z~V
:
•
~
work 3.4
40
The result
of this comparison
We call ~
the force and ~
field.
This quantities
programming
device
the stress tensor of the electrostatic
measure
applies
the real forces which the potential
to the piece
We refer to this construction tensors
is
for a general field.
in Section
of the electrostatic 25 where we define
field. stress
II. Nonrelativistic
particle
dynamics
5. Preliminaries
The fundamental
geometric
space in our considerations
nifold Q of all possible
configurations
times.
M is identified
The time manifold
corresponds re bundle denoted
to the choice
figuration
iF
/cf. Appendix
space
of Q is assumed not defined. vity.
of a standard
over the time manifold
by
of a particle
system at all
with ~q. This clock.
identification
The manifold
M. The bundle projection
A/. Each fibre Qt =
~-~(t)
of the system at time t. No standard
We call Q the configuration
bundle.
Q is a fib-
Q--
~ M is
is the con-
trivialization
which means that the rest of the particle
This point of v~ew is similar
is the ma-
system
to that of Galilean Fibres
is
relati-
of Q are assumed to
be simply connected. We now introduce
the notion
of the phase bundle
The phase bundle P is also a fibre bundle bundle projection called the phase figuration
P
, M is denoted by
over the time manifold. ~
. Each fibre Pt =
space at time t, is the cotangent
bundle
The
~-~(t),
of the con-
space
5.~
Pt
=
T~Qt
The cotangent
bundle projection
dle structure
of P over M can be characterized
Pt
~ Qt is denoted
bundle V~Q of the bundle VQ c TQ of vertical
tion
of the system.
The family
of projections
ffC : P
~ Q. The diagram P
~g
M
ff~t defines
~q
by
ffgt" The bun-
as that of the adjoint
vectors
tangent
in an obvious
to Q.
way a libra-
42
is commutative. Histories sections
of the particle
of the phase bundle P over time.
tible with the physical shall
system aredescribed
laws governing
Not all sections
we {ive formulations
are compa-
the motion of the system.
refer to these laws as the dynamics
sections
by differentiable
of dynamics
of the system.
We
In subsequent
in terms of symplectic
~e o-
merry.
6.
Special
symplectic
In the general
structures.
case the cotangent
is equipped with a canonical ~i, ~ >
of
~
__~ : T~-----+~
the tangent mapping
=
from T~ to ~
@ . ~he evaluation is g~ven by
bundle projection
/cf.
Appendix
and
$~u is
B/. ~he definition
6.fl
to
O(p) Q(p)
~-form
of a manifold
< _z. u,p >
is the cotangent
6.2 where
buudle ~ = T ~
differential
<~,e>
is equivalent
functions
on a vector u tangent to ~ at a point p ~
6.~
where
Generatin~
is the value of
the covector p e T ~
-- _~'(p) 0
at p and
to @*~ /cf. Appendix
The differential
denotes
the pull-back
of
C/.
form
6.3
is called the canonical
gO
2-form
=
d@
on P. It is a standard
manifold P together with the 2-form
(~, ~)/c~. b ] , b l / -
~p
~
define
result that the
a symplectic
manifold
43
In the case considered in Section 5, we have a family of canonical l-forms Pt = ~ Q t
0 t and canonical 2-forms
~ t defined on each phase space
separately.
If a coordinate system
(qJ), j=Q,...,n,
every pQint of ~ the differentials
dq J
is chosen in ~ then at
form a linear basis for co-
vectors at this point. The components p~ of a covector p with respect u
to this basis together with the coordinates q~ of the point define coordinates
~(p) e
(qJ~pj) in the space T*~. The local expression for
in this coordinate system is
6.~r
~
=
pjdq J
.
9be Einstein summation convention will always be used. The local expression for 60 is consequently
6.5
~
=
dpj A dq J
In the case of the configuration bundle Q we shall use coordinate systems (t,q j) which are compatible with the f~bration
~ ~ i.e.
(t,q j) = t. The construction introduced above leads to the coordinate system (t,qJ,pj) in the phase bundle P, compatible with both fibrations
~
and
~
:
%(t,J,pj)
=
t
6.6
(t,j,pj)
= (t,J)
The fundamental geometric concepts used to describe dynamics will be that of a lagran~ian submanifold of a symplectic manifold /see[55],
[57]/. Definition
: A lagrangian submanifold of a symplectie manifold
44
C~,~)
is
a
submanifold
The condition bivectors
tangent
called isotropic.
N c p such that
CoiN
@ IN = 0 means that to N. A submanifold A simple algebraic
sion of an isotropic
submanifold
@
of generating
argument
Lagrangian,
the action,
nifolds. tion
Let N be a lasrangian
~IN of the canonical
We consider
when evaluated
manifold
submanifold
objects
functions
submanifold
1-form
only simply connected
~
is
is thus an iso-
in terms
the Hamiltonian~
of lagrangian
to N is closed
the
tensor in field
of ( T ~ , a o ) .
lagrangian
on
~,
submanifolds
and also the energy-momentum
theory will be shown to be generating
dim ~.
shows that the dimen-
lagrangian
Such important
I ~
=
this condition
N of a symplectic
aim at describing
functions.
0 and dim N
vanishes
satisfying
is not hisher than ~ dim P. A lagrangian 2 tropic submanifold of maximal dimension. We will always
=
subma-
The restric-
since
submanifolds.
It follows
that there is a function S on N such that
This function Suppose
is called a proper function
that N is a section of a bundle T*~ over C c 2-
case the proper a projection submanifold
function ~ definies
N which
completely
the
is simply the image of the section
dS
In a coordinate
In this
a function S on C which is simply
of ~ onto C. The function S determines
6.8
manifold
of N.
system
N is described
:
C
(qJ,pj)
~ T#~
.
the above statement
by equations
means that the sub-
45
6.9
pj
The f u n c t i o n
S is called
Generating
S
--
% qJ
a generating
functions
function
can be also used
w h e n N is no lon~er a section
of N.
in more
complicated
cases,
of T ~ .
Let C c ~ be a s u b m a n i f o l d
of ~ and let S be a f u n c t i o n
on C. It
I ~ (p) ~ C ; 4 u , p >
for
can be easily shown that the set
N
{
--
6.70
p ~ T*~
each vector
is a l a ~ r a n g i a n ~enerating
submanifold
function
S is a p r o j e c t i o n
of N. Also
u tangent
. The f u n c t i o n
in this
to C at £t(p)}
S is called
case the g e n e r a t i n g
a
function
onto C of a p r o p e r f u n c t i o n S given by 6.9. L a g r a n -
gian submani:Folds which characterized
of (T*~,cO)
=
can be g~erated:~, in this way can be l o o s e l y
as those whose p r o p e r
£unctions
are n r o j e c t i b l e
onto
~ub~ani~olds of 2 /cf.[~@/. Generating
functions
up to an a d d i t i v e
functions
are d e t e r m i n e d
constant.
If the s u b m a n i f o l d
6.77
in a c o o r d i n a t e
as well as p r o p e r
C is ~iven by e q u a t i o n s
G~ 6 q j )
system
t h e n the s u b m a n i f o l d
o ;
:
(qJ)
~ =7,...,k,
and i f ' S is any c o n t i n u a t i o n
6.70 is given by equations
G ~ (qJ)
/cf. [6]/
o
--
6.42
~ Pj
=
%qj
+
2~
~ G - - - ~~
~qJ
of S to :
46
Only the symplec%ic lagrangian submanifolds.
structure
of a manifold
is used to define
To define proper functions
and generating
functions we needed much more structure namely the structure tangent bundle. one encounters
In applications symplectic
bundles but are isomorphic be called special plectie
structure
of symplectic
of a co-
geometry to dynamics
manifolds which are not directly cotangent to cotangent
symplectic
manifolds.
bundles.
Such manifolds will
More precisely
in a symplectic manifold ~ P , ~
a special
sym-
is a £ihration
and a symplectomorphism
such that
where
~
: T*~----~
a symplectomorphism from T ~
is the cotangent bundle projection. the pull-back
~
to ~ is equal to the symplectic
The presence manifold ~ , ~ )
of a special
form
symplec%ic
ring functions
are constructed
~
some lagrangian
on submanifolds
aP
submanifold
subma-
of 2" GeneraThe l-form
is a differential
of a function called again a proper function of a manifold. ting function is the projection
is
in a symplectic
as in cotangent bundles.
restricted to a lagrangian
2-form
~
.
structure
makes it possible to describe
nifolds by generating functions defined
= ~
of the canonical
Since
A genera-
of a proper function to ~ .
Thus we see that the objects of a special symplectic used to define generating functions
are the projection
_~
structure onto a ma-
47
nifold ~ and the q-form
~
such that d ~
= ~
One symplectio manifold may be equipped~ with several special symplectic structures in which case one lagran~ian submanifold may be generated by several different
~enerating functions.
We will find that
the Lagrangian and the Hami!tonian /in field theory also energy-momentum tensor/ ~re generatin~ functions of the same legrangian submanifold with respect to different special symp!ectic structures /cf. ~@/.
7. Finite time interval formulatiou of dynamics
As
is
usual in canonical
formulations
a~sume the existence of a differentiable
Qf particle dynamics we
two-parameter
family of dif-
feomorphisms
g.1
R(t 2,tq)
:
Pt I
~ Pt 2
satisfyin~
7.~
-~(t3,t2) ° ~ 2 , h )
Dynamics : M
--
~t3,h)
is expressed in terms of this family as follows.
A section
~ P is dyn.amic~]!y edmissible if and. only if
7.3
~(t 2)
=
R(t2,tQ ) (~{(tfl) )
for each (tl,t2). It is assumed that mappings R(t2,t~ ) are symplectomorpbisms
:
g •~
This formulation
R ~(t2,tl)
6Or 2
=
6Ot~
of dynamics is equivalent to statin~ a system of first
48
order differential cally admissible
equations.
histories.
the sytem of equations.
Solutions
of particle dynamics can be described to field theory.
We define a
family of submanifolds
D (t2'tq)
graph R(t2,tq ) c
=
p(t2,tl)
The manifold
Pt 2
×
Pt I
=
P (t2'tQ)
law 7.2 reads
D(t3,tl )
•
will be called the boundary phase space corres-
pondins to the time interval [tj,t2] c M. In terms of graphs the
position
of
We return to this point in the next section.
in terms suitable for ~eneralizations
7.5
are dynami-
The family R(t2,tq ~_ is the resolvent
Th~s classic formulation
two-parameter
of the equations
com-
:
=
{
(~3~,p)~ ~4,, P (t3,tl) I there is a ~p @ p t 2
7.6
such that (c~ ,py (~ m D(t3't2 ) , ,(~)(I)\ [p,p) ~
We wil]_ denote the right hand side by O me way we introduce multiple
(t~'tN-~)
.....
(ts,t2)- ° (t2,t~)
compositions
-
(t2'tl]] .
. In the sa-
of relations
D(t3't2 ) o n(t2't! )
there is a sequence
7.7
D
= { ( c ~ $ 0 E p(tN'tl) I
(%'[ ...,p) (~'
~ p
×
...x
iN_ ~
Pt2
such t~at (T',{') ~ D (ti+1'ti) , i .< N-n]
corresponding
to divisions
of the time interval (td,tN)
into N-I sub-
intervals.
In terms of this definition we have the composition
7.8
D
(t N, t~)
=
D
(~N'~N-¢
.....
D
(t2, t~)
law
49
corresponding to
2.9.
R(tN,tl )
=
R(tN,tN_I )
... oR(t2,tl)
o
The property 2.& of the resolvent is equivalent to D (t2~tl) being a lagrangian submanifold when an appropriate symplectic structure in P
(t2,t 1)
2.10
is chosen. This symplectic structure is given by the 2-form
(u-) (t2'tl)
~he "minus-sign"
=
M
~
M -
t2
dO
tI
is defined by
<(v2,v I) A (w2,wl),
got2
~
dot1
2.11
where v~,. wA! are vectors, tangent(.t2~t~)~ to Ptl, and v2, w 2 are vectors tangent to Pt " The submanmfold D fold
is a lagrangian submanifold of mani-
~P(t2'tl) ~¢t~ t11)" since ¢t2 t~ is the image of the mappin~
2.Q2
(R(t2,tl) ' id)
:
> Pt 2 × Pt I
Pt I
and
,t~),
± ~
~
cot i
(t2,tl)
~°tp_ - c o t I = o.
A coordinate system (t,qJ,p.) in P gives rise to a coordinate ~ , " Lt~,tA~J syste~ (qJ,pj,qJ,pjj =n P f o r each ( t 2 , t l ) . I~ this eoordinate system
7.1#
6o ( t 2 ' t l )
=
dpj A
-
dpj A
50
If the mappin~ R(t2,tl ) /or equivalently the submanifold D
(b,%)
.
is described locally by
7.~5 ~Pj
Pjtq ,P j )
=
then
\ 9 ~Z
d pc,~j
-
+
A
d~J
<,~ dP i Pi
A % Pk
=
7.16
....~T%• y~k - - d%')- ^ d~ k + ~ ~i"> ~'~k d%. ,, d~'k + '~ a~ +/~% ~'~~
- a ~~" .
=
~'~)
~
d%_ ~ d~ ~
-
oP i
0
since
a,,~.j =
Pi
Pk
~'.
a~k
0
oP i
and i
D oP i
k
d%^
d'~ --
51
due to 7 . 1 5 being a canonical transformation /see[6]/. We will refer to the lagrangian submanifold D (t2'tl) as the dynamics corresponding to the time interval [tl,t2] ~ M. ~here is a natural special symplectic structure in
(p (t2' tl )
60 (t2't~)) defined by the projection
(t2,t 1)
7.q?
(t 2 ,t 1)
and t h e diffeomorphism
7.18
o(
~2,tI)
:
p
(t2,td)
~ T#Q
(t2,t I ) =
~*Qt2 ~ T~Qtl
where
The manifold Q (t2'tl) will be called the boundary configuration space corresponding to the interval The 2-form
7.20
60 ~2~t~)
{D(t2, tl )
bl,t2].
is the exterior differential of the l-form
=
~ t2
M__
0 tQ
defined by
7.2~
<(v2.,v~),~t 2 _M eta'7
--
where v~ is a vector tangent to P
and v~ is a vector tangent to Pt "
It is easy to see that the form of the canonical l-form in T~Q (t2'tl) pj,q ,pjj we have
2'tii
is the pull-back by o(C~a'~2
In the coordinate system ~ J
52
=
pjaq- - pjaq
..|tl,t2| r
It will be assumed in the sequel that for each interval
(t2,t¢
the lagrangian submanifold D
i
is generated with respect to the
above special symplectic structure by a function W
(t2,t 1)
defined on
a submanifold c~t2,t~j{~ c Q Ct2'tl) called the constraint submanifold. Using coordinates
str~int~-(c~t~'t~)
(~J,p.~q :
~p.) and assummng that there are no con-
Q-, ~, , t ~ , t ~ ) ~]
. . . . ~aav . . . ~ne . . . suomanm f ' "o±a u we Imna
is described by the equation
7.23
pjdq~
-
pjdq~
=
dW
equivalent to the familiar formulae
~
,~J]
:
,~ pj
=
~,~j
w
pj
=
- ~-~
w
(t2,t ~)
(t2,t~)
The composition law 7.8 for lagrangian submanifolds is reflected in composition law for their generating functions. law in the simplest ease of no constraints (C (t2't~) each
[t~,t2] ). Theorem Let a function wktN,t~)i be defined by
7.25
We state this
= Qkt2,tl)fo r r
53
where for
each ( ? , ~ )
the
sequence
~ QtN_ fl ( ~ " ,...,q,q) '~ ~
is a stationary point of the right-hand W (tN't~)
X
.-. Qt 3 x
Qt 2
side. The function
is a generating function of D (tN'tq).
We give the proof of this theorem in the case of N = 3.
Proof
:
?.2~
wCb't~)(~,~)
= w(t~'t~)("q,q)"
+ w(t~'t~)t'q,q)"'
where ~ is the stationary point of the right-hand side. For given
(,,, ~,,,
(b,t~)
~,,.,
q~qj E Q via
(b,tO
denote by (p~p) ~ D
~c(t3'tl)
unique point which projects
onto (q,q).~3~ Take the unique point ~ e
(p,p) ~ ~'~ ~ D ~t2't~)
simplicity we use coordinate descriptions =
such that
/or equivalently (~) ~)~ ~ D (tg't2) /. Pt2 p,p) For the sake of : p
=
(
,pj),
=
,Pj).
It follows from 7.24 and 7,6 that the equation
has the unique solution q
a
7.28
a~ j
w
(t2,t~)
= I' = ~ t 2 ( F )
~., ,,,.
(q,q~
=-
a
_,,,j
for whioh w
(t~,t2) (~
,~)
=
.,
pj
dq ~,¢here p =
7.29
,pj
. Treatin8]
+
C,q) as a f u n c t i o n
of (~,~')
we have
54
Similarly
B~O
7 . 3 0
w
~q,q)
which proves that W (t3'tg)
=
>j
-
iS a generating function for D (t3'tq).
Under specJ_al conditions which are not stated here the same composition law holds in the presence of constraints.
In this case the
sequence ('~,~4',...,~)must be compatible with the constraints so t~at the right-hand
side of 7.25 is defined.
The constraint
c(tN'tl)cq (tN'tl)
is the set of pairs (~,~) for which stationary points (¢~),...,~{,({) exist /see[g5]/.
Example I The configuration
bundle of the harmonic
oscillator
is the tri-
vial bundle Q = M × R I and the phase bundle P can be identified with M × R 2. In terms of coordinates
mh
(t,q,p)
=
p
=
-kq
the equations of motion are
7.3d
Integrating these equations we obtain the general expression for dynamically admissible
q(t)
sections
A cos~.t
+
B
sin~.t
T~ 7.32
p(t)
-Af~sin~-~.t
+ Bcos~.t
(t2,%) The manifold D
is described by equations
'p) sin ~ (t2-h) 7-33 4)
=
_~
55
(tf,tl)
to
In order
find the p r o p e r
by c o o r d i n a t e s
(q,p)
function
we p a r a m e t r i z e
D
:
dW _ {if' tl)('~,~)
@{t~'t~) I (tf,t 1)
=
aq
-
?. 3~
'"
+ P---- sim
G
(t2-t ~
I}
~-~,
~ p~q
Hence
?.35
.~o~~ (~,_-~ To obtain the g e n e r a t i n g Q
(t2,t 1)
. We c o n s i d e r
(i) I f sin ~ ( [ t 2 - t j )
7.36
Hence
p
C
function
three
~
cases
~, ~
:
c~ cotan ~ t 2 - t l )
= Q
(t2,t I)
and
~_~
we project w (if't1) to
0 then
=
( t 2 , t ~)
W (t2'tl)
_ ~
56
( t 2 ' t q ) r~a~{4, t~,q)
WEt2'tfll
=
_
('~,~'(~,<~))
=
7.37 =
_
- 2qg + qq cos
_
2sin~(t2-t
(ii) If ~ ( t 2 - t q )
7.38
Hence that
(iii)
=
q
(t2,t q) (t2,%) W If
q
=
P
of
=
×
Q~2 Qtq
7.33
P
imply
•
and it follows
from 7.3g
O,
=
(2n + q)W
c~)
q
equations
,
'
~-~(t2-tq)
7.39
then
is the diagonal
C
.
I)
2nUF
=
(t2-tl)
(~) =
then
g,)
-q
P
,
(4) =
-P
( t 2 , t fl) ( t 2 , t fl) <,, and W = 0 on C described by the equation ~ = -q, Do i l lustrate
the composition law f o r generating functions
~3) ~,,o~ t~t
sinC
o an~ sing(t~-t~)
we take (tQ~t2~
~ O. ~hen
~e(t3,t'2)(<,,q,q)<~>, + w(t2,%)(q,qj=,c,>~
7./#0
~_ ~:mls
<_~
in
@~qq + qqjcos
(t3 -t2 )
-
+
(t3-t2
[('~ + <~)eo~(t2-t
~) -
sin{~(t2-%) The derivative
with respect
to (~ is
-
-
57
If sin~(t3-tl)
7.1.2
~ 0 th~n there is a unique stationar+v point
~> =
1
f <,,
-
v
sin~(ts-t O]
Substituting 7.42 into 7.40 we obtain the value
TEl
f"" oos~(b-t ~)
2¢# +
<<
2sin~mk-'(t3-t 1) which a~rees with 7.37. If s in~(t3-t~).
=
0 which means that ~ ( t 3 - t q ]
=
ng[ then sta-
tionary ~oints exist if and only if
7.~3
Using the equality
the equation 7.43 reads
=
0
i.e.
which agrees with 7.38 and 7.39. The substitution of the above equalities into 7.40 gives the value O.
58
8. Infinitesimal description of dynamics
Dividing the time interval the dynamics
T
l
[tl,t2] into subintervals we obtained
D (~2 ',.~1) as a composition of dynamics corresponding to
the subintervals.
This procedure can be carried to the limit of the
number of subintervals increasing to infinity and their lengths decreasin{ to zero. ~his limit is expressed in terms of ~ets of sections of P. The term "jet" is taken from modern geometry but it denotes a geometric object known already in traditional differential geometry. Objects of this type are described for example by Schouten in [45] /see also Appendix D at the end of our notes/.
Definition
: The space Qt =
J
Q) of l-jets of sections of the
bundle Q at t is called the infinitesimal configuration space at t.
~he s ~- a c e p it = J_~( P) is called the infinitesimal phase space at t. The corresponding bundles qi = jq(q) and pi = j1(p) are called the infinitesimal configuration bundle and the infinitesimal phase bundle, respectively.
Intuitively the jet jlp(t)
of a section
is a limit of the pair (p(t+b),p(t)) E p(t+h,t) when h -~ O. In this sense Pti is the limit of p(t+h,t) when h - - ~ O. The same interpretation applies to Q~. There are the canonical jet-target projections
8.2
and
bp
:
pi
~ p
59
8.3
Cq
•
qi
,
,
Q
There is also the jet prolongation
8.g
of
cgi
~ : P
=
j!~g
:
jqp
~ jIQ
~ q. We have thus the commutative diagram ~p
pi
8.5
P
96i
~q
Q~
The restriction
of
~
Q
~i to a .f~b~e . . . . . Pit will be denoted
~
i 0%. i : D~t---~
We show that each infinitesimal phase space Pti is canonically diffeomorphic
to the cot,:~no~nt bundle T Qt" For this purpose we use
the canonical identificatioD
vat(q}
8.s
=
al(vQ )
The symbol VQ denotes the bundle of vertical ~'ectors tangent to the b~ndle q
~M /cf. Appendix E/. Similarly VJO(Q) denotes the bun dle
of vertical vectors tangent to the bundle J~(q) tion 8.6 means that each vector tangent to Qt = J
} M. The identifica, Q) may be represen-
ted by the jet: of a section
8.7
for u~
M ~
~c = t. Now let g
~
=
Y(~)
~
TQz
j ~ p f t ) ~ P ti be the jet of the section 8.4. Let
T qti be a vector attached at the point
~gicg ) and represented by
60
the jet
j~x(t) ~
J~[VQ) of the section 8.7. This section is chosen
in such a way that for each the point
8.8
~(p(~))
=
~ ~M
the vector X(T)
is attached at
. The evaluation
< jqx(t),jqp(t) >
= -~<x{z) ,p(~)>
I~ m
defines a covector
~t
<- ,g~ ( T ~ i associated with g. Using local coor~t
dinates we show later that the mapping
8.9
pi
~
t ~g
is a diffeomorphism.
o
t(g)
= , <.
,g>
E T e qti
Due to the existence of this diffeomorphism the
infinitesimal phase space pi is a symplectic manifold with a canonit cal special symplectic structure, The pull-backs by
i of the cac< t
nonical d-form and the canonical 2-form from T~Q~ to pi will be det noted by ~ and COti respectively. i Let v, w be vectors tangent to Pt and attached at the same point. Using the identification
8.~o
vJq(P) = S(vP)
similar to 8.6 we represent v and w by jets of sections
8.qq M
~
~ Z(Z)
e
TP~
We chose these sections in such a way that for each ~ Z~] of
vectors Y ~ ) ,
are attached at the same point. It follows from the construction C< ti that
~it and
60 ti satisfy the following equalities
61
=
lira h-,O
_-
8.12
~{-~
=
O(t+h,t)>
.im
h-,O
and
~ --t 8.13
=
lim
=
lim h~O
~{
Z(t+h),oOt+h~-
~
~<
Comparing the above formulae with
7.11
tain intuitive sense the forms ~ I M- dOt ) when h ~ O . and 5(60t+h
.
and 7.21 we see that in a cer-
and cot are limits of This interpretation
Ot+h
M
justifies the
following notation i
Md
@t
=
i CO t
=
d-V ~t
8.14 Md ~
C°t
Let (t,q j) be a coordinate system in Q compatible with the bundle structure and let (t~qJ,pj) be the corresponding coordinate system in P. These systems lead to coordinate systems (t,qJ,~ j) and (t,qJ,pj~qJ,pj)
in Qi and pi respectively.
jet jlq(t) of the section
The coordinates ~J of the
62
are calculated from
qJ
8.16
dqO(~)
=
d~
=t when q3(~)
are coordinates of the point q(~) . Since M is identified
with R 1 the jet jlq(t) can be interpreted as the vector tangent to the curve
8.17
R I ~z
~
(~ ,qJc~)) ~
Q-
The coordinate expression for this vector is
8.18
jlq(t)
-
~t
+
~qJ
Coordinates ~j are defined in a similar way. Coordinate expressions for mappings
bQ, Cp and ~ i are :
bQ(t,qJ,~ j)
8.19
=
(t,q J)
Lp (t,qJ,pj,~J,~j)
=
(t,qJ,pj)
~ui(t,qJ,pj,~J,~j)
=
(t,qJ,~ j)
.
For a fixed t a M systems (qJ,~J) and (qJ,pj,~J,~j) are coordinate sysi The coordinate expressions of forms @~ and ~ ti are tems in Qti and Pt" Q ~
Md
Md
•
dt @t
=
aT(Pj ~q3)
Mdd--~ ~t
=
Md -g~#dpj^dqJ)"
.
=
Pj dq° + Pj d~°
8.20
~
=
=
" d~j^~qO + dpj^dqO
63 These expressions
follow easily from formulae 8.12 and 8.13.
In Section 6 we described the canonical construction nates in the cotangent bundle from coordinates this procedure we introduce ponding to coordinates
coordinates
(qJ,~J)
of coordi-
of a manifold.
(qJ,~J,rj,sj)
of Qi. The coordinate
Using
of T*Q~_ corresexpression for
i O( t is thus
,i. j ,Pj,qJ,t~j) O
8.21
=
(qJ,~a,rj,sj)
where rj = ~j and sj = pj. It is clear from 8.21 that O( it is a diffeomorphism.
Definition: tem is specified
We say that the /infinitesimal/ if a lagrangian
dynamics
of the sys-
submanifold Dit c Pti is chosen at each
time t e M.
If the infinitesimal : M -
dynamics
is specified then a section
, P is considered to be dynamically
if j1~( t)& D ti for each t E M. Conversely admissible
sections
admissible
if and only
if the class of dynamically
is known then infinitesimal
dynamics D ti can be
obtained as the set of all jets of these sections at t. We recall that in terms of finite time interval dynamics tion
~: M - - ~ P
is dynamically
(tl)) e D
admissible
if and only if
for each tl,t 2 e M. cConversely
sections are known then D ~t2'tl)
set of pairs
(~(t2) , ~(tl) ) e P (t2~tl)\ where
8.22
D (t2'tl)
can be obtained as the ~ is admissible
section.
can be obtained
dynamics: { (~
=
\-(~(t2),
if the dynamically
We conclude that the finite time interval dynamics from infinitesimal
a sec-
0~, ~t2,tq)there is a section :~--+P} ,p) 6 such that ~(tq) = ~ , .
~)
41
e
64
The above formula
can be considered
the number of subintervals Conversely interval
8.25
Dt
tends to infinity
the infinitesimal
dynamics
=
as the limit of formula 7.7 when
dynamics
and their lengths
can be obtained
to zero.
from finite
: there is a section ~:M--*P such that ) g = j q ~ ( t ) and (~Ct+h)~ ~(t))~D(t+h't)~.
g E
for each h Originally defined
we postulated
the finite
the existence
interval
lowing theorem guaranties of the infinitesimal
Theorem
the correctness
Through
satisfies
The fol-
of the above construction
all conditions
by 8.23 is a lagrangian
stated in Section 7,
submanifold
i of Pt"
each point p ~ Pt there is a unique dynamically
section
This implies that g(p)
•
that
in terms of resolvent.
2
then D ti defined
admissible
R(t2,tl ) . and
dynamics:
If the resolvent
Proof:
dynamics
of a resolvent
Cp(g(p))
= jl ~ ( t )
= p. It follows
that the mapping p - - ~ g(p) of this mapping
element
from the properties
of D i such t
of the resolvent
is smooth.
and therefore
to that of Pt and hence
is the unique
The manifold D i is the image t i is equal is smooth. The dimension of D t
equal to
3 dim
Pt"
It remains
to prove that
i i = O. gO t IDt Let v e TP
be a vector tangent
8.10 we represent
i Using the identification to D t-
v by the jet jIY(t)
of a section
~ --~Y~)
. Since
v is tangent to D ti the section Y may be chosen in such a way that (Y(z)
~Y(t))
is tangent
to D (~ ,t) for each Z
. Let w be another vec-
6B
tor tangent to D i and t
the corresponding representant.
, z(~l
It
follows from 8.q3 that
< v ^ w , co ti > = l i m h~O
{t+h),zCt}),
because D (t+h't) is lagrangian.
If the dynamics certain additional
=
0
This completes the proof.
is introduced
in terms of a family
ID~
then
conditions have to be satisfied in order that the
formula 8.22 defines a lagrangian submanifold. We assume that for each t e M the infinitesimal a generating function L t :
Qit '
dynamics D i has t > Rq" The c o r r e s p o n d i n g proper function
of Dit is denoted by ~t" Both L t and ~t are defined up to an additive constant.
The family
~Lt~
called the Lagrangian.
defines a function L: Qi
The Lagrangian
~R q which is
is defined up to an arbitrary
additive function depending only on t. Using coordinates
(qJ,pj,~J,~j)
and the formula 8.20 we find that the generating formula for D ti reads
8.2~
pjdq j + pjd~ j
equivalent
to the familiar formulae
f~j
=
=
dLt(qJ,~J )
~qj
s(t,qJ,~J)
8.25
T,(t,q4,44)
p4
It is interesting to note that also the composition for generating functions has its infinitesimal give in the case of no constraints
:
Theorem 3 Let a function W
(t2,t I)
be defined by
law 7.25
formulation
which we
66
8.26
w(t2'tl)(
)
--
t2
tI where for each
C~' ,~)
the section
M ~ ~
,
q(z)
~
Q~
is a stationary section /in the sense of the calculus of variations/ of~the right-hand side, such that q(tl) = ~ , q(t2) = '~. The function W
(t2,t fl)
is a generating function of D
(t2,t fl) .
Equations 8.25 are obviously equivalent to Euler - Lagrange equations. This implies that stationary sections are precisely projections by ~
of dynamically admissible sections.
Due to lack of constraints the Theorem 3 can be stated in terms of proper functions
:
Theorem 3 ~ (t2~t I) be defined by
Let a function W
t2
w(t2, tfl)
8.27
, p,~, )
=
i
,
t~ where for each C~ ~ ,~) e D (t2'~q) the section
~--~pg~)
unique dynamically admissible section such that p(tl)
is the = p ,
P(t2) = ~ . The function ~ (t2'tq) is a proper function of D (t2'tl)
Proof: Let (~ ,u) ~I, e TP (t2' tl) be a vector tangent to D (t2'tl) and let
%
~C~(~),~(~))
its tangent vector at such that for each
~=
be a curve in D (t2'tl) such that (~ ,u) c~), is O. There is a unique mapping
~ the mapping
(u,%]-~p(~,%)
67 l
is a d y n a m i c a l l y
admissible
section
(I)
and p ~ (tl)
= p(%),
p,(ty)
~)
= p(&).
Then
<(T
,~)) , d I f l ( t 2 ' t l ] >
-
(ty'tl)
dZ
'~" " "' (p(~t;,p(;t))
_W
t2
d--~ L~(jlp~ (~))
I
~=0
t1
t1 t2 0
=
;t =0
i
dI
=
I I
t2 P
= \
="
d'~
\d~
?
o
tI where %
v(t)
tI is the v e c t o r
= O. The last
tangent
equality
follows
i of D~ . Using the i d e n t i f i c a t i o n the
jet jly(~)
vector Y(tq)
tangent
of the section to the
= u ~ Y(t2)
mula 8.12
= u
curve
from L ~
~t
,jlp~(~)
being
a proper
8.10 we replace
~
• Y(~)
~t --~p~(~)
are o b v i o u s l y
e a
TP~ P~
satisfied.
function
the v e c t o r
v(~)
where
is the
at
Y(~)
t2
t2
tI
tI
tI
completes
Example
Or1>
2t= O. R e l a t i o n s
<(~ ~',, Q/ty,tl/#
the proof.
2
Equations
of m o t i o n
of the harmonic
1 p
=
m~
=
-kq
by
Now we use the for-
t2
Example
c- D@4 at
:
-- -<~
to the curve
oscillator
considered
in
68
define a two-dimensional submanifold D ti of the four-dimensional infinitesimal phase space Pt" i We restrict the form ( ~ = }dq + pd~ t o Dti and obtain
8.28
Gi~ I Dti
--
-kqdq + m~d~
=
d ~(m~i 2 - k q 2 )
=
dLL_t(q,~i)
It follows that D it is a lagrangian submanifold generated by the Lagrangian
8.29
L(t,q,~)
:
~(m~ 2 - kq 2)
.
The stationary section of the functional
~2 8.30
I L~(jlq(~)) d~ tI
satisfying the boundary condition q(tl) = ~ , q(t 2) = ~q
is given
from formula 7.32 by
q [~ sin~'(1# -t I) (~ s i n ~ ( ~ -t2) ) sin ~(t2-tl) ~q - q Substituting this expression into 8.29 we obtain
k 8.32
Integrating this over
8.33
W
(t2'tl)d ,
=
~
[tl,t2] we obtain
"
69
which
is the same as formula
9. Hamiltonian
description
The description
spaces.
special
on the choice there
special
The h a m i l t o n i a n
symplectic
This confirms
structures
symplectic description
interpreted
as the choice
of particle
mechanics
a single hamiltonian motivated
was obtained
in infinitesimal
is associated
with different
canonical
of the configuration descriptions
but depend
bundle
of a trivialization
of a rest frame.
In standard
description.
is always
phase
with can be
formulations
assumed which leads to
The generality
by later generalizations
Q. Thus
associated
of Q. The choice
a rest frame
Each infinitesimal
structure
which are no longer
of a trivialization
trivializations
3.
in terms of a Lagrangian
is a whole family of hamiltonian
different
Theorem
of dynamics
of dynamics
by using the canonical phase
7.37.
of our approach
is
to field theory.
space Pti is already fibred
over Pt" The a
symplectic
structure
it remains
o(~ : Pit
, T*P t. This construction
zation of Q. A trivialization
to construct
in the time manifold
M. Sections
of P are mappings
are vectors
tangent
jets of sections
Since
a section composed
with the projection
ping,
vectors
to the section project
g ePti is a jet attached
of a triviali-
a trivialization in P. At 8 horizontal vector - ~ - w h i c h projects
from M = R I to P ; hence
tangent
a choice
in Q induces
each point p e P there is a unique onto the unit vector
utilizes
a diffeomorphism
at p then g -
~
to P.
is the identity
map-
onto the unit vector.
If
~St is a vector which projects
8
onto zero.
9.1
It means that g -
t
is a bundle
~ g
~t
~@t(~)
e TPt /i.e.
=
g-
is vertical/.
~-~
~
The mapping
~Pt
isomorphism.
Due to the existence
of the symplectic
form
~t
the tangent
bun-
70 die TP t is canonically isomorphic to the cotangent bundle T~P t. The canonical isomorphism
~t
: TPt
>
T*Pt
is defined by
9.2
<w, Tt(u)# -- < u ~ w , ~ t >
The composition
=
°
is the required diffeomorphism Let
~t
from Pti onto T~Pt
: T~Pt---+ Pt be the canonical cotangent bundle projection.
Let
@ ~ and ~ht be the canonical Q-form and the canonical 2-form in h ~ h T*P t. The condition ~ t = JUt°O(t is obviously satisfied and the equa^ h will be proved using local coordinates. The q-form tion dO ti = o
dinates (t~qJ~pj~J,~j)
introduced earlier. The jet g~ Pti with coor-
dinates (t,qJ,pj,~J,~j)
is the vector
at the point with coordinates
g
(t,qJ,pj). Hence
9.3
@t(g)
If coefficients
(qJ,pj) together with coordinates
as coordinates
:
~- +
~t
~qa
+ ~j-~qJ
~PJ
~pj (qJ,pj) are used
in TP t then the coordinate expression for
identity. Coordinates
attached
~t is the
(qJ,pj) in Pt give rise to coordinates
(qJ,pj,
mj,n j) in T*P t by a standard procedure described earlier. The canoni-
71 ^h
cal forms
~'h
~t and 60 t are expressed in terms of these coordinates ~h @t
9.~
=
mj dqj + nJdpj
=
d m j A d q 0 + dn J A d p j
by
and
z%
9.5
h
COt
'
If ue TP t has coordinates
(qJ,pj~q
"
~ ~j) then u =
--. + ~j ~qJ
•
Let
~t(u)
have coordinates
~t(U)
=
(qJ~p~mj,
nO). Then
9p~
mjdq j + nJdpj
For each vector w = a j - - . + ~qJ
b.--
J
the equality 9.2 reads
~pj
9
aim. J
+
b.n j J
= < (/p i " a k - ~ a b ~ - - ~ - - , d p j ~qk )Pi
A dqJ>:
9.6
=
pja j - ~Jbj
Since a j and bj• are arbitrary we conclude
that
follows that the coordinate expression for ~
mj
=
p j,
nJ
=
-~J
.
It
-- 7to~t is
~(qJ,pj,~J,~j) = (qJ,pj,mj,nJ)
where
9.7
Substituting
mj•
=
~j ,
nJ
-q"J
9.7 into 9.$ and 9°5 we obtain the following
expressions 9.8
=
O~
=
pjdq J - ~Jdp j "
coordinate
72
h~ ~ h O(t cOt
9.9
=
d~
j,\dqj
- d
The last equality proves that 04
~JAdpj
=
i cot
"
~h t defines a special
together with
Through each point p ~ Pt there is exactly one dynamically
admis-
sible section of P. The jet of this section is the unique element of D it attached at p. It follows that D ti is the image of a section of the h pi bundle 0"6 i : t ~ Pt" Since we also assumed that fibres of Q and consequently
fibres of P are simply connected,
each lagrangian subma-
nifold Dit is generated by a function F t on Pt" Functions H t = -F t define a function on P called the Hamiltonian.
The Hamiltonian
is defi-
ned up to an arbitrary additive function depending only on t. Using coordinates
• "'[qJ,pj,~J,~j) we find that D ti is described by
the equation
9.10
~jdq j - ~Jdpj
=
- dHt(qJ,pj)
analogous to 8.24 and equivalent
Pj
=
qJ
=
-
to the formulae
_~_~ H(t,qJ,pj ) Dq3 3pj~ H(t'qJ'Pj)
known as the Hamilton canonical equations.
Example 3 Equations
of motion of the harmonic oscillator
in the form I
9.12 =
-kq
can be written
73
Restricting the form
~hl ~t I Dit
~
= ~dq - ~dp to D ti we obtain
=
- kqdq - ~pJdp
:
_
=-d
2~pJfJ2 + kq2~}
9.13 t
(q,p)
It follows that D ti is generated by the Hamiltonian
H (t,q,p)
9.Jz~
=
q#1 2
2[~P
kq2)
+
In the preceding section we constructed a special symplectic structure in the symplectic space (P~,CO~). The fundamental objects i pi i of that structure were the projection ~ t : t ~ Qt and the J-form i satisfying d ~
@t
i . In the present section we constructed ano= cO t
ther special symplectic structure in ( pit , ~ ti) depending on the choice of a trivialization in Q. The fundamental objects of this structure are the projection
~U~ : pit----*Pt and the J-form
~
which again
= @ ti " With respect to the two special symplectic structures the same objects D ti are described by two sets of genera-
satisfies d @
ting functions. The two descrptions are parallel. The formula 9.10 \
has its counterpart in the formula 8.2~. The difference
@i_ ~ h is a closed J-form. Due to our topolot t gical assumptions it is also exact. We define a function ~t on Pti by
9. 5
If
g
9.16
Yt(g)
=
<@t(g),Ot3
has coordinates ( q J , p j , ~ J , ~ j ) ~t(g )
=
~J
•
then B
~qj
+ ~j ~Pj
and
74 9.q7
~jt(qj p j , ~ j , ~ j )
aqa a
= < ~j
~J
+
a ,pjdqJ > = ~jpj DPj
Hence 9.q8
dVt
=
pjd{J =
~OdPa +
0{ -
O~
The function ~t on Dit defined as
9.q9
tlt
~t l Dti - _Lt
=
satisfies the equation
i t
h
-
=
=
Hence -_Ht is the proper function corresponding
-
i
et
to the generating fun-
ction -H t. Our approach to hamiltonian description
of dynamics
from, though equivalent to, the standard approach.
is different
Since for each point
peP
there is a unique vector /jet/ in D ~p) i
attached at p
the family
ID~
defines a vector field in P which will be denoted by ~ t " The
difference
9.20
xh
-
d
dt
_
~__ at
is a vertical vector field in P. In coordinates dd-~ and X h are :
9.21
and
d
d--t =
a
8--t +
~j
a ~qJ
+ Pj
a ~Pj
"-(t,gJ,pj) the fields
75
9.22
Xh
where
~J _ _~ + ~j ~qJ 3pj
:
qJ and pj are functions
fields
on P are usually
on P given by 9.11.
called time dependent
Vertical
vector fields/cf. [1]/.
i It can be shown that the fact that D t are lagrangian equivalent se of
to X h being a locally hamiltonian
[1]/. In our ease X h is even globally
ar from equations
9.11 that our Hamiltonian
vector
submanifolds
is
vector field /in the senhamiltonian
and it is cle-
H is the Hamiltonian
for
X h in the usual sense.
10. The Legendre
transformation
We assumed that the infinitesimal fibration
i p~____~ i ~t : ~ Qt" It follows
there is a unique
locity/
element
D ti is a section
dynamics
that for each element
g e Pit such that
~(g)
of the
i v ~ Qt / v e -
= v. The map-
ping
i Qt
is called the Legendre the fibration title
/of.
Pt
~ v
Since D ti is also a section
transformation.
~ ht : pit----~Pt
the Legendre
transformation
of
is inver-
[9],[53]/.
11. The Caftan form The disadvantage in the dependence of combining
on the choice
the different
trivializations /cf.
a way manifestly
ponents
object.
corresponding
This object
of dynamics There
Hamiltonians
of trivializations.
are extracted
to different
lies
is a method
to different
is the Cartan form
this form in terms of the Lagrangian
independent
with respect
description
of a trivialization.
Hamiltonians
into a single
[TJ/. We define
how different
of the hamiltonian
and thus in
We show subsequently
from the Caftan form as com-
trivializations.
76
Definition: l-form
~
The C a f t a n form a s s o c i a t e d
w i t h the d y n a m i c s
is a
on P such that
i~.I
---
for each v e c t o r u E T P
~.~
tangent
< ~,~
to the fibre Pt'
o ~ ( (~ ~i ) )d
~
and
•
Theorem For any t r i v i a l i z a t i o n
is a H a m i l t o n i a n
Proof
:
Since
~t d dt -
~.5 Since
corresponding
We c o n s i d e r
11 .~
~t + Xh'
~
the f u n c t i o n
:
-<~,O?iPt
h
~t +(x~,~)>l~t i : Dt
vector
field,
we may write
id
:
-~t(~(~))+<x~,ot> •
"~ Pt is a d i f f e o m o r p h i s m
a lift --Hi of the above f u n c t i o n
~t~i¢~!~
to this t r i v i a l i z a t i o n .
where X h is a v e r t i c a l
~t : ( - ~ , ~ > the m a p p i n g
of Q the f u n c t i o n
we may define
i to D t. The lift of the f u n c t i o n
gives obviously -~t Since xho~th = ~ t it follows t~at the
lift of t~e funotion <X ~, et> is equal to <@t' Or> IDit = ytlD ti /cf.
formula
9.15/.
Thus
77
11.6
Ht
We noted earlier /formula miltonian
=
9.19/ that this function
Ct,qO,pj)
_-
11.7
from 1 1 . 1
which follows )
The Cartan's
form
on the choice
an additive
is a lift of a Ha-
function
the Caftan form is
pjdqO
_ ~,lt
and 11.5. ~
is independent
of the Lagrangian of time.
of trivialization
which is determined
It is clear from formula
a function f (t) is added to the Lagrangian by an additive = d@
"
; thus the proof is complete.
In local coordinates
pends
i ~t I Dt
- Lt +
term f(t)dt.
, which
all information
Theorem
only up to
11.2 that if
the Cartan form is changed
It is interesting
is uniquely determined
but de-
to note that the form
by dynamics,
also contains
about dynamics.
5
A section
of P
is dynamically
admissible
if and only if for each vector field
X in P
11.s
(x,
)is
--
o
where S~:P is the image of ~ .
Proof
: Using coordinates
(t,qJ,pj)
of P we obtain from 11.7 the
78 following expression for
11.9
~
~
:
DH dpj )A dt
= d p j A d q j - \{~qJD~HdqJ +
~pj
Let the field X be described by
11.10 and let
X
=
A .-~-~ +
+ Cj
~qJ
~Pj
~ be given by
11.11
~(t)
(t,qJ(t)
=
,pj (t))
•
Then ~H
xJ~
=
A (
~H
--
~qj
dq j +
~PJ dpj)
- BJdpj +
11.12 + Cjdq j - (B j ~H + Cj ~ j ) dt ~qJ the X J b'~
restricted to S is zero i2 and only i f < j l ~
In terms of coordinates where
qJ
~ t j(t)
jl~ pj
is the vector
~-~ +
,XJ~> - •+ ~qJ
= 0.
~j
~pj ,
dpj(t) dt . Hence
-~j
+ ~j ~py
11.13 _
+
H
~Pj for all values of A, B j, Cj. This is equivalent to
~Pj 11.1LI-
%qJ
)
=
0
79
and ~H qJ ~J +
11.15
~H pj
" Pj
=
o
•
The last equation is a consequence of 1 1 . 1 ~ and equations 11.1% are precisely the conditions for
~ to be dynamically admissible.
Theorem 5 is stated in a form specially suitable for generalization to field theory. Descriptions
of dynamics in terms of the Car-
tan form may be found in references
[9], ~q], ~6], ~7], ~8], ~9], ~0],
[24],
[26],
J2. The Poisson algebra Using diffeomorphisms
~ t introduced in Section 9 /formula 9.2/
we can assign to each differentiable function f on P a vertical vector field X f defined by
where ft = f IPt ' Xft = xf IPt• . One-parameter groups of diffeomorphisms generated by
X~ are symplectomorphisms of symplectic spaces ( P t ' ~ t )"
Differentiable functions on P form a Lie algebra with respect to the standard Poisson bracket defined by
{f,g)
=
Xfg
.
Theorem 6 The field X H associated with the Hamiltonian H : P
, R q is
identical with the field X h defined by the formula 9.20.
Proof - obvious from 9.22 and 9.11.
III. Field Theory
15. The configuration The fundamental : Q
bundle
and the phase bundle
geometric
~ M over a manifold
called the configuration
space for field theory is a bundle
M of dimension
bundle
is called the configuration
of the field.
space.
For dynamical
lativistic
theories
The manifold
or elastostatics
field theories of gravitation
M replaces
The fibre Qx =
space at x ~ M. The manifold
sical space in which the field is defined. such as electrostatics
m. This bundle will be
M is the phy-
For static field theories
M is 5-dimensional
physical
such as electrodynamics M is the 4-dimensional
the Q-dimensional
~ -1(x)
time manifold
and re-
space-time.
of particle
dynamics. As in particle corresponding
dynamics
to the configuration
the field will be sections the behaviour
admissible.
of time segments
sider domains /usually of segments
As in particle
in terms
V ¢ M. Instead
of symplectic
each domain V c M the symplectic This space replaces of dynamically
the space P
admissible
In most physical
BV
(t2, tQ)
field theories
BV
the laws of
equations.
of P 8 V for a wide class
of end points
(t1~t2)
In order to desc-
we will associate
of mechanics.
subspaces
we will con-
of boundary values
define D
of domains
of
which will be
mechanics
of domains.
geometry
space P
histories
In the sequel we consider
or states
[t~,t2] used in mechanics
compact/
~ : P--~M
laws governing
a class of histories
we will have boundaries
ribe dynamics
in M.
Physical
take form of first order differential
Instead
nifolds
a phase bundle
bundle Q. Histories
of this bundle.
of the field select
called dynamically dynamics
we shall define
on
8V.
Boundary values
a subspace BV
with
D 8Vc P BV
are lagrangian
subma-
V.
only curvilinear
polygons
as domains
81
Definition
: A curvilinear /oriented/ polygon of dimension k in
M is a diffeomorphic lygon
image V = ~ ( V ) o f
a k-dimensional
oriented po-
g c em °
N
A polygon V has an orientation which is transferred from V by the diffeomorphism ~ . If V is a polygon of dimension m then polygons of dimension
m-q . Segments
8V = ~ (D~)
is a sum of
[tl,t2] considered in particle
mechanics were l-dimensional polygons and their orientation was determined by the direction of the time axis. The boundary segment
(t2,tl)
of the
[tq,t2] is the sum of the positively oriented point t 2 and
the negetively oriented point tq. The formulae 7.10 and 7.20 are interpreted as integrals over O-dimensional boundary of the segment and the negative sign corresponds to the negative orientation of t I. The generalization of these formulae to more dimensional manifolds M will require the use of objects which can be integrated over boundaries of m-dimensional polygons.
Such objects are differential
(m-1)-
m-1
forms, which are sections of the bundle
A T*M of (m-1)-covectors
M. The vector space o£ (m-fl)-coveotors at x e M Differential
is denoted by
(m-~)-forms will be also called vector densities
and their values at x ~ M
in
/~T~M on M
will be called vector densities at x. The
above remarks lead to the following constructions. Let P q denote the tensor product oi
where q~ Q and x = ~ (q) ~ M. The collection of all spaces Pq form a vector bundle
fg : P
> Q whose fibres are equal Pq. The bundle
structure of P is that of the fibre tensor product of the bundle V~Q (adjoint to the bundle V Q c TQ of vertical vectors tangent to Q ) a n d the pull-back
~(
/~ T'M). Because Q is already fibred over M we may
82 treat P as a fibre bundle we have the commutative
over M with the projection
~=
~oSU
. Thus
diagram
OZ
~Q
P
13.2 M
similar
to the diagram
considered
primarily
5.2 in Chapter
Ii. As in Chapter
as a fibre bundle
II P will be
over M. The fibre
x ~ M will be denoted by Px" The restriction
of
~
~1(x)
over
to Px induces
the
mapping
13.3
~x
Definition the field.
: Px
~ Qx
: The bundle P over M is called the phase bundle
The space Px is called the phase space at x.
An element p * Px can be treated tangent
space T~Cp)Qx
Consequently, vectors
of
to the space
as a linear mapping from the
/~ T x~ M
we shall call elements
of vector densities
at x.
of Px vector-density-valued
on Qx" The value of p on a vector u ~ T%(p)Qx
co-
is denoted by
mv~
13.~
(u,p~q
The symbol
~
' ~I
~M " ~ / ~ Tx
can be interpreted
as the contraction
tor u with the first factor of the tensor product An element p e Px can also be interpreted from the space cotangent
/ ~ TxM of hypersurface
bundle T*Qx.
/hypersurface
volume
The value
element/
volume
of the vec-
13.1.
as a linear mapping elements
at x e M to the
of p ~ Px on an (m-1)-vector ~ ~ / ~ T x M
is denoted by
8S
13.5
The
2
symbol
-vector
<
' > 2 can be interpreted
n with
the second
In p a r t i c l e factor
in
~ ~Qx
15.1
factor
dynamics
13.6
of the tensor
m = 1 and
is trivial
as the c o n t r a c t i o n product
/°~ T x~ M = R 1
of the
(m-l)
13.1.
Hence
the second
and
Pq
=
T ~q Qx
Px
=
T*Qx
or
15.7
as in formula Let
5.1.
(x~),
%=l,...,m,
A=I,...,,N, dle
structure
be a coordinate
be a coordinate
system
in M and let (x~,~A),
in Q compatible
with the bun-
:
13.8
A)
A vector
system
density
=
(x
in M can be expressed
as a linear
combination
of
~-l)-forms
(-1)k-ldxl^ . . . . . . ^
15.9
where
the symbol
follows
that
A.%
eA
=
that the
coefficients
;9
]dxl~... A d x
~t-th factor
p e Px is a linear
dqOA@( !
The
=
m
~x ~-
means
each element
13.1o
Adx m
p~ t o g e t h e r
~ _Jdx I ...
has been
combination
dx m)
omitted.
p = P A e ~A where
.
~ x~
with the coordinates
It
(X "%, ? A ) o f
the
84
point
g~fp) define a coordinate system (x ~, ~A,pA~) in P. There is a canonical vector-density-valued
l-form
0 x on each
fibre Px defined by the formula
13.11
@x(P~
=
~xP
analogous to 6.2. Here
~ *xp denotes the pull-back of the first fac-
tor of the tensor product 13.1 from Qx to Px" If v is a vector tangent to Px at p then the value of
~x
on v is a vector density at x given
by the formula
13.12
-- <~Xx,~V,p>l ~ ~ T"Mx
analogous to 6.1. We define the canonical vector-density-valued
13.13
Co x
=
X
d @x
Here the symbol d denotes the exterior differential -density-valued/
2-form on P
of the /vector-
form on Px" This means that if ~ is an (m-1)-vector
at x then
13.14
2
= d<_~, e x > 2
where the differential on the right-hand side is the differential of the scalar-valued l-form ~ n , ~ x ~ 2 on Px" In the coordinate system
(x ~-, fO*,p~ we 13.15
and
have
e,: -- p~d?* ® (~
Jdxl..,
.,dxm)
85
13.q6
:
14. The symplectic ,sPace of Cauchy .data on a boundary Let 8V be the boundary of a domain V c M .
We define objects~QaV
paV,@~V, g o ~ corresponding to objects Q (t2'tl)
P (t2'tl)
0 (t2'tl)
CO (t2'tl) of particle mechanics. At each point we first define objects ~V ~V aV 8V Qx ' Px ' @ x ' 60 x derived from objects Qx' Px' @x' 60 x by projecting the second factor in 13.1 onto the hypersurface projection takes elements of elements of
/~T~ M
8V. This
/vector densities at x/ into
/~ T x* ( B V) /scalar densities on 8V at x/.
The space Qx does not contain the factor
/~ T*x M. Hence we take
~V
I¢.I
Qx
=
The space Pq = T~q Qx ® /~Tx M
Qx
"
becomes the space of /scalar-den-
sity-on- 8V-valued/ covectors on Qx :
14.2
p vq = mq Qx eAT x*(av)
and pSV =x
U p~V qc Qx
The projection
q4.3
rSV P x
is defined by the formula
:
Px
~ pSV x
86 for each (m-1)-vector ~ tangent to ment on
$V at x /hypersurface
volume ele-
8V at x /.
If coordinates of x the surface
(x ~) of M are chosen so that in a neighbourhood
8 V is described by the equation x I = const then
(x2,...,x m) is a coordinate
system on
$V.
It follows that each element
p~V e p~V is a linear combination
lZ~.4
p~V
=
p~V~A
where
fl~.5
~A
Hence
(~ A ,p,8V ) is a coordinate
=
d ~oA®(dx2~... i\ dx m)
.
system of Px v. It follows from 1~.~
that
14.6
8VIeA Prx ~ k )
Hence the coordinate
=
for k = 2,3,...,m.
0
expression of the mapping pr x~V is
( _ A,p~V T A
Px~T
,
where
bV PA
1L~. 7
The form @ ~xV 8V on Px d e f i n e d
=
I
PA
is a /scalar-density-on-8V-valued/ by the
formulae
differential
form
87
1~.8
~ a V . aV, x
aV~
O[x
=
P
aV
or
<w,@~V~
14.9
~V~\
~V w , p D V ~
where
grxav : Px8V
14-.lO
is the canonical projection.
The form ~ x 3V is defined by
CO ~ v x
14.11
The forms
@~V
av Qx
'
d 0 aV x
=
and CO " x aV are obviously projections
(~x onto the surface
of forms
~ x and
~ V. Coordinate expressions of ~ x8V and co aV x are
thus
14-.12
O~xv x
/We remember that
:
=
ple?A~(dx~...~,dx pA • d
@
dx2..,
m)
,
n d x m)
3 V is described by the equation x d = const./.
The above formulae closely resemble formulae 6.4 and 6.5. The only difference
is the factor
dx 2 ...
dx m
which is trivial in ca-
se of m=q. We now define spaces QaV and paV. The space Q3V is a space of sections
88 and is called the space of Dirichlet data on
BV.
The space pSV is a space of sections
14.15
~V
~ x
~
p~V(x)
and is called the space of Cauchy data on
e
PaxV
~V. These definitions must
be completed by specifying the class of the sections and choosing an appropriate topology. We do not make such specifications
in the pre-
sent paper except in one special case given in the next section. Elements of pBV are in a natural way covectors on Q~V. Let
14.16
8V
~ x
~
s(x)
~
DV Qx
be an element of QgV. The space Ts QgV of vectors tangent to QaV at s consists of sections
14.17
3V
~ x
BV
~ x
X(x) e
Ts(~Q x
~(x)
p~V
If
la.18
~
~
is an element of p~V such that
then the formula
14.20
( x, g> --
~(x(x), ~(x)> I ~V
defines a linear functional
<',~
on Ts QBV
/we remember that
89 (X(x), ~(x)~fl is a scalar density on
~V at x /. If topologies
and p~V are defined in such a way that all continuous
of Q~V
linear functio-
nals on TsQSV are of this type theathe space p~V may be identified with the cotangent bundle T*Q ~V. Assuming that this is the case, we have the canonical projection
~v
14.21
the canonical
1-form
: pSV
....
Q~V ,
@ ~ V and the canonical 2-form
co BV = d @~V.
The
formula 14.20 implies the formula
14.22
< ~, e~v)
=
I(y(x)'6}x~v~ I DV
where Y is now a section
14.23
representing
~V
~ x
~ Y(x)
~
T ~x)x
p~V
a vector tangent to the space of Cauchy data p~V at g
Similarly
,,
,,)
>
--
, (x~ ,,, , ~ ( x ) , ~ x
8V
2 1
9V
for two such vectors Y and Y. In the coordinate based on a coordinate
system
(x ~) such that
system (x ~,~A,pAq)
B V is described by the
equation x I = const, we have
14.25
=
I
plA(x ) " ~ A ( x ) d x 2 . . . A
dx m
8V and
d ~V
xo
m
.
90
where
the f o l l o w i n g
coordinate
expressions
~(x) 1#.27
Y(x)
.~
~cx~ ~ ~ x ~
~ ~p~ + ~q~x~ ~
, i=I ,2;
are used. The r e s t r i c t i o n
q~.28
to
M
3 V is an element If
~
to
~ x
of
M
3V
milarly
q~v d e n o t e d
and p r o j e c t i n g
e
Qx
I
by s I~V.
:
~ x
~ i~V will denote
s(x)
,
is a s e c t i o n of P
44.29
then
of a s e c t i o n
~
~(x)
an element
its v a l u e s
e
of p S V
Px
o b t a i n e d by r e s t r i c t i n g
from Px to PxDV by means
of pr ~x V . Si-
if
~.30
Is a s e c t i o n will denote
M
s x
of the bundle the
section
,
Y(x)
of v e r t i c a l
e
TP x
vectors
tangent
to P t h e n Y I D V
91
obtained
by restricting
of the tangent
Y to
~ V and projecting
~V mapping pr x .
. The section
its values by means
Y I8V
is thus the object
of the type I#.23 and may be treated as an element ~SV
evaluated
on the vector
av, O
1#.32
of TP 8V. The form
Y I ~ V gives
:
SV
~bv
Similarly
14.33
x/1 '~V r
=
\ ( Y ( x ) A Z(x), ~ x > I ~v
if
I#.3#
V
is a section
~
x
~ Z(x)
such that Y(x)
and Z(x)
~
TP x
are vectors
attached
at the sa,!
me point
~(x~• ~ Px for each x ~M.
Formulae
I#.32 and 1#.33 are ana-/~J ~ f
logous
to formulae
7.21
and 7 . 1 1 .
In analogy to formulae
7 . 2 0 and
7.10 we write M
14.35
~v 4 ¸
~V M I#. 36
60 8V
= f(~Ox ~V
15. Finite domain description Finite
domain description
of dynamics of field dynamics
can not be presented
92
with a rigour matching the finite time interval description of particle dynamics.
Field dynamics is based on the theory of partial dif-
ferential equations which is not as well developed as the theory of ordinary differential equations used in particle dynamics. ly we give only heuristic considerations
Consequent-
as an introduction to a ri-
gorous infinitesimal description of field dynamics given in the next section. We begin with the discussion of electrostatics
in a 3-dimensio-
nal manifold M which is assumed to be a riemannian manifold with a metric g. The configuration space Qx at each point x e M of values of the electrostatic potential
~.
is the space
Hence Qx = R1 and Q is
the trivial bundle Q = M × R 1. The value of the electrostatic potential ~
together with coordinates
(x ~) in M define a coordinate system
[x~,~) , ~ = 1,2,3 in Q. The first factor in the formula 15.1 is tri& $ vial in this case. It fellows that Pq = /~T~M, Px = R ~ / ~ T ~xM and
p : ~1× / k ~ M
~lements of Px are thus pairs (%p) where ? ~
the value of the electrostatic potential at x and p ~ T ~ M value of the electrostatic coordinates ( x ~ , ~ )
i~
is the
induction field at x. Corresponding to
in Q we have coordinates ( x % , ? , p %) in P. In terms
of these coordinates
15.1
@x
: d W ® ( P ld~d~3 + ~2dx3Adxl + p3dx~dx2)
and
dJ x 15.2
+ (d~3AdT) ~(dxIAdx 2) If the coordinates
(x~) are chosen in such a way that the boundary 3V
of a domain V is described by the equation x I = const, then is a coordinate system in the space P~Vx " Coordinates
?
(~,pl)
and pl
93
are interpreted
as the value
of the induction
on
of the potential
8 V /interpreted
and the normal
as the surface
component
charge density
on
~v/. The equations
of electrostatics
~5.3 where
p ~
with the metric
15.4
sity.
=
- ~
is a 3-form /scalar density/
In terms
and
* is the Hodge
operator
g, and
Vp
~
:
= .V T
is the exterior differential
associated
where
are
of coordinates
,
representing
the above equations
a fixed charge denread
:
~X ~
15.6
~P~ ~x m
=
- ~
where r is the scalar function equivalently,
--
~
A?
~=
r dxhdx~dx3
r = , ~
, or
.
15.5 into 15.6 we obtain the Poisson
15.8
where
defined by the equation
by
15.7
Substituting
~Fg-~r
~ g ~ ~ i s
=
- ~
the Laplace
equation'
r ,
- Beltrami
operator
associa-
vo
ted with the metric Poisson
equation
g. Applying
the formula
15.8 we generate
15.3 to solutions
dynamically
admissible
of the
sections
of
94
P.
It follows is the normal on
from the formula
derivative
8 V determines
cifying the value
of ~ .
15.5 that the normal Specifying
the normal
component
of p
component
of p
thus the Neumann data for the Poisson
equation.
of
data.
~
on
8V determines
the Dirichlet
Spe-
We con-
clude that the space pSV which we called the space of Cauchy data is the space of combined Dirichlet Dirichlet
and Neumann data on
of the Poisson
equation.
tions to the boundary D ~V is composed
space Q~V of Dirichlet
to any solution admissible
~ V form a subspace D ~V of pSV.
regular
of Dirichlet
sec-
The subspace
and Neumann data
domain V it can be shown that if the
data is the Sobolev
submanifold
(PaV,co~V).
of dynamically
combined
to solutions.
For a sufficiently
space
Restrictions
of
In general
BV do not correspond
of those special pairs
which do correspond
is a lagrangian
and Neumann data.
~ V then D ~V
space H 1/2 on
of the infinite
dimensional
symplectic
In this case
15.9
paY
=
T~Q~V
=
H 1 / 2 × H-1/2
which means that the space of Neumann data is the Sobolev
space H -1/2
dual to the space H 1/2. A lagrangian
submanifold
of an infinite
and maximal
We use the Green's
formula to prove that D aV is isotropic.
M
~ x
M
x
and
15.11
~ (~(x),p(x))
sense /of.
symplectic
space is isotropic
15.10
in a certain
dimensional
[8] and
~ Px = R 1 × /~ Tx* M
Let
[57]/.
95 be sections of the bundle P. We denote by ,~ the vector tangent to pS~ at the point q5.3 and 15.4 and if
(T,p)J~V. If
((~T,~p)
{~,
~p) [ 8V
(T,p) satisfies equations
satisfies corresponding homogenous
equations
15.~3 then ( ~ + ~
V(,~p)
= o
!,p+ ~ip) satisfies again the inhomogenous equations 15.3
and 15.4 which means that the vector Y is tangent to D 8V. It is obvious that all vectors tangent to D ~V can be obtained in this way. Let Y be a second vector tangent to D 8V corresponding to the section
Applying the Stokes formula to 14.26 we obtain
l!~¢~.~p¢~l ~V
V
15.15
V
V
- ~®¢x).~pcx~I_-
96
For any pair of functions re the symbol by the metric
f,h on M we have
~ i ) denotes
VfA
the scalar product
of covectors
defined
g. In local coordinates
7 f A ~Vh The above expression
g~
--
is symmetric
15.16
Df 8h dx~dx2Adx 3 . D x ~ 8x ~
in f and h. This implies
-
,,
___
that
o
Consequently,
15.17
<,~,,,,,,~,~v~
which means that D 8V is isotropic. Green's
formula
for the Laplace
Finite dimensional tropic
submanifolds
apply to infinite valent
criterion
sional
cases
Formula
corresponding
Neumann data.
space tangent
to the fibre
does not
which does apply to infinite
submanifold
N is lagrangian
of
if and only comple-
D ~V is the image of
given Dirichlet
15.8 and subsequently
It follows
dimen-
to N has an isotropio
here the submanifold
equation
as iso-
There is however an equi-
a section of the bundle p~V over D ~V because can solve the Poisson
are defined
This definition
submanifolds.
of N the space tangent
In the case considered
15.q5 is the familiar
submanifolds
dimension.
for maximality
: an isotropic
if at each point ment.
dimensional
o
equation.
lagrangian
of maximal
_-
data one
calculate
the
that at each point p % V ~ D~V the
p~V
complements
to D 8V. Both D 9V and the fibre are isotropic.
the space tangent
We conclude
that D ~V is
lagrangian. As a second The manifold
example we shall use the theory of a vibrating
M is now a 2-dimensional
pseudo-riemannian
manifold
string. with
97
a metric
tensor
g of signature
configuration
bundle
tic potential
is replaced
equilibrium
is the trivial
position.
are formally
~+,-).
and the same symbols.
bundle
of the string from its
that the construction
We use the same constructions The coordinates
time t and the distance
(xl,x 2) are interpreted
tensor has components
15.18
and g = det(g~y)
=
°1
o
-lj
= -I. The dynamical
p
equations
are
: ~V T
15.19
Vp
=
0
.
In terms of coordinates
~x ~ 15.20 9P~
=
0
,
or
pt
=
8~ at
~
x
P
=
9x
'
15.21 ~pt Dt
It follows
that
~(t,x)
+
%pX 9X
satisfies
=
of Q~V and p~V
of coordinates
x along the string respectively.
(g~)
example the
Q = M × R 1. The electrosta-
by the deflection
It follows
the same.
As in the previous
0
.
the wave equation
as the
The metric
98
15.22
82~ t2
Also for this example compatible
82~ ~ x2
0
.
-
one can prove that the space D $V of Cauchy data
with dynamics
is a lagrangian
submanifold
of p~V for a wi-
de class of domains V. To prove that D ~V is isotropic
one uses the
Green's
of maximality
formula
is much harder. the Dirichlet
as in the previous The argument
problem
example.
The proof
used for the elliptic
is not well posed.
case fails because
The general
idea of the proof
is however
the same. The space pSV is split into two complementary
components
and it is shown that the dynamics
component liptic richlet
of the Cauchy data from the first
equations
of electrostatics
data and Neumann data.
more general different
splittings
mixed boundary
determines one.
the second
In the case of el-
we split the Cauchy data into Di-
In the case of hyperbolic
must be used.
Such splittings
value problems
be used in the proof
of maximality
D
D
equations
correspond
to
which are well posed and may
of D BV :
N
V D,N
D,N
The symbols each piece
D - Dirichlet, of the boundary
ated by similar methods.
D,N
N - Neumann
indicate
of V. Some non-compact
the data given at domains
can be tre-
The Cauchy problem
D,N
is one example.
We return to this example
later.
The following
charac-
99 teristic
boundary
value problem t
"x
although
well posed fails to establish
maximality
can be shown that D 8V is not lagrangian characteristic.
whenever
We limit all considerations
of D ~V. In fact it a piece
to domains
of
8 V is
V for which
D BV is lagrangian. The lagrangian
subspaces
milar to the composition
D BV exhibit
properties
Let a domain V o be subdivided
expressed
law
7.7 and 7.8.
Vi, i=l,2,...,N
:
: f
45.23
si-
Vo
composition
there
properties
by formulae
into smaller domains
~ We define the following
composition
is a sequence [p
~V.~ N l)i=q
, p
~V. V. ~p l
such that i f V i and V have a common IY ~Vi ~Vj O wall then p and p~ are equal on tha~ I
o
i=1
common wall for i,j = 0,1,2~...,N
In the case of M = R 1 this law reduces
to the formula
shown in many cases that
~V o
15.2~
D
=
N o
i=1
8V i D
7.7.
It can be
100
Lagrangian subspaces D ~V are generated very often by generating functions W 8V defined on constraint subspaees caVe QaV. In the case of electrostatics functions W aV are defined on the full space of Dirichlet data of class H 1/2 /there is no constraints/ and can be expressed in terms of Green's functions. The composition property q5.2~ implies the composition formula for generating functions : N
15.25
w~V°(q 9V°)
=
~
W 9Vi'iq9Vi
i=q where ~ q ~
i=I is a stationary point of the right-hand side. The { ~Vi] N right-hand side is treated as a function defined on sequences ~q ~=1 such that if ~V i and
~Vj have a common wall then q~Vi and qgVj are
equal on that common wall for i,j=O,1,2,...,N. /The boundary value q aVo on %V o is thus fixed/. In the case of M = R I the above formula reduces to the formula 7.25.
16.
Infinitesimal description of dynamics
Considering a domain V shrinking to a point x one obtains infia i pix' 0 ix' gOx' i Dxi which are in ^ sense limits of nitesimal objects Qx' corresponding objects Q a V
Definition:
pSV
09V, o o 8 ¥
DSV.
The space Qxi = j1 x(Q) of l-jets of sections of the
configuration bundle Q at x is called the infinitesimal configuration space at x. The 1-jet bundle Qi = jq(Q) is called the infinitesimal configuration bundle.
If (x ~, ~A)
?*, 6.1
is a coordinate system of Q then a coordinate system
of Qi is obtained by taking
,f*
--
101
and A
as coordinates
16.3
of the jet s = jlqO(x) of the section !
M
~ y
T(y)
>
Intuitively the jet j 1 ~ ( x )
(y~,TA(y~,
=
is a limit of
c- Qy
•
the boundary values
~I~Va
i Qsv when the domain V shrinks to x. In this sense Qx is the limit of Q~V. The canonical identification 8.6 applies also to the present cai se. Thus we can represent vectors tangent to Qx as jets of vertical vector fields in Q. Let
16.4
M ~
, X(y)
y
be a section of VQ such that X(y)
~
TT(yjQy
is a vector tangent to Qy at
~(y).
i The jet jqX(x) represents a vector u tangent to Qx at the point s = = j1~(x).
Intuitively the jet jIX(x) represents the limit of the boun-
dary value X I $ V which is a vector tangent to Q~V at the point If in the coordinate system the field X is described by
16.5
x(Yl
=
xA(y)
~TA
then the vector u is described by
16.6
where
u
=
u
A
~
+
u
A
~I~V.
102
U
A
x A ( X) .
=
16.7 A
u~
~X A
:
(x)
.
x m
It is obvious
that each vector
tangent
i to Qx may be represented
in this
way. The construction cated in the general
of the infinitesimal
phase space is more compli-
case than in mechanics.
that the infinitesimal
phase
One expects
space at x is related
intuitively
to the jet space
J!(P). Let there be a section x---
16.8
According
M
~ y
, p(y)
to 14.20 the evaluation
e
Py
of the covector p l S V ~ T * Q ~V = P
8V
on the vector XI~V equals
16.9
<xl~v,plmv>
=
<X, p ~ I 8V
The surface using Stokes
~6.1o
integral theorem
on the right can be replaced
by a volume
integral
:
<Xl~V,pt~v~
=
I diKX'P2 q V
The limit of the right-hand
side /per volume
density d i v ~ ( y ) , p ( y ) ~
I y=x
tion of the evaluation
for jets
" We take this expression
The above definition
associates
sity-valued
<. , g ~
covector
element/
is the scalar as the defini-
:
with each jet g = jlp(x)
i The situation on Qx"
a scalar-den-
is however different
103
than in particle mechanics where the evaluation for jets was defined by the formula 8.8. In the general case many different the same covector.
jets define
We illustrate this phenomenon using local coordi-
nates. Let the section 16.8 be described in coordinates by
~A = ~ACy) 16.12
We define coordinates ((~OA,p~ , t~9A ,pA/~) '7~ of the jet g = jlp<x) by
Cx , %
^,~
=
P~p-
f~(xl
,~ x r-
.
The formula '16.11 reads
=
div
XAp~
~J~...~dx
m
=
16.13
=
~ ( x A PA%) dxl^''" ~ dxm
+
We see that only combinations components p ~
=
PAZ) dxlA'''A dx m
~ A = P~%
/summation over
~
!/ of
appear in the result. Two jets having the same value
104
of
i ~A define the same covector on Qx" We shall consider these two
jets equivalent. This equivalence relation leads to a quotient bundle ~l(p) of the bundle jl(p). The equivalence class of a jet jlp(x) will be denoted by ~lp(x)~ Jq(P). The "generalized jet" ~lp(x) /cf.
[54]/
does not contain the information about all derivatives of p~ but only a about the divergence ~A = ~-x~ PA" It follows that (x%~ ~ A p~, ~ ' ~ A ) may be used as coordinates in jd(~. The fibre of ~l(p) over the point xe M is denoted by ~ ( P ) . A
PA'
It can be parametrized by coordinates
(~A,
%, %)"
Definition
: The quotient bundle pi = ~q(p) is called the infi-
nitesimal phase bundle and the fibre Pxi = ~ ( P )
is called the infini-
tesimal phase space at x.
The reason why infinitesimal states of the field are equivalence classes of jets and not jets themselves is that only the normal component of p e Px entered in the construction of the boundary-value space p~V . We denote the canonical jet-target projections by
bQ and
respectively. The jet prolongation of ~C is denoted by ~i.
Up
We have
thus the commutative diagram Lp
16.1LI
pi
~ p
Qi
~ Q
.L
-
bQ similar to the diagram 8.5 in Chapter II. In terms of coordinates (x~, ~ A p~, ~,A ~gA) the above mappings are described by formulae analogous to 8.19 :
(
:
105
16.15
o~6iCx~,
A ' P~A ' ~ 'A~ g A )
The formula ~z. u , g > = < u , g > class ~
=
(x%,~A, ~A)
, where g 6 J
is
.
a representant
of the
~ C~ , defines a mapping from Pxi into the space of scalar-
i -density -valued covectors on Qx :
i
16.16
Px
a
N
g
X
•
The above formula is analogous to 8.9. It follows from the coordinate expression 16.13 that the mapping (Xix is a diffeomorphism. of
i may Px
thus be treated as scalar-density-valued
Elements
i covectors on Qx"
A / ~ T x M of scalar densities at x e M is one-dimensio-
Since the space
nal it follows that the space of scalar-density-valued
covectors on
i Qx has the same dimension as the cotangent bundle T*Oi~x" Although the space
(TWQ~) @ / ~ T~x M
is not the symplectic manifold in the strict
sense there are canonical /scalar-density-valued/
1-form and 2-form
defined by formulae analogous to 6.1 and 6.3. The pull-backs by of these forms are denoted by
0 ~- and dO i
X"
Similarly as in p&rticle
mechanics we give an alternative construction of forms by differentiation of forms
i c< x
0 i and CO i X
X
~ x and COx . Let v, w be vectors tangent
i to Px and attached at the same point• Using the canonical identification similar to 8.10 we represent v and w by classes of jets of sections M
~ y
~ Y(y)
c- TPy
M
~ y
~ Z(y)
~
,
16.17 TPy
We choose these sections in such a way that for each y vectors Y(y)
106 and Z(y) are attached at the same point. tion of ~ x
i
that
i @ x and 0 ~
It follows from the defini-
satisfy the following equalities
di~< Y(y), 0y>1 Coi
< v A w, ¢Ox~>
y=x
=
16.19
div
~6.2o Q6.2Q
=
i div1 v
~(YI~V)~(Z!~V)'0JDV>
=
,
I div 1
"
V We see that forms
0 i and ~P i are limits of forms
0 ~V and CO ~V when
V shrinks to the point x similarly as v and w are limits of YI~V and Z!~V. The above remarks justify the following notation
16.22
0~
: Mdiv @~
16.23
CO i x
=
Mdiv 60
,
x
similar to 8.14. Now we find the coordinate exT~rssions for
~
Y(y) and Z(y) are expressed by
~Cy)
=
yA(y)
~T A
~P~
and
°Oi'x If vectors
107 16.2a
z(y) = zA(~Y) ~-~ + ZA(Y)~PA then the c o r r e s p o n d i n g
jets are
~yA(x )
YA(X)~p~
~ x ~, 3~pA
+
x~
16.25
~ PA#
%
jlZ(x) : Z~(x) 8 • ~?~ -
Consequently
coordinate
-
+
Z~(x) 8 ~p~ ,
expressions
+
8ZA(x) ~ ~.~ ~?~
for equivalence
+
~x ~ 8 P A~~"
classes
v and w
are
~X~(x) +
~x,~ 16.26
~Z~(x) +
•
~x ~ The
equation
~x~
~A
16.18 reads
x%
"" "
y=x
16.27 --
Comparing
1~.~
These
dx m
.
with 16.26 we see that
oix
Similarly
16.29
16.27
(Y~ p~ + YA ~ A ) d x ~ . . . A
=
(~f~
~ ?~) ® d x l ^ . . .
+ p d
~dx
m
we show that
CO x
expressions
+ dPA~\d
are formal
derivatives
®dxlA... A d x m .
of expressions
13.15 and 15.16.
108
The space and the form
(pix' oo~) together with the projection 0 2 is a scalar-density-valued
~ x i:
pix
~ Qxi
special symplectic space.
This means that for each m-vector ~ tangent to M at x /volume element i at x/ there is in Px a special symplectic structure in the ordinary sense defined by the forms
~,
~ x ~ 2 and ~ ,
In particle me-
chanics the standard volume element ~ = ~-~ was used to convert all scalar densities to ordinary scalars. The same can be done in any field theory if some standard volume element is present at each point x e M. In the present approach we do not assume the existence of standard volume elements because of intended applications to General Relativity. The unit volume element in General Relativity
6.3o
•
t
=
where g = det(g#~)
~
A
~
A
~
3
A
can not be used for this purpose since the metric
g is itself a field variable and therefore m is not fixed. Since volume elements at x form a q-dimensional <~' ~ 2
space the forms
calculated for different volume elements differ from each
other only by a numerical factor. The same applies to the forms ~ m , c O ~ . It follows that lagrangian submanifolds in pi defined with X
respect to different symplectic forms ~ m , d o ~
2 are the same and will
be called lagrangian submanifolds of (pi coi~ \
X'
X /
Also proper functions "
and generating functions of lagrangian submanifolds These functions will be scalar-density-valued
can be introduced.
and are defined by for-
mulae 6.7 and 6.10. It must be remembered that the exterior derivative of a scalar-density-valued
function is a scalar-density-valued
ferential form. An alternative,
dif-
equivalent definition of a proper fun-
ction is given below.
Definition
: A scalar-density-valued
function S on N is called
a proper function of the lagrangian submanifold N of (Pix,ODi)if for
109
each volume element _m at x the function < _ m , S ~ 2 of N with respect to the special symplectic <~' ~
2 and ~ m , g o ~
is a proper function
structure defined by forms
2" A scalar-density-valued
function S is called
a generating function of N if ~ ~ , S ~ 2 is a generating function of N with respect to ~ ,
Definition is specified
@~2
and ~ _ m, C O x i~ 2
"
: We say that the infinitesimal
if a lagrangian submanifold D i c P i X
X
dynamics
of the field
is chosen at each point
X ~ M.
The condition ~Ip(x)~ Dix for a section M g x - ~ p ( x ) the field equation.
The field equation is actually a system of l-st
order partial differential
equations.
We assume that for each x & M the infinitesimal nerating function noted by ~ x "
Both
~x" ~x
dynamics has a ge-
The corresponding and
proper function of D i is deE ~ x are defined up to an additive constant.
In all examples
of physical field theories
however natural
choices of unique generating functions
There is a unique distinguished describes
the physical vacuum.
is the one which vanishes tinguished plectic
6 Px is called
considered by us there are of the dynamics.
state of the field in each theory which The distiguished
generating function
on that state. We shall always use such dis-
generating function also with respect to other special sym-
structures.
The family
~x~
defines the scalar-density-va-
lued function
16.31
~] : Qi
which is called the Lagrangian. we may write ~
16.32
~
ATOM
Using coordinates
as
~ =
L dx~... ^dx m
(x ~, ^ A
110
where L = ~ ,
Z~
16.33
m
_
2 and
=
~x I A...
A
The function L is scalar-valued.
~x m
The generating formula 8.24 assumes
the following form
+ p d1 .j
m
16.34
..A dx m
---
equivalent to the formulae
9 % x ~pA
16.35
The formulae 16.35, equivalent to Euler - Lagrange equations,
are coor-
dinate expressions of field equations. It is usual to postulate field dynamics in infinitesimal form from which the dynamics for finite domains is derived. The procedure is similar to that used in particle dynamics /formula 8.22/. We introduce the class of dynamically admissible sections of P which are solutions of field equations. The Cauchy boundary values of these solutions form subspaces D~Vc pgV :
there is a section M a y - ~ p ( y ) ~ P y
q6.gro
DgV = I P S V ~
pSV
)
such that pI~V = p~V and ~IP(X)~Dx I for x ~ V
If the composition property 15.24 holds then the above formula
"
111
can be considered subdomains
as the limit of formula 15.23 when the number of
V i tends to infinity and their dimensions
tend to zero.
In general the proof that the formula 16.36 defines submanifold
of p~V is difficult
few special oases. However,
a lagrangian
and has been carried through only in
the proof that D ~V is isotropic
is rela-
tively easy and is based on a non-linear version of Green's formula analogous
to the formula 15.~5. Let
16.37
(~,~,
be a two-parameter
x)
--
•
p~,~(x)~
family of dynamically
j~lP~,~(x}~D~- For each x c M
Px
admissible
denote by Y(x)
sections of P, i.e.
and Z(x) vectors tangent to
the curve
at
~
at
"
P~,o(X)~ Px
~
PO,~ (x) ~ Px
= O and to the curve
~ = O respectively.
We see that YIgV and ZIBV are vectors tangent
to the curve
• ( p ~ , o t~v) ~ P ~v
at ~
at
= O and to the curve
~ = 0 respectively.
These vectors
U s i n g 16.21 and 16.19 we obtain
are obviously tangent to D 3v .
112
16.38
((~lav)A (z I~v), co~v~
~(T1Y(x)Aylz(x), COx~ ,i 1
=
V The integrand tors tangent
at
at
vanishes
~ly(x)
for each x e M since
and
~']ZQx)
are vec-
to the curve
•
~lpm,0(x)
~
Dxi
~
J Po,~(x)
~
Di x
~ = 0 and to the curve
~ = 0 respectively,
and D i is a lagrangian
submanifold
of pi
X
16.38
((Yl~v)^(zl~v),co~v~
=
The formula
16.38 may be considered
as a generalized
which in the case of electrostatics If the submanifolds solution
sition law for generating chanics
= QBV~ /
can be proved.
o
.
reduces
D aV are lagrangian
of field equations
functions
In the simplest
as follows.
--
IZx(Jq~9(x)) v
where for each
~6.~o
M
~BV~ QSV the section ~ x
and if there
is a unique
~(x)
to 8.26 in particle
case of no constraints
W 8V be defined by
w ~ V ( ~ aV)
formula
to the formula 15.17.
analogous
Theorem 7
16.39
Green's
for each Cauchy data in D 9V then a compo-
this law can be formulated
Let a function
Thus
X °
~ Qx
me-
(C aV
113
is a stationary section /in the sense of the calculus of variations/ of the right-hand side, such that
The function W 8V is a generating function of D 9V.
The formula 16.39 can be treated as the limit of the formula 15.25 when the number of subdomains V i tends to infinity and their dimensions tend to zero. Euler -Lagrange equations 16.35 imply that stationary sections are precisely the projections by ~
of dynamically admissible sections.
Due to the lack of constraints the above theorem can be stated in terms of proper functions
:
Theorem 7' Let a function W ~V be defined by
V where for each p aV~ D~V
16.43
M a x
the section
~ p(x) ~ P
x
is the unique dynamically admissible section of P such that pIaV = = paV. The function w ~ V is proper function of D 8V.
The proof is practically the same as in mechanics.
Proof 16.44
:
Let q~
p~ Vt_~ L-~,;
D~V
114
be a curve in D 9v. There is a unique mapping
16.45
(~,x)
such that for each
16.46
~
- - ~
p~Cx) ~ P
the mapping
M ~ x
is the dynamically
x
~
admissible
16.47
p~(x) ~ P
x
section satisfying the boundary condition
P= i aV
=
p
8V
Denote by Y(x) the vector tangent to the curve
16.48
at
~
~
= O. Then ~ly(x)
>
p=(x) ~ P x
is the vector tangent to the curve
. ?Ip.(x) ~ at
o i
x
~ = 0 and YIaV is the vector tangent to the curve 16.44 at
~
= O.
Now d
-
w~V
d ~ --
-_
(P~
V(~))
d I~x(~lpz(x)) I d'~ ~ =0 V
=0
(~1
--
( ~l~(x)'d-~x)l
=
=0 V
16.49 =
V
I~flY(x), ~ i > I
=
V
I~
=
'~V
which completes the proof.
Idiv'Y(x), V
=
Ox~ 1
=
115
Example
q
Equations of electrostatics 45.5 and 45.6 define a 4-dimensional
Dxi in the 8-dimensional i n f i n i t e s i m a l phase space pix described by coordinates ( ~ , p ~ , ~ , ~ g ) where ~ = 4,2,3. The submanifold
submanifold
Dxi i s described by # equations
:
pX=
& g,~, ~,,
46.50 ae
Restricting the form
:
x
=
- #~
@i = C~d~+p~d~m)
{-~rCx)
® d x q A d x 2 d x 3 to Dix we obtain
VT(-~r~x)d~+ g ~ d ~ ) @
dx4^dx%dx 3 =
46.54 =
d
It follows that D i is a lagrangian submanifold generated by the Lagranx gian
Example 2 The Lagrangian
Z ( x~, ~9, ~gm)
where ( g ~ )
=
~F~o (F(~)- ~m2lj9 2 + ~g%/~n t ~ ) dxOdxqAdx~dx3
is a pseudo-riemannian metric tensor in the four-dimensio-
nal space-time M generates the dynamics
16o53
116
which
corresponds
to the scalar field theory with field
16.54
([] + m2) T
The infinitesimal = 0,q,2,3, equations
17. Hamiltonian
phase space described
description
rizontal
ohase
with different
no longer canonical
in terms
special
but depend
symplectic
on the choice
associated
with different
connections
description
space-time
connection
Before going further
into details
cussed in the present
over M.
hamiltonian
is it-
to General
Our
Relativity.
we have to draw the attention
generating
One of these objects section.
in space-time
as a fixed structure.
of the terminology
are two different
"Hamiltonians".
connection
by applications
of the reader to an ambiguity
called
is not
This is not true in the relativistic
and cannot be treated
There
descriptions
is a distinguished
self a field variable
is thus motivated
in
in Q induced by the given affine
M. Thus there
of such a theory.
a ho-
field theory in given pseu-
where the affine
literature.
in the
to the "rest frame"
in Q. This generality
theory of gravitation
generality
which are
M. In this case Q is a tensor bundle
is a distinguished
stru-
description
of a connection
of hamiltonian
in the case of the relativistic
connectio~in
symplectic
structures
at each point of Q, analogous
space-time
was obtai-
in Q enables us to define
Thus there is a whole family
There
special
X
mechanics.
do-riemannian
by
X
spaces pi. The hamiltonian
bundle Q. A connection
m-vector
necessary
D i described
of the Lagrangian
/scalar-density-valued/
in infinitesimal
configuration
The submanifold
(~,p~,~,~g),
of dynamics
of dynamics
ned by using canonical
is associated
by coordinates
is 5-dimensional.
The description
ctures
=
is now qO-dimensional.
16.53
equations
This object
which appears
functions
in the
which are
is the Hamiltonian
dis-
is little known by physi-
117 cists. It has been introduced in geometrical formulations of the calculus of variations /cf.
~0], [17], ~4], ~6]./. The name "Hamiltonian"
is usually given by physicists to the other object which we call "energy" and discuss in the next section. The ambiguity arises because in the case of dim M = 1 /particle mechanics/ the Hamiltonian and the energy coincide. Each infinitesimal phase space Pix is already fibred over Px" The fibration GCh ~X
:
P~----~Px is given by the jet-target projection
Op
Each connection in Q determines a diffeomorphism
h
17.1
C4x
:
pi x
T~Px ® ~ T x* M
which will be constructed~ later. Each element of pi may thus be treaX h ted as a / s c a l a r - d e n s i t y - v a l u e d / c o v e c t o r on Px" U s i n g ~ x we p u l l ^h ^ h -back canonical / scalar-density-valued/ forms ~ x and dO x from T*Px
®AT~x M to Pi.x We shall prove that in terms of coordinates
(~A p~, ~ , ~A) the result
ex
17.2
17.3
are
-
C~xh*'hOb x = (d~AAd~ A - d ~
dp
)
dxt,...*dx
m
dp~)@dxl~...Adx m.
The above formulae are analogous to 9.8 and 9.9 in mechanics. The equality 17.3 shows that C~x'h*~'hcOx= dO xi . Similarly as in particle mechanics, if the generating function of the dynamics D xi with respect to the above special symplectic structure exists then, taken with the opposite sign, it is called the Hamiltonian at x and denoted by
~x"
are denoted by - ---~x" The family valued function
The corresponding proper functions (~x}
defines the scalar-density-
118
which is called the Hamiltonian. For any coordinate
system
(x %) in M the value of ~ x ~
and ~ x
respectively)
is proportional
to the scalar density d x ~ . . . A d x
corresponding
proportionality
coefficient
respectively)
:
17.5
~'~
=
H(x, ~pA,p~)@dx~,
m. The
is denoted by H x ( H and H
--X
...
,a dx m
•
The generating formulae for D Xi are obtained from fl7.2 :
( ~ Ad ~9A -
1
A i, ?~dPA)®dx
A ... ~ d x m
=
-d~x(~°
A
~ ,pA)
=
17.6
_- _%(?~,pb~dxl~...
~ dx m
This is equivalent to equations
H(x%, SOA,p~) 17.7
~P~ analogous to Hamilton canonical equations
in mechanics /see
[9], [17],
[2~], [26]/. Now we give the construction structure
in coordinate-free
of the above special symplectic
language.
We first define a differential
m-form 1Y on P depending on the connection
in Q. Let ~ be an m-vector
tangent to P at p~ Px of the form
17.8
~
:
Y/, ~.
where Y is a vertical vector in P and fi is an (m-fl)-vector. the value of ~
on such m-vectors by
We define
119
12.9 where ~ =
~
is the p r o j e c t i o n
tor tangent to P such that sum of m-vectors not unique. of
l> 2
=
~
,
of ~ onto M. Let now ~ b e
any m-vec-
= O. Then ~ can be decomposed
of the form 17.8. This d e c o m p o s i t i o n
into the
is in general
We use the formula 17.9 to define by linearity the value
IY on such sums. We shall show later that this d e f i n i t i o n does not
depend on the choice of the decomposition the value of
~
on the subspace
tions onto M. This subspace the value of ~
of ~. This way we defined
of m-vectors
which have zero projec-
is of codimension 1. It remains to define
on a single vector which has a non-zero projection
to M. This vector will be roughly speaking the horizontal vector. ever,
in general the connection in Q does not determine
in P. We adopt the following definition
Definition ~
:
How-
a connection
of horizontality
An m-vector ~ in P is horizontal
on-
:
if its projection
onto Q is horizontal with respect to the given connection.
We set the value of ~
on horizontal vectors
equal zero. Since
horizontal vectors do not form vector subspaces the correctness this d e f i n i t i o n must be demonstrated. The above construction in Q. However,
of
We return to this point later.
of I~ depends
on the choice
the value of b~ on m-vectors
of a connection
which have zero projections
onto M does not depend on the connection. We denote the differential
17.1o
g2
The form
~
=
of
l~ by
d b~-
is used in the construction
~
:
•
h of ~ x
@
Let m be an m-vector tangent to M at x / a volume 1 For each jet g = j p(x)
of a section
element at x/.
r
120
17.11
M
~ y
~ p(y)
E
we denote by ~g the lift of ~ to the section.
P
Y
Thus ~g is an m-vector
tangent to the section and
17.12
~_~g
Let ~ be an element of Px =
--
m
.
P . For each vector Y tangent to Px we
set
17.13 \
where g is any representative We prove the correctness dinate expressions coordinate tors
~x
of l~ , ~
2
;
of the class ~. of the above definitions and
~hx. Let (x ~, @ A )
system of Q in a neighbourhood
by giving coor-
be a geodesic
of Qx" This means that vec-
attached at points of Qx are horizontal with respect to the
chosen connection.
Let (x ~, ~ A , p ~ )
be the corresponding
coordinate
sys-
tem in P. The reader may easily show that the form
17.14
=
PAdx A . . . A
...Adx m %
/the factor dx ~ in the product dxlA...A dx m has been replaced by d ~ A / satisfies
all requirements
17.15
Y
stated in the definition.
=
If, for example,
~q~A
and
17.16
fi
=
(_1) "%-1
"~ 8 xI
A...~
• ~ ,~ x%_1 A ~ - - ~ ^ . . .
A
'8 "8 xm
121 17.17
as is required in the definition 17.9. One shows in a similar way that the form 17.14 satisfies the remaining conditions stated in the defirich. It follows from 17.14 that the coordinate expression for
~
in
geodesic coordinates is :
=
q7.18
Let
d~,~,,dx~...,,
d2¢...
%
A d~ m
.
(~0A ,_D ~ , ?f,pA/~) A % be coordinates of the jet g = jqp(x) and let 9
17.19
m --
=
b. ~-----~A...A m ~x ~x
then
17.20
~g
=
b,/~xQ+ ~A ~+p~q ~p~)A-.-A(~xm+ ~Am ~A+P~m~ ~)-
If 17.21
Y
=
~_
,
AA
8
%
9
then
12.22
=
b( *pk
- BA(~%
)
•
The above equality proves that the right-hand side of 17.13 does not depend on the representant g = ( ~ A ~ p ~ A , p ~ ) o f A p , T~A=
the class
~=
p ~). This proves that the definition of o{ x is cor-
rect. In order to be able to give an explicit coordinate expression for °
( ~ A ,PA,rA,s%) ~ A in
responding to the unique representation
T*P x @
~ T~x M
cot-
122
=
17.23
( rAd~A
A % ® d x q - •. A d x m + s~dPA)
of an arbitrary /scalar-density-valued/
17.2~
~_m,<~,~>
covector
I ~ 2 : b • (~r,
+
@ on Px" Since
~~s,~A)
it follows from 17.22 and q7.J3 that
h A ~ ~x(~ ,p,, ~A ~,)
17.25
% A < T A 'Pa'rA's~)
=
where
rA
=
~A
17.26
This result is analogous to the formula 9.7. Substituting
17.26 into
17.23 we obtain the formula q7.2 from which 17.3 follows immediately. Similarly as in mechanics we define the /scalar-density-valued/ function
~x
i on Px by
It is easy to check that also in this case the right-hand not depend on the choice of a representative ~eP~.
17.28
In terms of geodesic coordinates
~x(~A
p~, ~ A , ~ A )
This formula is analogous
=
g ~ J~(P) of the class
we have
P %A ~ d Ax
side does
1. . . . A d X. m.
to 9.q7. It follows that
123
= (p~d~A + tf)Adp~)@dx1,,...AdX m
~17.29
d~#x
The function
_~x on D Ix defined as
~7.3o
~x
:
~
=
_ sh
.
4;xIDx~ - Z x
satisfies the equation
=
d %iD i - d~ x
=
i
i
Hence - ~
is the proper function corresponding to the generating
X
- -
:
-~I
D i
cL ~ x - -
(d ~x - 0xlbx
~h
~7.3~
.
x
•
function - ~ x "
Example Field equations 46.50 of electrostatics can be written in the form
~x/~
,~
17.32
~ x~ ~h = (3
Restricting
7.33
ex
- ~dp~)@dxIA...Adx
:
(- ~
:
- d{
~
r<x)dF
m to Dix we obtain
g~p~dp~) ® dx% dx~dx 3 : I g~ p~p~) dx% dx~d~ 3 ]: + -2g
=
-
d
~
x
Hence
~7.3~
~(x~,?,p~)
=
~( 4~r(x)T
+ 1
~
g~P~P~)dx~dx~dx3
124
Example 2 Equations q6.53 of the scalar field theory may be written in the form
g/~ p~ I?.35 (F'(~O)
=
-
m2~)
Restricting
i ~ x to D x we obtain in this case
17.36
~ ( x ~ , 7,p~l
=
In both examples the equation
17.37
~x = P~%l Dix - ~x
equivalent to 17.30 is satisfied.
18. The Legendre transformation We define the Legendre transformation assuming that the infinitesimal dynamics Dix is a section of the fibration
~ix : Pix
" Qx'i
This means that for each element v e Qxi /"velocity"/ there is a unique element ~ e Pxi which projects on v :
~xiC~)
=
v.
The mapping
i Qx
~ v
~xh(~ ) ~
is called the Legendre transformation /cf. also a section of the fibration
~ xh : pix
Px
[9]/. In many cases D Xi is - - ~ Px" This implies
125
that the Legendre transformation is invertible. In terms of coordinates
(~A p~, ~A, 8~A) the above statements mean that Dix may be parame-
trized by coordinates
(~A, ~ A ) or by coordinates ( ~ A
p~). The dyna-
mically admissible infinitesimal state of the field
is thus uniquely determined by its "position and velocity" / ~ A ~A/
or its "position and momentum" / ~ A
19. Partial Legendre transformations.
and
and p%A/.
The energy-momentum density ~A
The complete Legendre transformation replaces the velocities
~
by momenta p~ as arguments of generating functions. Partial Legendre transformations replacing only some components of the velocity by the corresponding components of the momentum can also be performed. We introduce below partial Legendre transformations corresponding to a vector field X on M and an (m-1)-vector ~ in M / a hypersurface volume element/ transversal to X. We begin with simple calculations in an adapted coordinate system. Let ~ be attached at a point x eM. We choose a coordinate system (x ~) in a neighbourhood of x in such a way that the field X is
a 19.1
X
=
~x I
and the volume element n is
19.2
Let
n
=
--
S x 2 A... A
~ X m
B(X,n) be the space parametrized by coordinates
(~A
re k = 2,3,...,m, and let 19.3
: pi
X
,
®
X
M
~ A p]) whe-
126
be an isomorphism density-valued/
19.~
such that the pull-back of the canonical /scalar-
1-form in T~B(X,~) @ / ~ T ; M
~ =
is
(SeAd#A - ~ lAd P A1 + PAk d ~ A ) ® d x l A . . A.d x.m
Since the exterior differential
d(~ =
19.5
(d~A^d~9 A + dPA~Ad~A)@dx<...
Adx m
is equal to dO~, we see that we have defined a special symplectic ture in p Xi .
Comparing 19.4 w i t h
196
-
we see that
~
transformation component
struc-
( AdmA + p dfL) dxl ... dxm
q and p~ have been exchanged. exchanges all velocities
The complete Legendre
and momenta.
of the velocity in the direction
Here only the
of X has been exchanged with
the projection of the momentum onto ~. Thus we have performed a partial Legendre transformation. We defined the scalar-density-valued closer analogy with the complete Legendre
form ~
transformation.
is more convenient to use the scalar-valued
19.7
e(x,n)
(X~n, e ) 2 =
obtained by contracting the form form
%dT
However,
C<(X,n)
A
A
G with the volume element X^~. The
by the isomorphism
: pi X
it
form
@(X)~) can also be obtained as the pull-back of the canonical
q-form in T*B(X,~)
19.8
in order to have a
,
TWB(X,n)
127
which results from composing the isomorphism
~
with the volume ele-
ment XAn : m
19.9
o/Cx,n)(~) We introduce in
=
<X~n,o<(~)>2
T'~B(X,n)
.
( ~ A (~?A,PA,mA,rA,s q k A)
coordinates A
n i c a l l y r e l a t e d to coordinates
1
( ~ A ~k,PA) in the
base
cano-
BCX,n). The
coordinate expression of (~(X,n) is
19.1o
~A)
o
(~A, ~,~, 1 k A, k,PA,mA,rA,s ]
=
where
19.11
mA
=
~gA
rk
--
pk
sA
=
V¢
,
k>
2
,
The generating function -E(X,~) of the dynamics with respect to the above special symplectic structure is obtained from the following generating formula
19.12
-~(X,n)
=
O(X,_~)tDxi
where -~(X,~) i s the proper function.
19.13
~k'PAI
=
In terms of coordinates
%d
-
÷
This is equivalent to the following form of field equations
.
:
128
19.1a
Pk~ =
a ~a
S ( X , n -)
,
k,~2
The minus sign in front of E(X,n) appears in order to give E(X,n) the interpretation of energy. The form
~(X,n) differs from <XAn, (~i>2
t~e ~ifferential of t~e function p]?~.
-- %d?~ + p~d ?~ by
We have
A
19.15
Hence
19.16
E(X,n)
=
p A1 ~ A D xi - L •
Example 1 We calculate the generating function E~X,~) for the scalar field theory. The field equations 16.53 can be rewritten in the form
p
~9.~7
%
k
=
=
~kl~l
~klglqpl
k,l=2,...,m
(gll - glkgklgll) pl + g l l g l k ~ k
(:F'(~) - m2tfJ )
'
.
Here we assumed that the matrix gkl is nondegenerate and denoted its inverse by ~kl. This assumption means that ~ is not tangent to the i light cone. Restricting the form 19.7 to D x we obtain
e(X,n)ID x
[~
(g11 - ~klgklgll)pl
129
+ gl~l~?~]ep I + ( f ~ l ? l
- ~l~llpl)d?l
19.18 d
~
1
2~ The corresponding proper function E~X,n) gian variables
~(x,~.)
( ? , ~,,)
is
:
(F~?) _-
expressed in terms of lagran-
12_2~
1
k~ ....
_
g
1
1'~
~Tk+E
=
19.19
-~m ~ +~g
~,t~ff)
.
This result confirms formula 19.16. Now we construct the space BCX,n) and the form dinate free fashion.
0(X,n) in a coor-
It will be shown that in the special coordinate
system the general construction reduces to our previous constructions. We take the volume element _m = X^n at x. Corresponding to the decomposition of m into X and n there is a decomposition of the space of jets i i i i Qx into two components Qx(X) and Qx(_n). The space Qxfn) is the space of 1-jets at x of sections of Q restricted to an arbitrary submanifold ~cM
i n ) contains information on tangent to n at x. An element of Qx(_
the value of the field to n /to Z / .
19.20
~
and its derivatives in directions tangent
We denote by
CQ(n)
:
Qx z(~)
the canonical jet-target projection.
>
Qx
If the coordinate system in M
i ) is described by satisfies 19.2 then a point of Qxf~ and the coordinate expression of
C Q ( ~ J is
(~A
~)
, k~2
130
A
A
The remaining information about derivatives of ~ A Lie derivative
~X~
of the physical field
is contained in the
~ with respect to X provi-
ded that the configuration bundle Q consists of objects for which the Lie derivative can be defined. The value of the Lie derivative is a vector tangent to the configuration space Qx /cf. Appendix G/. This justifies the following definition
~9.22
Q~<x) = ~Qx
•
The canonical tangent bundle projection onto Qx is denoted by
~Qcx~
:
Q~(x)
,
%
.
For the sake of simplicity we deal in this section only with bundles of geometric objects in M /more general cases will appear in Chapter IV /. In this case the value of the Lie derivative
calculated in an
adapted coordinate system 19.1 is equal to
~ x T (x) = TA
19.23
~A
•
As coordinates in Qx(X)~ " we may thus take the system rigA ~ ). ~ " terms of these coordinates
The space Qx can be identified with a subset of Q
~9.25
Qxi
=
{
i (sx'Sn)~%(x)
× i n
X)×Q
Qx(-)IL'Q(x)Sx = ~'Q(_n)s_n]
In
131
An element p ~ Px determines a covector PX ~ T~Qx given by the formula
19.26
PX
= ~'P~2
and a /vector-density-on-~-vaiued/
covector pn ~ T*Qx® /~ T ~
de-
fined by the formula
19.27
~ k,Pn ~ 2
=
~ -XAk,p~ 2
where ~ is any (m-2)-vector tangent to ~
19.28
PxCX)
=
,
at x. We take
T~Qx
and
19.29
Px(n)
=
T~Qx @ /~ Tx~~
Canonical projections onto Qx are denoted by
19.3o
~x
" PxCX)
' Qx
gun : Px (n)
~ Qx
and
19.31
m
If the coordinates of M satisfy 19.1 and 19.2 we have for an element p = (~pA p%) the equalities
19.32 and
px
=
p~d~ A
132
Pn
"19.33
this means that
-~d~... (P~d? ~)®¢ ~x
=
(~0A,pZ)
may
be used
~ dx m )
.
as coordinates in Px~X) and
(~A
pk) as coordinates in Px(_n). Obviously
and
~9.35
~nC~A,p~)
= (?A)
The space Px can be identified with a subset of Px(X)×Px(~)
19.]56
Px = ((px,psl~Px(xl~x~-~)
:
~x~p~l-- ~n~Pn~I "
i i Spaces qx(~) and Px(~) have been obtained by projecting spaces Qx and Px onto the hypersurface ~ .
These spaces can also be obtained by ap-
plying to the restriction of Q to ~
the procedure that was used to
i construct Qx and Px from Q. If the restriction of Q to ~ by Q I ~
is denoted
then
i
19.37
Qx(n)
=
J~x(QIz)
and
19.3s
sx(_~)
We denote by P ( ~ )
= e*(Ql~-,)x@ A e * ~ x
the union of all such spaces obtained for all
points x~ M. The tensor product in the above formula is understood in the sense explained in Section 13 and used in formula 13.1. We may apply the same procedure / or rather its simpler version
133
from Chapter II / to the restriction of Q to the parametrized integral curve
19.39
R
~ t
•
~(t) e
M
passing through x and tangent to X at t = O. The resulting spaces are
19.~o
i x) Qx(
=
J (Qf#)
Px (x)
=
T~[QI~)O
and
19.aI
Again we denote by P(~)
the union of all such spaces obtained for all
points of ~ . This procedure can be carried one step further. struct spaces P~(X)
and P~(~) /corresponding to pi
x/
i
1 9.~2
PxCn)
= JxP(~)
and
We have canonical projections
~X
i
~n
:
:
P~(Bt
~
i
Qx (~)
19.~z~
bp(x): pi(x)
~ Px(X)
bp(n) " P~(n)
, Px(n)
The following diagram is commutative
:
.
:
We con-
134
Pxi ( X )
LP(X)
, Px(X)
Px (n)
(
bP(n)
pi(n)
The standard constructions provide also the lagrangian special symplectic structures given by canonical diffeomorphisms
:
and :
,
Using coordinates (xk), k = 2,...,m, on ~ coordinates
( ~ A p~, ~ ,
~ A ) where
~=
use the coordinate t = x ~ on the curve used in Chapter II yields coordinates A =
Qx(_)~/%
x~
we may construct in P~(~)
P~k" In a similar way we may ~ ={ x k = const.3 (~A p],~A
. The method
~]) in P ~ ( X ) w h e r e
are coordinates of the Lie derivative
and PA = PAl
are coordinates of the Lie derivative of the momentum contracted with n
•
I PAl
19.48
= 2
In terms of these coordinates all projections in the diagram 19.45 are effected by omitting apropriate coordinates.
With each pair (~X,~n)~ P~(X)~P~(n) which projects onto the same element in Qx' i.e.
~9.~9
~x
° ~p(x)(~x)
=
~n
° bp(n)(~)
'
we associate an element of Pix denoted by ~(gx,gn). The following fori mula determines uniquely C(x(~(~X,~n)) :
135
~9.5o UX ,
:
i
+
i i is a vector tangent to Qx c Qx(X)×Q~(~).
where u = (UX,Un)
The vector u X is thus tangent to Q~(X) The element
and u n is tangent to Q ~ ) . i l~(~X,~n ) is completely determined since ~ x is an isomor-
phism. In order to obtain the coordinate
expression for
~
we take a vec-
tor
19.51
u
=
BA
8
Ux
BA
D
Un
BA
8
A
3
A
3
It is easily seen that
19.52 and
19.53
~ If ~X = ((]0A 'PA'
~A
< u X,
"I ,pA) and
gn_ = ( ~A pk (]Ok A, ~A ) then
i
_A. 1
= Jo PA
A.1
BlPA
+
and 19.55
<'n, < U2n , O i i-(_n ) ( ~_n ) > l )
The right-hand It follows that
=
BA~A
+ B AkpAk
A % + B~p A .
side of 19.50 is thus equal to BA(~AI + ~ # ~(gx'%)has
=
The special symplectic
coordinates
=
structure
(~A,pA~,~,~A)
=
PA%
, where
"
in Pxi which we are going to de-
136 4
fine will be a mixture of the lagrangian structure hamiltonian
structure in P~(X).
in P~(~) and the
In order to define a hamiltonian
cture in P~(X) we need a connection in the bundle QI~ tion of the section
stru-
, i.e. the no-
"constant in the direction of X". This connection
is given by the Lie derivative
: the section is constant in the direc-
tion of X if its Lie derivative with respect to X vanishes.
We used
this connection already in 19.22 identifying the space of jets ~ ( X ) with the tangent bundle TQx. As the base of the special symplectic
structure which we are go-
ing to define we take the space
i
19.57
described
in our coordinate
system by coordinates
space Pxi is fibred in an obvious way over B(X,~).
(~A
A 1 ). Tk,PA
The
Using local coordi-
nates one can prove that this fibration is given by
Px~
19.58
where
~(X,n)(~)
=
(~x,g~n) is any pair such that
(Op(x)gx,SUngn)eB(X,n)
,
~(~X,~n ) = ~.
We define an isomorphism
~(X,B)
19.59
: piX
> T*B(X,B)
by the formula
( v , (xCX,n) (~) ~ =
h x(X)
÷
19.60
where v
i
= (Vx,Vn) is a vector tangent to B(X,~) CPx(X)×Q~(~)
and
137
h pi Oi x : x(X)
19.61
,
T*Px(X )
is the isomorphism defining the hamiltonian structure in P~(X) / constructed e.g. in Section 9 /. Using local coordinates one can easily verify the formulae 19.q0 and 19.qi. This proves that the definition 19.60 does not depend on the choice of the pair (~X,~n). Formulae q9.50 and q9.60 may be interpreted in the following way. The element gl is decomposed into a pair (~X,~n). The value of the lagrangian isomorphism ~ on ~ is obtained by applying the lagrangian i i isomorphisms ~x(X) and ~x(~) to gx and gn respectively / formula 19.50 /. The value of the "mixed isomorphism" applying the hamiltonian isomorphism ~ ( X ) the lagrangian isomorphism
~(X,~) is obtained by
to the component gx and
i n to the component gn / formula 19.60/. ~x(_)
Corresponding to these constructions also the special symplectic forms ~
and
~(X,~) are obtained as combinations of lagrangian and hamil-
tonian special symplectic forms in the component spaces P~(X) and P~(~). We have
w,
_
<X^n,<w,O
:
>2
19.62 =
<w X, ~i(x)>
+
n < n , < w n, ~ix(_)~I>2 m
where Qx (i X) and Q~Cn)_ are
lagrangian special
symptectic forms in P~(X)
and P (~) respectively, w is a vector tangent to Px and (WX,Wn) is a decomposition of w into components tangent to P~(X) and P~(~) respectively. The pair (Wx,Wn) is a decomposition of the vector w in the m
sense that the value of the derivative of ~
is equal to w when cal-
culated on (Wx,W ~ . We have also
°
<w, O(X,n)2 --
i n exO>
22
138
The difference between these two special symplectic forms is
~w,<X^~, 0x>2 i
0~x,:)~I = <w X, e~(x)
h X Ox<)>
19.63
-- < wx,d T(x,_:)?, where, similarly as in the Section 9 / formulae 9.17 and 9.18 /, the function
~(X,n)
~(X,n) (~A,pA,
~9.6@
The function
19.65
defined on pi(x) is equal to
~(X,n) defines a function on piX by the formula
~](X,n) ( ~ ) =
<~(X,n)( ~X)
where gX is the first component in a decomposition te expression for ~(X,~) follows from q9.6a
19.66
0/('X,n)(~0 A, pAZ, ~A, ~ A ) =
of g. The coordina-
:
A
I
~lPA
•
We may rewrite this expression in a coordinate idependent form :
(x,_:)(~)
19.67
" < ~'""
=
~ A
PA~I
=
" ~ x-:' < ? : W :'p: ~" C~x~
Obviously
19.68
<X^n,
Oix > 2
_
O(x,:)
=
d ~,(x,_~) .
139
This implies in the general case the following generating formula
-~(x,_~)
:
(~(X,n) IDi~
=
{
<x^_~,O
xi> 2 - d ~ ( x , n
:
)]ID~
19.69
the We a l w a y s
assume
a normalization
o f __E(X,_.n) s u c h
that above A
mains valid when the differentiation sign is omitted
formula
re-
:
(x,~) = T(X,n ID xi - <x^_~, _Z~>2
19.7o
We recall that the Lagrangian is always normalized in such a way that it vanishes on the physical vacuum state. This gives the unique definition of E(X,~).
Substituting 19.67 into 19.70 we obtain
_z(x,,_~](~) 19.71
We see that ~(X,~)(~) depends linearly on the hypersurface volume element n. The / vector-density-valued / function
where for each element ~ e P ~
we denote by p the value of the p r o j e c -
tion of ~ onto Px' is called the energy-momentum density corresponding to X.
140
If (x #) are any coordinates
in M / not necessarily
and ~ as in 19.1 and 19.2 / the vector density ~ X )
adapted to X
may be written in
the form
19.73
~(x) --
--
E~(x)
~- ~ d x % . . . A d x ~x %
m "
The values of the components E%(X) follow from 19.72
19.7~
:
E~(X) ~ (ix T A)p~ _ X~_~
where X
=
X% Sx ~
and _~ =
~ dxlA... A d x m
•
Example 2 For the scalar field ~ nary derivative
the Lie derivative
is equal to the ordi-
in the direction of X :
19.75
~xT
= ~%
Thus % 19.76
E~(X)
In adapted coordinates The expression t ~ sor density.
X#(p~O#_ - ~
L)
.
19.1 and 19.2 this expression reduces to 19.16.
= p t~
-
~L
is called the energy-momentum
In this case the energy-momentum
the energy-momentum
19.77
=
tensor density by
E~(X)
= x~t~
ten-
density is related to
141
In the general
case the Lie derivative
of X and its derivatives bundle
up to some finite
over M then the Lie derivative
IV, the fundamental
in space-time. second-order
This situation
is represented
by the value
of X. The splitting
sor density t ~
objects
the second
contains
of t ~
deriva-
over M / tensor
only the first deri-
density ~(X)
into a part proportional
to its covariant
us to
derivative.
:
+
of the covariant
derivative
energy-momentum
tensor.
tensor.
of X. The tenWe call the ten-
A discussion
is given in Section 25. The interpretation
part
field
In this case the jet of X
is called the energy-momentum
sor density t ~
follows
from the following
of
of the anti-
example.
3
We consider cial Relativity.
a tensor field theory We use an affine
to this coordinate
components
of ~
=
~ X ~ are components
Example
on the
of the jet of X into two parts enables
formula
E Cx)
respect
field.
of X and a part proportional
9.78
symmetric
jet
connection
the classical
case when Q is a tensor bundle
We have thus the general
these
is the a£fine
of the field X and its covariant
to split the energy-momentum
where
~
occurs when we consider
field theory/ the Lie derivative
the value
on the first
of such a quantity depends
fixed gravitational
In the simplest
vatives
depends
in some detail the case when M is equipped with a con-
theory in given,
tives.
If Q is a tensor
jet of the field X.
We discuss nection.
on the value
field which will be described
quantity
The Lie derivative
depends
order.
of ~
of X. In the case of the gravitational in Chapter
of ~
g~.
in the flat spce-time
coordinate
system the Lorentz
Let X be a constant
vector
system metric
M of Spe-
(x #) in M. With g has constant
field X = X ~
d Then x ~"
142
~ X ~ : 0 and
19.79
Era(X)
The density E(X)
=
t~X ~
.
is in this case the component of the four-momentum
in the direction of X. Let X 0e a vector field X = ~
g~x~
_ _ 8 with x~ This field represents an
constant skew symmetric coefficients
~.
infinitesimal rotation of space-time.
The corresponding density E(X)
is given by
19.80
where the tensor
s~/~ = t ~ ~ ] ~
19.8~
= ZL~ f f ~ -
g~).
is usually called the spin angular momentum tensor. The density 19.80 is the angular momentum density in the direction of dO ~ .
It is com-
posed of the orbital angular momentum and the spin angular momentum.
We note that the quantity ~(X,~) defined by the formula 19.71 is the proper function for the infinitesimal dynamics only if X^~ ~ O. The formula 19.71 defines E(X,~) also when X ~ of ~(X,~)
= O. No interpretation
in this case is given there. In Section 21 we define conser-
vation laws for the energy-momentum density valid also when X^~ = O. In Section 16 we defined the infinitesimal special symplectic form
~
as a formal divergence
The formal divergence the general formula
of the form
0 x / formula 16.22 /.
can be decomposed into two parts in analogy with
143
49.82
XJdivc~
= ~
o~
- div~XJc~)
X This decomposition tial Legendre
is exactly
transformation
the one given in formula is a Legendre
19.62.
transformation
The par-
applied
to
the first term.
20. The Caftan form As in particle depends
on the choice
cription
of dynamics
in particle miltonians object.
dynamics
This object
the hamiltonian
of a connection. in terms there
corresponding
of the Lagrangian tions.
mechanics
description
The same is true about the des-
of the energy-momentum
is a method
to different
of combining connections
is the Caftan form.
how different
from the Caftan form as components
tensors.
Again as
the different
Ha-
in Q into a single
We define this form in terms
and thus in a way manifestly
We show subsequently
of dynamics
independent
Hamiltonians
with respect
of connec-
are extracted
to different
connec-
tions. We assume
that there are no hamiltonian
that for each element p ~ Px there L p g = P. Let g e J~(P) m be any m-vector
constraints.
is a unique
be any representative
in M at x / volume
element
element
~
This means D i such that X
of the class ~ and let at x /. The lift ~g of m
to the jet g is called the m-vector
compatible
recall
of ~ g was given in Section
that the detailed
coordinate
description
Definition m-form
~
definition
of ~ g
with the dynamics.
is given by formulae
: The Cartan form associated
17. The
17.19 and 17.20.
with the dynamics
is an
in P such that
for any m-vector ~ in P which projects
We
onto zero in M / the form
144
was defined of
~
for some connection
on ~ does not depend
20.2
<_~,e>
in Q but we remember
on the connection
-- < _ ~ , Z.-x ( ~ ) # 2
-- <
We show later that the definition in the sense that 20.2 is fulfilled fulfilled
for one representative
Theorem
if
that the value
~ @ ~ = 0 / and
~x(~)f2
~ , -~,g _
correctly
defines
•
an m-form
for any representative
g if it is
and if 20.q holds.
8
For any connection
of Q let a / scalar-density-valued
/ function
~ x on Px be defined by
20.3
where ~ is a volume
element
tor at p E Px which projects miltonian
Proof
corresponding
at x and ~ h is any horizontal onto ~. The function
~x
m-vec-
is the Ha-
to the given connection.
:
< m,~x(P)32
= -<_%,@~
- <_~g,e>
+
20.~
+<~g-~h'e>
--
<~*x(~)>2
+<~g--~h,-';'~
because
20.5
~ ( _ ~ g - --~h)
=
%~_~g -
~_~ h
The last term of 20.4 is equal to < ~ g , % ~ ce the mapping
gg h x : D xi___~p~
=
m--
m --
=
since < ~ h , ~ #
is a diffeomorphism
O.
= O. Sin-
we may define
a lift
145
--X
of the above function to D i. Applying the formula 17.27 we obtain X
20.6
<m,~x~22
-- < - ~ , T x ( ~ ) -
-Zx(~)22
or
20.7
_~x -- ~'xlO~--Zx
•
This completes the proof.
In coordinates
(x %, ~ A
p~)-- which are geodesic at x the formula
20.1 together with 17.14 give
2o.~
< i~x1A... ~ ~ ^ ' " ~ m
,~ >
-- ~2
%
The formula 20.3 implies
20.9
d_--~A...~,e~
9
-- - H ,
where
=
20.10
The form
a ~A
and
~
has to vanish on all other m-products of vectors
~ 3PA in order to fulfill
that i£ the form
20.11
H dx~... Adx m
0
=
~
the condition
20.1.
It
follows
exists it must be equal to
~ IA . . . A dA~ A^ . . . ^ dx m - H d x ~ . . . A d x PAdX
m
8
~x~'
=
146
It remains to be shown that this form satisfies the condition 20.2 for any representative ~'
~A
2o.~2
g = (~A
p~, ~ ,Ap A ~%) o f
the element ~ = (~A,pA~ ,
" Using formulae 17.19 and 17.20 we obtain
<~g,o>
=
b ( pA ~ ~ ,A
- ~(x~, ~A,,~))
_- <_~,S ~ ) > ~ .
This proves that the definition was correct. The Cartan form evaluated on an m-vector ~ g compatible with the dynamics gives the value of the Lagrangian.
Evaluated on a horizontal
m-vector ~ h / compatible with the connection / the Caftan form is equal to the Hamiltonian.
Also the energy-momentum density can be obtai-
ned by evaluating the Cartan form on an appriopriate m-vector in P. Let n be a hypersurface volume element in M at x and let g be any representative
of an element ~
Px" We define the lift ~ g of ~ to the
jet g similarly as the lift ~ g was defined for the volume element ~. If g is the jet of a section
20.13
M
~ y
~ p(y)
e P
Y
at y = x then by ~ g we denote the lift of ~ to the restriction of this section to the hypersurface
~-~c M tangent to ~. The coordinate des-
cription of ~ g is
-~ =C~d+ ~ ~
-~xm+ ~*m WA +
20.1¢
8 9~) PA
+ P~m
for
8 20.15
n
=
8 x 2 A...
A
~X m
.
147
Let X be a vector field in M. In the case when Q is a bundle of geometric objects ciated with
in M the field X has unique lift to Q. This lift is asso"dragging"
the geometric
A p p e n d i x G /,, If Q is a bundle
objects along the flow of X /see
of geometric
objects the same is true
about P. It follows that X can be lifted to P. The lift of X to P is denoted by X. The m-vector
20.16
~
is "horizontal"
=
in the d i r e c t i o n
in the d i r e c t i o n
~Afig
of X and compatible
with the dynamics
of n.
Theorem 9
(X,n)(¢)
20.17
=
-<~g,e~
where g is any representative
Proof
,
of ~ & D i x"
:
2o.18
<_~, @ ~
since
~(~g
: < m^g ,e>
-
- ~ ) : O. To calculate the second term of the right-hand
side we note that
2o.19
_~g
=
xg^_ng
,
where X g is the lift of X to the jet g. Thus
148
_~g_~
20.20
=
b:g-~')~g
,
and according to 17.9
<_~g - _ ~ , 4 - >
< n , < x g - ~, e~V 1> 2 --
=
20.21
h~
where p = ~dxg 6 Px is the point at which all the vectors used above are attached.
The vector
~xg-x)
is difference between the lift of
X to the section
20.22
M ~ y
, ~(y)=
and the horizontal lift of X to Q. Hence
~:(p(y))
~
~e(Xg-~) is the Lie deriva-
tive'of this section with respect to X :
20.
23
~,(~g
- t)
=
~o
/see Appendix G /. Thus, according to 19.71
-<_~,~# 20.24
--
_.s(x,n)(~)
.
This completes the proof.
Similarly as in particle mechanics the
20.25
~
:
d~
Qy
m+l -form
149
contains all information
about dynamics.
Theorem 10 A section
20.26
M
~
x
~ ~(x~
is dynamically admissible
e
Px
if and only if for each vector field Y
in P
(Y_lg)
20.27
s
: o,
where S is the image of the section / cf.[9]/.
Proof
:
Using geodesic o o o r d i n a t e s
(x~,~A,pA~)
a t x we o b t a i n
from 20.I~
the following expression for
~
dPA~A dx IA...A
=
d~gA... ^
dx
m
-
A
20.28
- (~8-~-~Ad ~ A + ~P~Hdp~)AdxqA...AdX m Let the field[ Y be vertical
and given by
20.29
8 a?A
y
=
BA
+
3 cA% ~p~
Then
Y_J ~,
=
B A{-dx'1~... ^ dp~A... A dx m - - -H d x ~ A dx m } + A-~A --%
20.30 + c
x~...^
d
~...
%
^ dx m - ~
. . .
A xm}
150 In order to calculate the section 20.26
2o.31
(Y~ ~)
IS we use the coordinate expression for
:
=
We set
dp Is is
S P Ak dx #
=
~ A dx~ x~
and obtain
20.32 ...A
dx m
~PA j Since equation 20.27 implies 17.7 the section 20.26 is dynamically admissible.
If 20.26 is dynamically admissible then equations 17.7
imply 20.27 for vertical vectors Y. For vectors Y tangent to S 16.27 is trivially satisfied.
Hence
(Y~ ~)
IS = 0 for arbitrary vectors.
This completes the proof.
Let a section
~ : M
, P be dynamically admissible / a solution
of field equations / and let X be a vector field in M. The(m-1)-form (Xd~)IS
defined on the graph S of the section can be projected onto
M. We obtain this way a vector density
g~X~)
in M. It follows
from Theorem 9 / formula 20.17 / that this vector density is equal to minus energy-momentum density ~(X)
:
The formula 20.33 can be used as a definition of E(X,~)
in the case
151
when X^n = O. Many important dynamical variables are also defined by the vector density
C * ( Y ~ 0)
for fields Y in P which are not necessa-
rily the lifts of vector fields in M. In this section we assumed that there are no hamiltonian constraints. There are, however,
important field theories where the projec-
i h of the dynamics D x onto Px gives a submanifold tion ggx denote by
~h c P the union of all these submanifolds
~ h itself is a submanifold.
h ~ x CPx.
We
and assume that
The conditions 20.1 and 20.3 / or equiva-
lently 20.2 / enable us to define the Caftan form at points belonging to
~h. Applying this form to m-vectors tangent to
m-form
@
on
~h.
~ h we obtain the
In all physical field theories which we consider in
Chapter IV the field equations for sections of
~ h over M are equiva-
lent to the condition 20.27 where Y is any vector field in neral, however, equivalence
~h. In ge-
this is not necessarily true. The question of this
is connected with the problem of formal integrability
partial differential
equations / cf.
~I]/.
of
We do not discuss this
problem in general and limit ourselves to the discussion of examples.
21. Conservation laws Let
C : M
~ P be a dynamically admissible section of the phase
bundle / a solution of field equations / and let X be a vector field in M. We calculate the divergence of the energy-momentum density
--
Using the formula 20.33 we have
21.1
.
152
The second term gives no contribution because of field equations 20.27 fulfilled by the graph S of ~ . Thus
21.2
div
Definition :
Ed(X )
=
- ~ ( ~ X
A vector field X in M is called an infinitesimal
i if the form symmetry transformation for the dynamics { D x] is invariant with respect to ~
~ = dO
:
Of special interest are infinitesimal symmetry transformations which leave also the Cartan form
21.&
~
=
~
~
0
invariant :
.
We call such infinitesimal transformations infinitesimal special symmetry transformations.
Theorem 11 The energy-momentum density corresponding to an infinitesimal special symmetry transformation X is a conserved quantity :
21.5
air
~(x)
=
o
.
The proof follows directly from 21.2. The field X generates a l-parameter / local / group feomorphisms of P such that N
{~w)
of
dif-
153
where
{~}
is the q-parameter / local / group of diffeomorphisms of
M generated by X. Diffeomorphisms
~
enable us to drag along X also
jets and classes of jets of sections of P. It can be easily seen that equivalence classes of jets are taken into equivalence classes when dragged along X. We obtain in this way the lift ~i of X to pi and corresponding / local/ group
~
of diffeomorphisms of pi. The follo-
wing theorem justifies the definition of symmetry transformations.
Theorem q2 An infinitesimal transformation X of M / a vector field in M / is an infinitesimal symmetry transformation for the dynamics {Di} if and only if the group the jet
{~m~ preserves {D i} in the sense that
~i(g) is dynamically admissible if g is.
Proof : The jet g = jq ~(x) of a section
~ : M---~P is dynamically ad-
missible if and only if the equation 20.27 is satisfied at the point ~(x) g P for any vector Y. This equation is obviously equivalent to an a n a l o g o u s
of ~
equation
for the
jet
with the f o r m
instead
. The condition 21.3 is equivalent to the equality
~j
for all
~ eR . Thus if X is an infinitesimal symmetry transformation
then the field equations 20.27 for the jet ~$Cg) are equivalent to the field equations for g. Hence ~~i ( g ) is dynamically admissible if g is. Conversely, suppose that
{~
back both sides of the equality
2
.8
i D yi
=
preserves the dynamics. We pull
154
from the point y = ping
~.
~(x)
to the point x by means of the adjoint map-
The result is
We note that
~i ~-~ D yi
21.10
=
Dx i
because the dynamics is preserved. Also
~.
because
~
preserves the entire internal structure of pl,this struc-
ture being canonical and
~
being the lift of a diffeomorphism in M.
Thus
21.-12
~x,MIDix -- d ( ~ _ ~ y)
On the other hand we have
--
x
.
Hence
21.13
d ( ~T~ ' i ' ~ L- - y -
~x)
--
o
i which means that the difference is constant on Px :
155
The collection of elements
~
at all points of M defines an m-form X
o( in M. We use the above formula to calculate the pull-back of the Cartan form
@
~
. If the m-vector ~ attached at the point p e P X
projects
onto
zero
in M /
i.e.
if
~
= 0 /
then
it
follows
from 20.1
that
21.15 < ~ , ~ e . s
° <9,~,¢#
__<_~a~
<~, ~ £ >
Here we used the fact that the form ~
<~,e>
belongs to the canonical struc-
ture of P and this structure is preserved by
9~
. Let the jet g e J~(P)
be dynamically admissible and let ~ g be an m-vector compatible with the dynamics.
We denote by f the jet
cally admissible.
f~(g). This jet is also dynami-
We have
21.16
#~.ag
= af
and ~f is also compatible with the dynamics.
It follows from 20.2 and
2 1 . 1 % that
~g ~~ e # -,
-- < 9~ - _~g, Q ~
-- <-#, Q #
21.17 =
<
~.m
,
(9 __xpeX ~i~
It follows from 21.15 and 21.17 that
^g
m
--
156
where
O( is some m-form in M. Thus
[:
which means that
de
~
--
dg
e
is preserved by
-
d(e
~
+
_-
, i.e. 21.5 is satisfied.
This completes the proof.
An infinitesimal symmetry transformation X of M can thus be characterized by the property that any solution of field equations remains a solution when dragged along X. The existence of the vacuum state enables us to formulate a simple criterion for an infinitesimal transformation to be an infinitesimal special symmetry transformation.
Theorem 13 A vector field X in M is an infinitesimal special symmetry transformation for the dynamics f ~D xi ] if and only if both dynamics and the vacuum state are invariant with respect to X.
Proof
:
We have ~ y ( ~ o ) = O, where go denotes the jet of the vacuum state at the point y 6 M. Hence the function The invariance of vacuum means that state at x =
~_t(y). Since
from 21.14 that
~x
~(~o)
~ ~y
vanishes on
~w(go).
is the jet of the vacuum
--~x vanishes also on this jet it follows
= O. Using 21.18 we see that the invariance of
the vacuum state is equivalent / for an infinitesimal symmetry transformation X / to the invariance of the Caftan form
~
. This comple-
tes the proof.
The energy-momentum density E(X) is an important dynamical variab-
157
le also in the case when it is not conserved. over the boundary
function
- Jacobi theory
is referred
of the Hamilton
is a fascinating
are discussed
laws are derived within the framework
formulation
of Noether's
21.2.
along
of div E(X)
One gets in this
- Jacobi
equation.
and difficult
[9],[10],[11].
a part of the Hamilton
tion of field theory by using Noether's
integral
chan-
subject.
in Section 24. The interested
to the work of Dedecker
laws may be considered these
analogue
of its consequences
reader
into the volume
with help of the equation
way a field-theoretical
Some
W ~V when the domain V is dragged
may be changed
over V and calculated
The Hamilton
of E(X)
8V of the domain V c M gives the infinitesimal
ge of the generating X. This integral
The integral
- Jacobi
Conservation
theory.
of the lagrangian
theorems
/~9],
~]/.
Usually formula-
Modern
t h e o r e m s can be f o u n d i n r e f . [ a 9 ] , ~ ,
~7]
and also ~ 2 ] .
22. The Poisson Let
Z
boundary applied forms
algebra
c M be a hypersurface
%V
ef a domain V. The procedure
to obtain spaces Q Z Oz, O0z associated
discussed
in M which is not necessarily
in Section
used in Section
the
14 can be
p Z of Cauchy data on ~- and canonical
with~_
. Under special
topological
conditions
15, we have pZ = T*QZ which means that (PZ,cO~)
is a syTnplectic manifold
in the strong sense / see [8]/. If (Pz,tOZ)
is a strongly
manifold
symplectic
22.1
analogous
~z:
TP z
then the mapping
~ T*P z
to 9.2 defined by
is an isomorphism.
It follows that for each smooth function
f on P x
158
there is a unique vector field 3~f in P~ such that
22.2
<-~,df>
-- ~ ' ~ 36f , ~ = >
for an arbitrary vector field ~ .
We define the Poisson bracket ~f,g~
of two smooth functions f and g on pZ by the standard formula
22.3
-If,g]
.
,~f~
.
Let (x %) be coordinates of M such that ~ const, and let vectors ~(4) and ~
=. x
is described by xl
be represented by sections :
.~¢x~. ~ x ~
~q
~ ~ x~
22.~ ~)
=
~
+
1
p (x) ~PA
~
TPz
x
Then
22.5 ~ ^ ~ ,
gOz~=
I~p~(x)'~A(x)
~
]
dx m
exactly as in formula 1~.26. Let f be a smooth function on PZ. The derivative of f in the direction of the vector ~
22.6
>- ~ x
represented by the section
xcx) = 8 ? A c x ) ~
+ g PA(X) I ~~_p ]
a Tp~ x
is a continuous linear functional on the set of pairs of functions (~A
~pq). Hence this derivative can be written in the form of an
integral :
22.7
~f
<~dO=
I{~A~x~ qx) ÷ O~¢x~p~Cx~]dx~ Adxm
159
The coefficients functional
B A and C A defined by this formula
derivatives
are usually
called
of f and denoted by
Sf
BA (x)
8~0A(x)
22.8
cA(x)
Comparing
6f
=
Sp]cx
22.5 with 22.2 and 22.7 we see that
~f
is represented
by
the section
22.9
~-,x-~x(~)
This ket
implies
_
~p~(x) ~TA -
the following
coordinate
~ * : x ~ ~p~ expression
for the Poisson brac-
:
dxm The construction to a non-compact tions vanishing
hypersurface at infinity
of all integrals for hyperbolic
involved.
require
a dense
:
where
Z
an extension
important
of the above constructions
to
only on dense subsets
by the energy which is defined
only on
smooth Cauchy data.
with the infinitesimal
transformation
22.11
~
pZ
of sec-
is a Cauchy surface.
The energy / on the hypersurface
p~
also
the convergence
is specially
and differentiable
is provided
set of sufficiently
Definition
if the space P ~ is composed
This construction
which are defined
of pz. An example
Z
given here applies
fast enough to guarantee
field theories
Applications functions
of the Poisson bracket
~
/ associated
X of M is the function
, ~(pZ)
~
R
160
defined by
-- I ~.(x) (s(x))
22.12
where for any section
22.13
~_~ x
~ Pz(X) e Px
and a hypersurface volume element n at x on M the element s(x)a B(X,n)~ i n ) is defined by the formula C Px(X~×QxC
22.1~
s(x)
=
(p~(x),jl gg~ (p~ (x))1
@
The above formula makes sense only for elements pZ such that the section
22.15
Z
equal to
~(pX) e
~ x
Q~
~ ~ xz( p ~(x)) e Qx
is differentiable.
Otherwise the jet in the for-
mula 22.14 is not defined.
23. The field as a mechanical
system with an infinite number of deg-
tees of freedom In this section we consider field theories in a pseudo-riemannian space-time M. Field equations are assumed to be hyperbolic and the Cauchy problem is assumed to be well posed on space-like hypersurfaces of M. Let
Z t be a q-parameter family of space-like hypersurfaces
tained by applying to the hypersurface
~=
~o
ob-
a l-parameter group
of transformations
generated by a time-like vector field X on M. We
define a bundle
over R I. For each t e R I the fibre
~
~
over t is
161
the space Pzt of Cauchy data on ~ t "
We also introduce the bundle
over R q whose fibres are spaces
= ~t
Under special topological each fibre re
~t"
~t
~t
conditions,
of Dirichlet data on ~ - t "
which include integrability,
is the cotangent bundle T * ~ t of the corresponding
For each t ~ R q there are canonical forms
= C O ~t. Sections respectively.
of ~D and ~
~t = @ z t
are generated by sections
fib-
and CO t =
of P and Q
For example if
M
9
x
,
p(x)
~
Px
is a section of P then
R I ~t
~ Pt
=
P I~t
e ~t
= ~t
is a section of ~D. The situation is formally the same as in particle mechanics.
The difference
lies in the fibres
~ t and
~t
being infi-
nite dimensional.
Since the dimension of ~ t
is interpreted
number of degrees
of freedom we can speak of the field as a mechani-
cal system with an infinite number of degrees of freedom. dynamics
in P induces a dynamics
as the
The field
in ~D : the dynamically admissible
sections in #D are those generated by the dynamically
admissible
sec-
tions of P. Since the Cauohy problem is well posed each element p e P Et = ~t
determines
uniquely a dynamically
hence a dynamically
admissible
admissible
section of P and
section of ~ .
We assume that the bundle Q is composed of objects for which the Lie derivative
can be defined.
lization of ~ .
Horizontal
In this case there is a natural trivia-
vectors are jets of sections with vanishing
Lie derivatiw~ with respect to X. We denote by tor field in
~which
denote by d
the vector field in
projects
projects
B ~ the horizontal vec-
onto the unit vector field in R 1. We ~
compatible with the dynamics which
onto the unit vector field in R I. Similarly as in formula
162
9.20 we introduce
the vertical vector field
~h
23.1
d -
There is a theorem analogous
dt
-
9t
to Theorem 6 in Section 12 :
Theorem 14 The field
~ 6 generated by the energy
is equal to
Proof
~ in the sense of 22.2
~h.
:
For the sake of simplicity we assume the existence coordinate
system
(x %) in M such that ~ t
the vector field X is equal to
of a global
is described by x I = t and
~--~ . We define the function E by the
equation
=
23.2
If a vector
~
233
zt ~ x
'PA (x) ,
• • •^dx m
is represented by the section
+ ~p~x~ ~~
•~c~ = ~?A~x~ ~
~ TPxz t
then
25.~
!
8E
A
~__~.~PA(X) ~ dx 2. .^dx m
where
23.5
~
Assuming that functions
?~ ~A
--
~ ~ i?A(x~ ~x
vanish at infinity sufficiently
fast we
163
by
integrate
23.6
]parts
~~E
I
~~' ~
~
ACx)
S~A(x) ~x 8~
=
zt
~8E
~t
Now we use equations qg.la. The result is
~-t + ~OACx)'~p~(x)tdx2...Adxm
23.7
~A rx~.~
8 PA
=
A
dx m
This implies
8~A t(x) =
23.8
Sg ~p~(x)
and
P~ (x)
23.9
~g
Bt
where M ~ x
is the dynamically admissible section determined by the Cauchy data p~t / the vector ~ is attached at a point pZtm ~ /. Formulae 22.9, 23.8 and 23.9 imply that ~
is represented by the section
23.~o
B~A c~ 8 ~ ~_.. ~
x~(x)
=
~P~ ÷ -~-¢x) ~p]
It is easily seen that ~ h is represented by the same section. This completes the proof.
164
We see that the part of the Hamiltonian picture
is played by the energy.
analogous
to equations
9.11.
Equations
in this time-evolution
25.8 and 25.9 are formally
This is the reason why the energy is usu-
ally called the Hamiltonian
of the field.
term for the object defined
in Section
more natural
of the Hamiltonian
generalization
The Lagrangian
We prefer to reserve
this
17 which in our opinion is a
in the time-evolution
of particle
picture
dynamics.
can be shown to be
the function
Lt ( T t ,
: Zt
where
~t : T l~-t" We note that this Lagrangian
gy are defined
and differentiable
as well as the ener-
only on the dense subsets
of smooth
Cauchy data. This section energy introduced the
"canonical
is meant to give one application earlier.
Hamilton
The energy appears
equations"
equations
ty. Moreover,
define
23.8 and 23.9.
the heavy mathematical
assumptions
of the time evolution picture
cability
of this definition
equations
of energy.
to physical
element
at x ~ M
necessary
drastically
are
We note that
for the for-
reduce
the appli-
and let s = jl~(x)
be the jet
situations.
action and the Hamilton-Jacobi
Let m be a volume
These
in
only the total energy and not the energy densi-
mulation
24. Virtual
of
as the "Hamiltonian"
usually used as one of the standard definitions these
of the concept
theorem
of a section
24.1
M
of the configuration change
of the value
y
Qy
bundle
at the point x. We consider
<~, ~(s)~when
the rate of
both m and s are dragged along the
165
vector field X in M. For this purpose we assume that Q is a bundle of geometric objects in M. In this case the group
~}
of diffeomorphisms
of M generated by X defines a group of diffeomorphisms
[~
of Qi. The
generator of this group is the lift X of the field X from M to Qi. Similarly the group of the bundle
~T~M
{~z) can be lifted to the group of transformations of volume elements.
of volume elements and the family {s(~)) ing the apprepriate lifts of the group Both ~ )
and s(r)
of jets obtained by apply{~z] to ~ and s respectively.
are attached at the point x(~)
the / scalar-density-valued
24.2
We consider the family { m ~ ) ~
/ function ~ ( X )
<_~, ~ ( x ) ( s)>2
=
~z(x). We define
on Qi by setting
d
In elastostatics / see Section 5 / this quantity / multiplied by $ ~ / is the virtual work which is performed in dragging the piece m of the elastic medium along the the field X. In dynamical field theories we shall call this quantity the virtual action corresponding to X. The right-hand side of the formula 24.2 can be treated as a definition of the Lie derivative
of the scalar-density-valued
function
on Qi with respect to the field X. We write
2~.3
v(x)
iv X
The following theorem shows the relation between the virtual action and the non-conservation
of energy.
Theorem 15
s~.4
%F(X)(s)
=
- div
~(x)(g)
where the value of the right-hand side has been calculated on the
166
jet g of a dynamically configuration
of Noether's
is the infinitesimal
A special
version
by the
of the Hamilton-
case of the Hamilton-Jacobi
theorem is one
theorem which states that if the virtual
then the energy E(X)
is conserved.
thus be characterized no virtual
section of P determined
jet s.
The above theorem Jacobi theorem.
admissible
Symmetry
as infinitesimal
fields
action vanishes
of the theory can
transformations
of M which
"cost
action".
Proof
of Theorem 15 :
Let M
be a dynamically equations
~ y
~
admissible
/ such that 24.1
div
e
P
Y
section of P / a solution is its projection
We denote by g the jet jl ~(x).
24.5
g(y)
~(X)(g)
According
=
-
div
of ~
into part ~,
The rest is obviously
the
2~-.s
~
tangent minus
=
onto Q, i.e.
~=~
@~(~J@)
,
the values
of X on the
to the image and the vertical Lie derivative
%,, - ~
.
to 20.33
where X is the lift of X to P. We decompose image
of the field
of
~
rest.
.Thus
g~ X
Let ~ g be an
(m-Q)-vector
be its projection
tangent to the image of ~
on M. This means that
N
24.7 Xij "h _
~
at
= ~g. Thus
~(x)
and let
167 b e c a u s e X~j~n g i s 20.2/,
It
follows
an m - v e c t o r
compatible
with
the dynamics"
/
see
that
g(x,,JO)
2~.8
=
xJZ(j~T)
Similarly we obtain from 20.1 and q7.9 the formula
X 24.9
-- ( 4
X
?,-,_
, @>
?IP 2
=
--
X
•
Obviously
24.1o
~,Zx?
= 4x~F
-- ZxF
Thus
24.11
g~
The divergence sity
~(j1~)
of X ~ ~ ( j q ~ )
is the Lie derivative
defined in M. This Lie derivative
of the scalar den-
corresponds to drag-
ging the volume element m in 24.2 along X and dragging the jet s along the lift of X to the image of j1~ . We may express this Lie derivative using the decomposition of the field X similar to 24.6
24.12
X
=
:
Xl, X
V
where X~r 24.13
is the field tangent to the image of j1~ . Thus
168 The divergence
of 24.11
tion of the L a g r a n g i a n
can be calculated
from 16.11
and the defini-
:
•
X
24.14
The last term is the Lie derivative cal f i e l d
Y = ~
X
of ~
with respect
~1(~). Thus
- div ~,(x) :
div C~(~_]O)
: £~f
-£..f
Xil
w h i c h completes
to the verti-
-- £ . Z "
Y
: 1~(x)
X
the proof.
25. E n e r g y - m o m e n t u m
tensors and stress tensor.
A review of different
approaches In Section f19 we introduced the energy density as a generating function of dynamics.
This definition
of energy provides not only the
total energy but also its local distribution. lished a physical
interpretation
m e - e v o l u t i o n picture. the energy-momentum
of the total energy based on the ti-
In terms of the local energy d e n s i t y we defined
tensors.
In the simplest
theory the energy density depends
23.1
tensors
t~/~ and t ~
defined as
in the expansion formula
z%Cx)
Both energy-momentum gy density.
case of a tensor field
on the first jet of the field X.
Hence there are two energy-momentum coefficients
In S e c t i o n 23 we estab-
=
t~x~ + t~9
Zx~
tensors must be known in order to know the ener-
This definition
of energy is new. The usual approaches
are based either on conservation laws or on the d i s c u s s i o n
of sources
169
of the gravitational nitions
field.
We give a brief review of different
of the energy-momentum
Definitions
tensors
and relations
between
based on conservation
laws (Noether
theorem)
ne only the total energy. ently the energy density
The localization are not uniquely
used to obtain an expression components
determined.
for the energy density
involving
tensor whose component
rection
of X is the energy density.
In terms
from conservation
of our energy density E(X). equivalent
only the
in the di-
of our definition
in the following
of ener-
way. The total
laws is equal to the integral
The formula 25.1
is
of X. This procedure
in a single energy-momentum
ergy obtained
determi-
This freedom
results
this procedure
them.
of this energy and consequ-
of the field X but not the derivatives
gy we may interpret
defi-
can be rewritten
en-
22.12 in the
form
In the simplest
case when the second energy-momentum
symmetric,
t%~/~ = - t~/~
i.e.
tensor is anti-
, the second term is a complete
diver-
gence
25.3
:
The integral replaced
of this term over a 3-dimensional
by the 2-dimensional
tegral vanishes
if appropriate
thus gives no contribution
integral
surface Z
"at spatial
boundary
conditions
to the total energy.
infinity".
is the "improved
7~
=
energy-momentum
and
In this case one says
t~
tensor"
This in-
are fulfilled
that the quantity
25.4
can thus be
and the quantity
170
25.5
~
which depends
=
~x
on the single
~
tensor ~
is the "true energy-density".
In the case when t~Y/~ is not antisymmetric because ~
the integral
construction
fails
of the second term of 25.2 over the 3-dimensional
does not vanish. Our definition
energy density. not contribute
of energy gives the unique
Subtracting
used in particular
variant.
the electromagnetic symmetric
of "improving"
"canonical
energy-momentum
A procedure
based
192q by Bessel-Hagen
variant
The result
tensor"
unacceptable
of this improvement
is
which turns out to be gauge-ingeometric
leads directly
interpretation
to the definition
This procedure
tensor t ~
is
a certain naive approach
energy-momentum
[5] . We describe
tensor
of the
was given already
this procedure
obtained
of
in Section
in 28.
in this way is gauge-in-
it needs no improvements.
Related problem
to the problem
of energy-momentum
of finding appropriate
tational
field.
and momentum
Such sources
ergy-momentum
localized
have again been attemped
tensor.
ved by subztracting find that sources
expressions
are believed
of matter properly
of these sources
We consider
divergences
localization
for sources
by energy Constructions of the en-
this problem too serious
to be sol-
or any other ad hoc modifications.
deformations
field depend
and are related
of energy rather then to the distribution
We return to discussion
of the gravi-
to be described in space-time.
is the
by modifications
of the gravitational
of matter to space-time vation
where
tensor"
tensor.
Since the energy-momentum
even if they do
the energy-momentum
on the correct
potential
energy-momentum
25.1 for
energy is not allowed.
of being gauge-dependent.
"symmetric
expression
divergences",
in electrodynamics
leads to the so called on account
"complete
to the total
The above procedure
the
the above
of sources
We
on the response to non-conser-
of conserved
of the gravitational
energy.
field in Sec-
171
tion 29. The following sider the virtual
construction
action ~ ( X ) .
the virtual work performed
The elastostatic
in dragging
the case of energy-conservation Hamilton-Jacobi
25.6
is used in this discussion. analog of ~ ( X )
We conis
the elastic medium along X. In
the virtual
action is zero.
Using the
theorem we obtain
~2(X)
=
- div E(X)
=
W(X)dx~...^
=
- E ~ ( X ) ~ dxlA...^ dx m
dx m
=
,
where
25.7
w(x)
The value second
of W(X)
first
with an affine
into three parts covariant
cond covariant
derivative
=
All these components
for tensor field theories
since E(X) connection
~ X '~
~
1
jets.
of the second covariant
on the first jet.
If
then the second jet of X ( given by components
X~),
part of the se-
+
v,
and may be chosen as coordinates
They determine
derivative
uniquely
For example
all derivatives
the antisymmetric
is given by the formula
25.9 2
w h e n R is the curvature
on the
:
of the field X up to the second order. part
depends
and the symmetric
are independent
of second
.
: the value of X
derivative
258
in the space
- F)(x),~
at a point depends
jet of X at this point
M is equipped splits
=
and Q is the torsion
of the connection
172
The splitting of the jet of X into three parts enables us to expand the virtual action ~g)X
25.1o
~(X)
into parts proportional to X e,
~ with uniquely defined coefficients
~#(x)
-- -(T~x~ + T ~ V~ X ~ + ~T I
The coefficient T % ~
~ X ~ and
:
~'t ~(~,~)X~) dxl ,,...,, dxm
is a symmetric tensor density
.
:
The interpretation of the vector density Tj~ is obtained by setting
V~ X ~ = o and
~g~X~=
0 at the point x ~ M
along X is locally a parallel displacement
At this point dragging
in the direction of X. The
quantity T~ measures thus the quantity of action which is necessary to make a local parallel displacement of the physical field ~ . By analogy with elastostatics we call T~ the force. In order to obtain the interpretation of the remaining terms in 25.10 we put X ~ = O at x e M. Dragging along X at x ~ M is now a local deformation of the field without displacement. and
~ V ~ ) X ~ . By analogy with elastostatics
the tensor density T t sor)
The deformation is described by derivatives
and T ~
the first stress tensor
~X ~
(formula 3.~) we call (or simply stress ten-
the second stress tensor.
The detailed analysis of the structure of General Relativity which we give in Chapter IV
suggests that the stress tensors and not ener-
gy-momentum tensors are sources of the gravitational field. Although these quantities have completely different physical meaning there are relations between them. It follows from the formulae 25.1 and 25.7 that the stress tensors are completely determined by the energy-momentum tensors. To show this we calculate the covariant divergence of 25.~. For vector densities the covariant and normal divergences cide. Thus
coin-
173 - w(x) 25.12
The last term can be decomposed into symmetric and antisymmetric parts. Using 25.9 for the antisymmetric part we obtain
-
wCx)
=
(~/tt~
+ 1 t~
+ (tt
+ V~t%
R~)
X~ +
25.13
+ o
.
~)V~x"
+
Comparing 25.13 with 25.10 we obtain the following formulae
25.14
Tff
25.15
T~,,
=
V~t~
=
t~/~ +
+ 2
¢ R ff~
~ ~, t ~ /~
,
+ t [~]/ ~ ,n~~
In the case of a symmetric connection
( Q~
:
,
= O)
the stress tensor
is equal to
25.~
~t
--
tt
+ ~
t~t
•
It is interesting to note that if the second energy-momentum tensor t~t
is antisymmetric then the stress tensor 25.17 is equal to the
"improved energy-momentum tensor" 25.4. This complete coincidence
is
probably the origin of much of the confusion in this domain.
For a wide class of field theories dynamics is determined only by the geometry of space-time M, i.e. by the affine connection
~
and the
174
pseudo-riemannian metric g. For such theories the Lagrangian ~ on the values of
~
of the jet of
:
~
and g in addition to the coordinates
We do not assume any relations between
~
depends
~ A and
~A
and g. Such relations be-
long already to the dynamics of the geometry, i.e. to the General Relativity Theory and will be discussed in Chapter IV. In the present section we consider a tensor field theory in a given geometry. For the sake of simplicity we assume that the connection
~
is symmetric. The
reader may easily generalize all results to the case of a non-symmetric connection.
We now define an invariance property of the theory usually described as the invariance of the Lagrangian under coordinate transformations. This property states that the value of < m , ~ ( s ) ~ 2 is invariant under simultaneous dragging along X the following three objects
: q. physi-
cal field ~ , 2. the volume element ~ and 3. the geometry of space-time represented by
~
and g. The rate of change of < ~ , ~(s)~2 due
to I. and 2. with the geometry fixed is just the Lie derivative of the Lagrangian which appears in the Hamilton-Jacobi theorem. This Lie derivative is equal to the virtual action ]~(X). The rate of change with respect to the deformation 3. of the geometry is equal to
25.19
2
~~
1_
X
~
ag if'
The invariance condition thus implies the following
25.20
~f(X)
1
~£
~%
2 ~Ji~ ~ x ~
_ 1_
2
~£
~g~
~ X g/~
=
0
175
equivalent to
25.21
W(X)
I
BL
iX
~
I
3L
~X
where
25.22
~
=
L dxlA... ^ d x m
.
A field which fulfills this property will be called a relativistic field theory.
To calculate the stress tensor for a relativistic
theory it remains to substitute the Lie derivative
of
~
in 25.21 the following expression for
and g :
25.2~
'~ Xxr;~
--
:~t-
25.2~
i
:
x ~ ' ~ g : ' - 2 V (:x~
g:'
field
+ V~V~
x"
=
- XCR ~(#~')~
+
%' ~ ) X~',
X where
25.25
V~ =
g:~V~
The result is
w(x)
=
- x~[~ R ~::
~c,;',
- Z
~-~-:~j
-
~L
Symmetrization Yes
~L
~ g~v and
in the indices /~ and ~ can be omitted since derivati-
~L
~ C~y
are symmetric.
Thus
SL
~g/~)
_
176
25.26 L
g#~ ~ X ~
+
1
8L
Comparing 25.26 with 25.10 we obtain
25.27
T~
where T ~
=
$ g#~
denotes the symmetric tensor density
25.28
T#~
=
~ L
=
_ ~L
and
25.29
(cf.
T#~
;
[2q] where the "hypermomentum" is introduced by the similar for-
mula). Moreover
25.3o
~
=
1 (T#~ R ~
-
+ T/~ ~ g#W)
Comparing 25.50 with 25.14 and 25.16 we get the following identity
~tl~
+ lt~
Ri~#y
=
-t(~)~ R ~
- 1T ~ g#~ 2 #~
=
25.51
lt#~
~
1
Now we use the so called second identity for the curvature tensor :
25.32
The result is
R%,
+ R~
+ R%g~
=
0
.
177
I T ~ g ~
For
metric
a
symbols,
connection
the covariant
25.34
~t~
+
Using the property
~
, where
o
are the Christoffel
derivative
of the metric vanishes
t~ % R~
=
and we have
0
'-TV~g T M = 0 we m a y r a i s e the index g . We use proper-
ties of the curvature tion
=
=
tensor fulfilled
in the case of a metric
connec-
: ~t
~
+ t~
R~
g~
=
25.35 :
~ t ~" + t ~ R ~ g ~
We recall that the tensor
g~[~ t ~ ]
the spin angular
momentum
tensor
25.36
t ~['m]
=
S ~m
=
V~t ~" + t. ~ [ ' ~ J R ~
was denoted by S ~ (see formula
=
S~
19.81
:
o .
and called
). Thus
g~ g~'m
We obtain finally the identity
~ t~
25.37
valid
in the case of metric
this identity
reduces
25.38
A second identity obtain
+ S ~'~ R~/~%
=
0
connection.
For the flat Minkowski
space
to
~
t~
=
0 .
can be obtained by comparing
25.28 with 25.17.
We
178
T/~
=
g~t~
+ g~
~ t ~%~
=
25. }9 t/~ + ~
( g/~t ~ ~,) - t ~
~g~
The antisymmetric part of the stress tensor T#~ vanishes. Thus
25.40 O.
For a metric connection
= ~
, the covariant derivative
of the
metric vanishes and the above formula reduces to
25.44
+
=
o
Equations 25.41 are often called the angular momentum conservation laws. It may be shown that identities 25.33 and 25.40
(or equivalen-
tly 25.34 and 25.4~ for the ease of a metric connection)
together
with 25.30 are necessary and sufficient conditions for the theory to be relativistic.
The formula 25.38 is often called "the General Rela-
tivistic expression of energy-momentum conservation"
(see[42]).
Sin-
ce in general this formula is not fulfilled in curved space-time attempts were made to "improve" the energy-momentum tensor t ~ re in order to obtain a "conserved quantity" T ~
once mo-
for which the equa-
tion 25.42
V ~ ~T ~
=
o
holds. This approach is so popular that in some textbooks equations 25.42 are treated as the unique reason for some quantity appearing in calculations to be called "the energy-momentum tensor"
(see e.g. [32]).
In our opinion equations div E(X) = 0 are conservation laws and not
179
25.42. In traditional tional
variational
field is represented
formulations
by the metric
fers from ours.
The formulation
the connection
~
cal formulation
of the traditional
defined the sources sity satisfying
presented
of gravity the gravita-
tensor g. This approach in Chapter
treated as the gravitational approach
of gravitational
25.42.
The Hilbert
IV is based on
field.
The mathemati-
is due to Hilbert
field as a symmetric
tensor
dif-
is defined
[2~
who
tensor den-
by
%L 25.43
TH
~ g~
where L is the matter Lagrangian ditionally
on the gravitational
9 g~
assumed to depend on field g ~
~
and its derivatives
and ad:
25.~@
In order to establish and stress tensors calculate mation
relations
and energy-momentum
the derivative
of the geometry
For a relativistic
between
the Hilbert tensors
of the Lagrangian
tensor
on one side
on the other side we
with respect
to the defor-
:
field theory the reasoning used earlier yields
the
equality
x
2
$
25.~6
On the other hand we may rewrite
equation
25.21 using the formulae
:
180
and g~
25.4.8
=
- g~ g~
X valid
g~ X
in the case of the metric
connection
~/~ =
~
. Using also
25.28 and 25.29 we obtain
w(x) X
X
X
25.4.9 I T~6 g~ 2 X
2
_ ~ _~
X
2
X The covariant
derivative
normal derivative sity.
Comparing
for the Hilbert
V~ in the last term has been replaced
since the expression
in the bracket
tensor in terms of stress
tensors
:
1 T/~yZ )
Consequently
w(x)
is a vector den-
the last equality with 25.4.6 we obtain an expression
25.50
25.5~
by the
= -Ez
£x g~
-~
2
x
181 U s i n g the f o r m u l a
X we write
25.5"I in terms
of d e r i v a t i v e s
~'
w(x)
_ [(m~
of X
:
'~ _
25.5~ =
The H i l b e r t tensors
- THw
tensor
~
X~
- [(T~(~w)
can be also
expressed
with the help of i d e n t i t i e s
T H~
=
t~
+ V%t%/~
:
t~'
+
__
t~
+ 5V~ (t ~
V ~
- 2
-
of e n e r g y - m o m e n t u m
25.16 and 25.17
V%( t(~)~ +
-
~
in terms
~ (t (''~
t (~~/~
:
- t (~w~
- t ~'~)
)
=
--
25.54
_
+ t~
_
Finally
E~L~
25.55
=
TH#~
=
t~
_
V~(S~/~
The above f o r m u l a i s the B e l i n f a n t e - R o s e n f e l d [42] ) . case
We draw t h e a t t o n t i o n
of ah a n t i s y m m e t r i c
the s e c o n d stress that the H i l b e r t tum tensor"
of t h e r e a d e r
second
tensor T
~
2 5 . 4 w h i c h turns
+ S~ff~)
theorem t o the f a c t
(cf. that
[ 3 ] , [4], in the
tensor
( t~ ~ = t~
and the f o r m u l a
25.50 shows
energy-momentum
vanishes
tensor reduces
+ S~
again to the
"improved
out to be s y m m e t r i c
:
energy-momen-
)
182
25.56
TH
=
T~
=
~
.
It follows from 25.29 that this happens when the L a g r a n g i a n not depend on the connection
~
but only on the metric.
~
does
The triple
equality 25.56 seems to be result of pure coincidence.
We calculate
the divergence
of the Hilbert tensor by first rewritting
the formula 25.55 in the following form
=
t~'
-
V~ (-s ~
:
+ S A~'~ + S ~
)
=
25.57 _-
+
_
.
Thus
V~ ~E
25.58
=
=
t~
~t~
+
(2
+ -I
s r~l~
S~
~ R ~~
2
=
~t ~
+ S~/~R~%A~ ~
Hence
25.59
TH
follows from the identity 25.37.
=
0
- S ~])
- R ~~
~I R .~
-
~
=
)
183
We have not found a direct physical tensor.
In Chapter
ficantly appearing
different
interpretation
IV we give a formulation from Hilbert's.
in this formulation
Sources
of the Hilbert
of gravity
theory signi-
of the gravitational
are the stress
tensors.
field
IV. Examples
26. Vector field The configuration bundle Q for the covariant vector field is the cotangent bundle T~M, where M is the space-time with a pseudo-riemannian metric tensor g and a symmetric affine connection
~
. No a prio-
ri relations between the metric and the connection is assumed. Such relations belong already to the General Relativity Theory and will be discussed in Section 29. We give a local coordinate formulation of the theory. Let ( x ~ be a coordinate system in M. In the present Chapter coordinates of space-time will always be denoted by x ~, ~ = induces a coordinate system
(x#,~)
0,1,2,3.
The system (x ~)
in the configuration bundle Q =
= T*M. The components of the metric tensor will be denoted by g#~ and g~.
The connection
I~
is described by components
phase bundle P is the union of tensor products 13.1
~~~
~=
. The
:
3
26.1
where q 6 Q, x =
Pq
=
Tq
Tx
Tx
,
~ (q)~ M. The space dual to the cotangent space is
the tangent space. Hence 3 26.2
Pq
=
TxM ® A
Tx M
It follows that the bundle Px over Qx is trivial
•
As a bundle over M, P is the Whitney sum off two bundles 3
:
185
The first bundle is the configuration bundle Q. The second bundle is the tensor product of vectors by vector densities. We call such objects contravariant tensor densities of second rank. Tensors
26.5
e~
=
~xf
~x~
3 form a basis in the vector space TxM @ A
TxM. Every element of this
space can be uniquely represented as a linear combination of elements 26.5. It follows that the bundle P has coordinates (x~, ~ % , p ~
where
the coordinates p~/~ are components with respect to the basis 26.5. Forms
~ x and oDx at each point x ~ M are given by formulae
~x ~ 26.7
%
:
dT
) ® (
~
~ d x O, "" . A d x ' )
"
The infinitesimal configuration bundle is the first jet bundle jIQ. A coordinate system ( x ~ , ~ (x~). The coordinates
,~)
is induced by the coordinate system
~ / ~ represent derivatives
26.8
?~b/~ = ~A~?%
"
The infinitesimal phase bundle is the quotient bundle pi -- ^~YP of the first jet bundle jIp. Coordinates in pi are (X ~, ~ , p ~ , ~ % / ~ ,
~)
,
where
26.9
~
=
~p~m
.
At each point x e M the infinitesimal phase space Pxi is a symplectic manifold with the symplectic form
186
i 60x 26.10
=
(dae°v,\d ~
+ dp~/*A d ~ z , ) ®dx0A... A d x 3
.
The symplectic form is the differential
6o xi
26.11
= dO~
of the q-form =
Mdiv
0
X
=
[~(~>~)]~
~o... ,,~x~ =
26.12
-- ( ~ d %
+ ~ d W ~ , ) ® d x O . . . A dx3
.
It is convenient to parametrize jets of sections of P by covariant derivatives of the sections. We thus introduce coordinates ki/x#'~'P%#' ~L~' ~'%) by setting : -i¢
g
26.13
it must be remembered that P is
a
tensor density ) T h e o
formulae
26.12 and 26.10 can be rewritten in terms of new coordinates. The result is
~.~¢ 26.q5
0k = ( ~ v ~ CO xi
=
+ ~,~9~)~
~xO... ~ x ~ ,
(d~,,d,¢~ + d~d%~)®dxO,..
~dx3
The dynamics D ixC Pix of the field is expressed by field equations •
26.16
p~/~ =
~
g~g/~
~
,
187 26.17
~=
_ m2 ~ L - ~ g ~
In the case when
.
~ = 0, i.e. when
~ = /~
, field equations
are equivalent to the second order system of equations
( [ - ] g + m2)~O~
=
0
,
26.18
The vacuum state is the zero-section of P : ~
=
0
p~/~
=
0
26.19
The dynamics and the choice of the vacuum completely determine the Lagrangian 26.2o
£
=
T, d x O . . . ~ d x 3
such that d-~ x
=
26.21
Using the condition L(O,O) = 0 we obtain
1 ~(
g ~ g F ~ 7~/~ 7~ ~
m2g%/~F)
=
26.22 =
I_ ~ 2
g~g~p(~$/~ _ ~ ) ( ~ -
~
~)_m2g~/~
.
Since the bundle Q is equipped with the affine connection we may perform the Legendre transformation and pass to the hamiltonian description of the theory. If coordinates(x~)are geodesic at the point x
188
i.e. if
I~%~vanish
at x) then hamiltonian special symplectic
form
is
o~ = (~e~d% - %dp~0 ® d~O,...A dx3
26.23
Since in geodesic coordinate system
~=
~
and
~=
~$% , we may
write in general
o~
26.2~
_~,~,~)~xO...~x3
- - ( ~
This formula is valid in all coordinate
.
systems, not necessarily geo-
desic. In order to find the Hamiltonian we rewrite the field equations in the following form
! 26.25
Inserting formulae 26.25 into the equation
~.~
_ ~%
=
e~x
I~X
we obtain 26.27
~
=
H dxOA...^ dx 5
where 26.28
Let ~ be a hypersurface element an infinitesimal transformation of M to ~. We use coordinates
(3-vector)
(vector field in M)
~
=
transverse
(x ~) adapted to ~ and X, i.e. coordinates
which 26.29
in M and let X be
~ x I ^ ~dx
A ~x 3
,
in
189
26.30
X
= x°
The energy ECX,~) k = 1,2,5.
is a function of variables
C(~,' (-~,k 'p ~0 )'
where
It is determined by the equation
26.3~
- ~(X,n)
~(X,n)lO xi
=
where 26. 32
O(X,_n)
=
~d%
+ p~kd~k-
~0dP ~°
In order to calculate E we should solve the field equations 26.16 and 26.17 with respect to the variables
~
p~k and ~ 0
and insert the
results into formula 26.32. The corresponding proper function ~(X,~) may be directly
obtained by the Legendre
transformation
19.72 or 19.74.
Thus 26.33
_E(X,n)
(~,~(x)>2
-
=
n~E~(X)dx°... ~d~ 3
where 26.34
Now we pass to a general coordinate ted to X and ]~. The Lie derivative
system
Hence
26.37 and
t~ 6
=
tensors are P # ~ ~--
-
adap-
of the covector field is given by
26.35
The two energy-momentum
(x ~) not necessarily
S~
L_
190
26.38
t~/~
=
p ~
The stress tensors may be calculated directly by expanding the virtual action W(X) = - ~%E%(X) with respect to X ~, q X .~ and
~(%~)X ~. Ins-
tead we use equations 25.q7 and 25.q6 :
26.99 =
p ~
+ p~6"~}~6. - m 2 ~Ig~(lO~.~O/~
I ~%(p~ ~ 2 26.~0
T~/~
=
tm~
m2 ~
+ t~L
=
-
~(~
2p(~)~p
.
It is easily seen that the first stress tensor is really symmetric as it should be by virtue of equation 25.28 :
26.41 - I ~ g~(g ~ g~
~
Now we calculate the Caftan form
~
~
- m 2g~ O ~ p )
.
in P. The formula 20.11 gives
the Cartan form in terms of coordinates (x~) which are geodesic at the point x
2 6 ~2
@
=
P~dx°
We derive a formula for ~
~ i~
^ ^
d~3 - ~ d x O
~ dx3
valid in a general coordinate system.
Transition to general coordinates requires the replacement of covectors d~O~ in the above formula by covectors
191
26.¢5
The second term is related rection
of x -axis.
to the parallel
Covectors
transport
26.¢5 vanish
of
on horizontal
~
in the di-
vectors
in
P. Hence
@=
p~,~dxO...,, (d~, ,-rj ~. -H~xO
~dx 3
dx~),,... ,, dx5 _
= p~dX~...~dS~*...^dX
3-
26.44
_ (p~.,,_ q [ ~ p~dx0A
=
+ ~)~ o,,... , , ~ dx 3
...~d~,...A
-
~
_~ dxOA.., i, dx 3
,
where - 7 ~"
=
26.45
p~l
Using calculations (formula
20.271
~ ~,
1
'1
+ ~ (~
g~ g~p~p~ +
similar to these used in the proof
we find that field equations
m2
f ~ g~i~).
of Theorem
may be written
10
in the
form
26.~6
~%
26.47
~p~
--
~ p~
,
=
It is easy to check that the above system
is equivalent
to 26.16 and
26.17.
27. The Proca field The Proca field is a covariant rent from that described example
in the preceding
of Proca field to illustrate
the hamiltonian
vector
description
field with dynamics Section.
the appearance
of dynamics.
diffe-
We have chosen the of constraints
in
192
Bundles Q, P, Q i
pi and forms
~ x , ~ x , ~ix' cci x
for the Proca
field are the same as for the covector field. The dynamics DXi is described by equations
27.1
p %AL
27.2
~%
2~
~ g ~ ~or<
=
where
I (~
27.3
The derivatives
and ~
~
covariant derivatives
27. ~
~j
~
in the above formulae may be replaced by and ~ % since
: ~ ?~
for the antisymmetric tensor density p ~ . i.e. when
~2~= {~v}
_)~
In the case when
, equations 27.~ and 27.5 imply
27.6 =
- m2 ~ ( g ~ / ~ )
=
- m2 ~
~
Hence
277
Vo~:
o
This equation is called the Lorentz gauge condition. Using this condition we write
193
27.8
because of the equality 25.9. By R@~ = R ~ # ~
we denote the Ricci cur-
vature tensor. This implies that in the case of metric connection the system 27.1 and 27.2 is equlivalent to the second order system of equations
: (~g
27.9
+ m2)~+
~
R ~
=
0
= O,
,
We note that the lagrangian submanifold D xic pix described by equations 27.1 and 27.2 is bigger than the space of jets of solutions of field equations. No condition involving the symmetric part of the covariant derivative
~
is contained in the definition of Dix" However,
a condition follows from 27.6. In the case when
~g#~
such
= 0 this con-
dition reads
27.~o
o = V~ g ~
= g ~
= g~~
This does not mean that the description of field dynamics in terms of D xi is incomplete since the additional equation 27.10 is automatically
194
satisfied
by any s e c t i o n
The v a c u u m
state
of P whose
is the
jets b e l o n g to D i at each x ~ M. x
zero s e c t i o n =
0
=
0
of P
27.11 p~ The d y n a m i c s a n d t h e
choice
of the vacuum completely
determine
the
Lagrangian 27.12
~
=
L d x 0 A . . . A dx 3
such that dL x
=
+ p~/~d~/~
~md~m
27.13
U s i n g the c o n d i t i o n L(0,0)
= 0 we obtain
I
27.1~
N o w we p e r f o r m
the L e g e n d r e
nian description. tum is always
There
transformation
are h a m i l t o n i a n
antisymmetric
The h a m i l t o n i a n
=
special
(~d~-
(2g ~ g~P
m
.
and pass to the h a m i l t o -
constraints
since the m o m e n -
:
symplectic
~
structure
is given by the form
~ p ~ ) ® d x ° .... Adx3
=
27.16
(~,~
_ ~
~
_ ~ ( ~ ) ~ p ~ ) ~ ~ O,...~ ~ 3
.
195
The Hamiltonian is defined only on
~h. The derivative of ~ x with
respect to p(~) is thus arbitrary. This implies that ry on D i
~)
is arbitra-
The field equations may be rewritten in the following form
X"
I
27.17 ~
- m2 k~-~ g ~/~~0/~
=
or equivalently
2
S~
-
m2 ~
27.18 ~m
=
g~P~
Inserting the equations 27.18 into the formula 27.i9
- d~x
=
~
=
O~ Dix
we obtain 27.20
H dx0A... A d x 3
where
I
Let
4
P~P~P
27.22
OF
g~F)
~_c:M be a Oauchy hypersurface in M. For the sake of simpli-
city we use adapted coordinates(x~)such that ~ x
+ m2 ~
@
The space pX of Cauchy data on ~
z
~
Not all elements of P
x
, (T~ (x),p~°{x))
is described by
is described by sections
~ P=x
are compatible with dynamics. Cauchy data cor-
responding to solutions of field equations satisfy the following condition implied by the dynamics :
196
p°°(x)
=
o
27.23 m2
~0
Sections 27.22 satisfying 27.23 form a subspace constraint submanifold.
of P
called the
The dynamics in the time-evolution picture is
determined by the energy defined on the constraint submanifold
~~ .
The construction of the energy density follows the pattern given in the preceding Section.
In adapted coordinates the form
~(X,~)
is
given by the formula
27.2~
8(x,_~)
--
~d~
+ p~kd?~ k -T~odP ~°
Equations 27.1 and 27.2 can be solved with respect to ~kO
when
~'~%k
and pk0 are given. The component
~,
~00
p~k and is arbitrary.
This corresponds to the constraint pO0 = O. The energy E(X,n) is determined by the formula 26.31. The corresponding proper function _E(X,n) may be directly obtained by the Legendre transformation
The two energy-momentum tensors are
and
27.27
tz~
=
p ~
•
The stress tensors are
T~
-g~+
:
197
27.28
and 27.29
=
T~
2 p(~)~m
=
0 .
The l-st stress tensor is symmetric
27,30
T,t~
:
g;~T%
=
4 ~
g~'~ %1~] ~OE/~]-m2~-~ % % - g ~ L
.
According to the remark given at the end of Section 20 the Cartan form
~
is defined on the constraint subspace
antisymmetric momenta.
~
=
p%~ ~~
since p Zm is antisymmetric
@ -- p~dx °
27.32
consisting of
In order to calculate the form
the same pattern as in the preceding Section.
27.31
~haph
...
and
A
i~
+ g
~z~
is symmetric.
^
~x3
we follow
We have
~
...
=
~
g
_
Hdxo
Hence,
...
~
dP
It may be easily checked that equations
27.33
(~Jde) J ~
for the section of
~ h over M are equivalent to field equations 27.1
o
and 27.2. The symplectic form in Pm M
27.3%
6o z
198
is degenerated when restricted to ~ z . vanishing vectors v tangent to
~
27.35
This means that there are non-
such that the equation
=
holds for any vector w tangent to ~
0
. Such vectors v are described
by sections
27.36
z
~x
~ (~x~,
~°~x~)
such that the only non-vanishing component is 27.35 is satisfied since dp O0 = 0 on ~ z . v form a distribution in ~z.
~kCX)
The equation
The set of all such vectors
This distribution is composed of all
vectors tangent to the foliation of ~ m fixing
~O"
~ TP~
and pkO(x) and varying
by submanifolds
~o(X).
obtained by
The space P ~ of leaves
of this foliation is parametrized by 6 functions
27 37
z
~x
~ ( Tk~X~ ,pk°CxJ)
We call this space the space of reduced Cauchy data. The form M
T
27.38
~
=
~(dpkOdT~)®dx~dx~
3
J induced on ~ z by cO~ is non-degenerate.
The pair ~Pz,c~z)
is a symplec-
tic space. The process of reducing the space (g~,oO =) to the spaze (~z ~ ) imitates a well known process of reduc/ing finite dimensional symplectic manifolds
(cf.
[44],[57]) •
28. The electromagnetic field The configuration of the electromagnetic field is the electromagnetic potential.
The field itself plays the part of velocity. The elec-
tromagnetic potential is not a covector field in space-time since dif-
199
ferent
c o v e c t o r f i e l d s c o r r e s p o n d to the same p h y s i c a l
The r e p r e s e n t a t i o n
of the p o t e n t i a l
as an e q u i v a l e n c e
t o t f i e l d s r e l a t e d by gauge t r a n s f o r m a t i o n s v e r a l reasons.
the d e s c r i p t i o n
n e c t i o n f o r m in a p r i n c i p a l
curvature
~ )
fibre b u n d l e
tensor
ferentiably fibres
a s s o c i a t e d w i t h the gauge
differential
g r o u p of r e a l numbers.
manifold, metric
let M be the spa-
g and let G d e n o t e
The g r o u p G is a s s u m e d to act d i f -
on the m a n i f o l d B, the m a n i f o l d B is f i b r e d over M and the
are the o r b i t s of the group action.
~
:
B
~
The f i b r a t i o n is d e n o t e d by
M
group a c t i o n is d e n o t e d by
28.2
G~B
~ (r,b)
~
r-b
e
We assume the e x i s t e n c e
of a class of m a p p i n g s
28.3
:
~
characterized
by c o n d i t i o n s
I°
B
is a d i f f e o m o r p h i s m
2°
~ b
B
~
B
G
:
(<
--
~ M×G
and
~(r.b)
=
r +
~(b)
f o r r e G. If
~q
28.~
and
as a con-
of the c o n n e c t i o n .
28.q
The
is also i n c o r r e c t for se-
of the p o t e n t i a l
e q u i p p e d w i t h the p s e u d o - r i e m a n n i a n
the a d d i t i v e
of c o v e c -
The electroma~etic field is then interpreted as the
L e t B be a 5 - d i m e n s i o n a l ce-time
class
interpretation of the e l e c t r o m a g n e t i c f i e l d
The m o d e r n
as a gauge f i e l d r e q u i r e s
group (see
configuration.
~2
b e l o n g to this class t h e n
(~2-
~I)
: B
~ G
200
is a m a p p i n g
constant
real f u n c t i o n
A
~2
-
)'1
is a d i s t i n g u i s h e d
image by the group 2 ° implies
~
and can be i d e n t i f i e d
with a
= /"~
vertical
= A ~
vector
f i e l d K on B w h i c h
a c t i o n of the unit v e c t o r
is the
field in G. The c o n d i t i o n
that
28.6
<
The m a n i f o l d
K,d
)
d
:
r
=
B w i t h the s t r u c t u r e
cipal fibre b u n d l e
1
described
.
above is a trivial p r i n -
with base M and the s t r u c t u r a l
K is called the f u n d a m e n t a l conditions
of
on M :
28.5 There
on fibres
group
v e c t o r f i e l d and m a p p i n g s
~
G. The f i e l d satisfying
1 ° and 2 ° are c a l l e d t r i v i a l i z a t i o n s .
A connection
form
~
on the p r i n c i p a l
fibre b u n d l e
B is a 1 - f o r m
on B such that
28.7
< K,~>
28.8
~K ~
The c h a r a c t e r i s t i c
28.9
O( b
of a c o n n e c t i o n tribution 28.10
distribution
form
= ~
=
I
=
0 .
:
VeTbB
:
is c a l l e d a connection.
are c a l l e d h o r i z o n t a l =
vectors.
do<
satisfies 28.11
K_~
=
0
~K ~
=
O
and 28.12
~ v , ~
•
= 0 Elements
The e x t e r i o r
of this dis-
differential
201
The condition
28.11
28.13
follows
from the identity
K_]do{
=
~
o~
-
d~K,c~>
.
K
The condition
28.12 follows
28.~4
Z
d~
from =
d£~¢
K
As a consequence M such that
K
of properties
,£ is the pull-back
28.15
~o
The form
~
~
<
~
is a 2-form f on
:
~~f
=
form of the connection
and f is
tensor.
be a trivialization.
28.16
and 28.12 there
of f by
is called the curvature
called the curvature Let
28.71
Then
~,~¢-d~?~
=
o
and
28.~7 K
Consequently
28.18
K
there
is a l-form
~
-dT
:
(a covector
field)
A on M such that
~*A
We note that 28.19 Hence 28.20 If ~ I
f an~ ~ 2
are two different
of 28.5 the corresponding differential
=
of a function
dA trivializations
covector :
of B then by virtue
fields A I and A 2 differ by the
202
-
=
o<
-
d
/2
-o<
+
d
/1
--
28.21 =
-d~*A
=
)6~dA
Hence, 28.22
A 2 - Aq
A connection potential netic
- d A
~
on B is i n t e r p r e t e d
and the curvature
field.
in space-time.
in B. Only when a t r i v i a l i z a t i o n
one a s s o c i a t e
with ~
a covector
is i n t e r p r e t e d
trivialization We have
of gauge
involving
potentials.
~
establishes fields
a correspondence
of Qx are
"values
between
to v e c t o r s
configuration
bundle
group
the correct of complex num-
descriptions
of
the f o r m u l a
connection
forms
and c o v e c t o r
has b e e n chosen. the c o n f i g u r a t i o n
forms
attached
Q spaces.
can be p a r a m e t r i z e d
from one
at x",
at p o i n t s
space Qx" E l e m e n t s
i.e. r e s t r i c t i o n s
of con-
b e B x = 9~-I(x) c B. The
Q is the u n i o n
28.2% of c o n f i g u r a t i o n
~
we d e f i n e
of c o n n e c t i o n
n e c t i o n forms
of t r i v i a -
d~ +~A
on M if a t r i v i a l i z a t i o n
At each p o i n t x e M
can
structural
group U(C,I)
lead to e q u i v a l e n t
In both cases
=
for the
complex wave f u n c t i o n s
q. The two groups
28.23
of B is chosen
and t r a n s i t i o n
of real n u m b e r s
choice would have been the m u l t i p l i c a t i v e
electromagnetic
The p o t e n t i a l
is a gauge t r a n s f o r m a t i o n .
c h o s e n the group
of m o d u l u s
~
electromag-
field A on M. Each choice
as a choice
to another
of B. In a p p l i c a t i o n s
bers
as an e l e c t r o m a g n e t i c
tensor f is the c o r r e s p o n d i n g
The field f is an object
is an object
lization
form
=
=
U Qx xeM
The f o r m u l a
by c o v e c t o r s
28.23 proves
on M a t t a c h e d
that
elements
of Qx
at the point x p r o v i d e d
203
a trivialization
~
has been chosen.
Given a trivialization the con-
figuration bundle Q can thus be identified with the cotangent bundle T~M and each configuration space Qx can be identified with the corresponding covector space T x M. A coordinate vialization
system
(x~
in M and a tri-
~ of B induce thus a coordinate system (x~,A~) in Q de-
fined by the formula C>( =
28.25 Let
~I
and ~ 2
+ A~dx ~
be two trivializations.
T~M associated with ciated with
d~
~2
~I
The identification of Q with
can be obtained from the identification asso-
by adding to each element of T * M the covector d A ( x ) X
The linear structure introduced into Qx by identification with T ~ M x changes with a change of trivialization.
However,
the affine structure
remains the same. We conclude that Qx has a natural affine structure. Vectors tangent to Qx are thus elements of T~x M and the tangent bundle TQx is the trivial bundle Qx @ T ~ x M. Consequently the cotangent space for Qx is the space TxM and the cotangent bundle T~Qx is the trivial bundle Qx@TxM. We conclude that the phase bundle P for electrodynamics is the Whitney sum of two bundles 3
28.26
P
=
Q
where the secnd term of the sum is the same as for the covector field Section 2 6 )
and for the Proca field
has coordinates
(x~,A~,p~)
where p ~
(Section 27). are coordinates
The bundle P of the contra-
variant tensor density of second rank and do not depend on the choice of trivialization
~
of the bundle B. Forms
~x and
x e M are given by formulae
28.27
0x
=
° \~x~
.^dx 3) ""
ODx at each point
204
28.28
=
o
The affine structure zn Qx implies that the differentials dA~ are gauge invariant objects and have tensorial transformation properties. The infinitesimal configuration bundle is the first jet bundle Qi = jIQ. A coordinate system (x~,A~,A~>) is induced in Qi by a coordinate system (x~,A~) in Q. The coordinates A ~ 28.29
A~
=
~
represent derivatives
A~
The infinitesimal phase bundle is the quotient bundle pi = ~Ip of the first jet bundle jIp. Coordinates in pi are (x~,A~,p ~ , A ~ , ~
~
28.30
=
~p~#
~) where
.
Canonical forms in pi are given by
28.3~
¢x
28.32
CO xi
= ~div e~ =
d 0 xi
=
=
(~dA~ + p~*dA~D~(dxO,...~dx3~
Mdiv dOx
=
The dynamics D xi c P xi of the electromagnetic field is described by equations p~/~
=
~
=
~
g ~ g / ~ (A~
- A~.O)
,
28.53 0
.
The quantities 28.34
f<~
=
~
A~ - 3~ A~
=
A6~ - A ~
are components of the electromagnetic field f = dA. Hence,
205
28.3~ We recognize
cribed
~ , ~ g~'g,~t' ~%
that 28.33 are Maxwell
Proca field equations
28.33
equations.
Similarly
imply a hamiltonian
as for the
constraint
~h des-
0y
28.56
p(~)
The existence
=
of this constraint
electromagnetic field
=
p~
field f
0 . is equivalent
to the fact that only
(and not all velocities
A#~)
is involved
in
equations. The vacuum
state is any solution
ing electromagnetic
field
28.37
(a flat connection f~
The dynamics
of field equations
and the choice
=
in B )
with vanish-
:
0
of the vacuum
completely
determine
the
Lagrangian
28.38
/j =
L dxO...Adx 3
such that
28.39
dL x
=
~dA
m + p~dA~
=
Hence
28.40
L x (A ~ , A ~ )
The hamiltonian similar
=
description
of the dynamics
to that used in the preceding
may be obtained
section.
The result
is
in a way
206
28.L~1
]~ =
defined on constraint
submanifold
H dxOA...AdX 3
Hx(A~,p~e)
28. ~2
=
~ h by the formula
p~'~A~,y_
-
Lx(A~,A~e)
1
=
I
1
g ~ g ~ P%~ p*~
Now we are going to calculate
the energy of the electromagnetic
field. According to the general discussion first define the Lie derivative
given in Section 19 we must
of the electromagnetic
potential with
respect to a vector field X in M. The value of the Die derivative each point x e M cotangent
will be a vector tangent to ~ ,
space T x M. Since a connection ~
at
i.e. an element of the
is a differential
form in
B and not in M we have to lift the vector field X from M to B. The natural lift is the horizontal
lift associated with ~
lift by X ~ . The Lie derivative
of ~
. We denote this
with respect to X~ has the fol-
lowing properties
=
0
and
28.44
L
o X~
and is thus the pull-back
of a covector field in M. We call this co-
vector field the Lie derivative
of the electromagnetic
respect to X and denote by ~ ~ X
:
28.45 X~
X
potential with
207
In terms
of local
coordinates
of B the Lie derivative the coordinate
X
induced by a trivialization
can be calculated
system (x~,y)
28.46
(x~,A~,A~)
=
as follows.
in B by setting y = ~ .
We choose
If
X/~
8x ~ then the horizontal
lift of X is
2s.~7
x~
x~
~
+ c
~x ~ where
the coefficient
c satisfies
~y the horizontality
=
+ c
condition
,dy + A ~ d x ~ >
~ X~
"
Thus 28.~9 ( cf.
c
=
- A~X 5
[51) . The Lie derivative
of ~
, with respect
to X~ is equal to
28.5O =
X~f~
dx ~
Thus 28.5q
Consequently
28.52
~(x~
= ~Zx~)~ p ~
The two energy-momentum
tensors
x~ are
= x~(%p~ •
s~)
208
28.53
t%~
=
f~/~p~m -
=
0
~L
and 28.5~
t~
.
The stress tensors are
28.55
~
:
tt~ + q
T~
=
2
t'~
=
t~
and 28.56
The covariant
28.57
,~
=
0
stress tensor
T~
is symmetric
t (~')
g~T ~
=
=
g~t~
=
and usually is called
sor of the electromagnetic
"the symmetric
traint
space
ten-
field".
The Caftan form for electromagnetic milar calculations
energy-momentum
field can be obtained by si-
as for the Proca field.
It is defined
on the cons-
~ h by the formula
28.58
O
The analysis
of the t i m e - e v o l u t i o n picture which we made for the Pro-
ca field applies
=
p~dx%
for the electromagnetic
adapted coordinates constraint
...A d?~^... ^ dx 3 - H d x % ... ^ dx 3
~zcPz
field.
If ~-- is described
by equation x 0 = 0 then field equations for Cauchy data
p
O0
:
=
0
=
0
28.59 ~ k pkO
.
in
imply the
209
The symplectic
form M
= I@P
28.60
is degenerate with respect quotient
when restricted
to
~m.
to this degeneracy.
The result
~_
~ x
Cauchy data.
tion of this reduction
29. The gravitational ~ccording
form
see e.g.
~.
For the detailed
Theory the gravitational
of the space-time
we described
the electromagnetic
various
field,
In the present
the Proca field )
other.
of the space-time ~
the
the "gravitational
formulation
momenta
ped with a linear structure meaningful.
and the notion
It follows
since there
connections.
"zero element"
nections. space
is no vector
equipped
with a linear
structure.
with each
conjecture
is
In our
belong to a space equip-
of "vanishing cannot
structure
is distinguished
On the other hand the metric
of
by two structures:
momentum".
that the connection
ted as momentum No
always
the dynamics
of the gravitational
field and the other describes of field dynamics
(the scalar
in a given geome-
g. The natural
"configuration"
field
. In the
interacting
is described
and the metric
that one of them describes
~5])
section we describe
of both matter and geometry
connection
(see
matter fields
the system composed The geometry
descrip-
field
by the geometry
try of space-time.
always
This space
[26].
to General Relativity
sections
the affine
is a
~ (fkl(X),pkOCx))
and equipped with the symplectic
preceding
may be reduced
by sections
28.61
is described
mE
of this reduction
space P ~ called the space of reduced
is described
field,
The space
momentum"
be interpre-
in the space of
among different
tensor belongs We conclude
is
con-
to the tensor
that the connection
210 shoul&play preted
the part of configuration
as momentum.
is the curvature
The third
tensor.
object
coefficients
curvature logous
is interpreted
also
[28] and
in the present ly symmetric
jet of the connection
of electrodynamics as configuration
vanish
formulation
This picture
that
the
is ana-
given in Section 28
and curvature
of General
~
= I~ ~
Given a symmetric
of these
at x. If
(by the con-
. We e x p e c t
Relativity
j
:
as velocity
coordinate
connection
(x~) and ( y ~
systems
we consider
on-
. This is not only for tech~
in space-time
fine at each point x & M a class of local coordinate is composed
structu-
~7] ).
connections
nical reasons.
Relativity
is not an independent
and their first derivatives)
to the description
should be inter-
in General
tensor will play the part of velocity.
connection (cf.
occuring
The curvature
re but it is defined by the first nection
and the metric
systems.
M we deThis class
for which the coefficients
are two coordinate
systems
belonging
~ to
this class then
8 2x~ 29.1
=
0
Y~ ~ Y~ x A class of coordinate a local inertial frame I~
vitational
~
= 0 in any coordinate
in space-time.
neighbouring Einstein's important
connection
is thus described The gravitational
tensor which
test particles.
original
ideas
a symmetric
may be treated
to the
as a field The gra-
by the field of local inertial field strength
This interpretation
role in understanding
connection
of space-time.
gives us the relative
("freely
if a local inertial
system belonging
frames defined at each point
potential
by the curvature
29.J will be called
frame at the point x. Conversely,
We see that a symmetric
of local inertial
frames
defined by equality
at each point x e M is given we may define
by setting
frame.
systems
falling
acceleration
follows
elevators")
the canonical
is described
structure
of
very closely and plays of General
an
211
Relativity. The configuration
bundle q describing
the gravity is thus the
bundle of local inertial frames in M. A coordinate ces coordinates
(x ~, I~~~)
system
in Q. The configuration
(x ~) indu-
space Qx composed
of local inertial frames at x has an affine structure.
Subtracting
J~
two elements
I~,~ and
riant ~ symmetric
Ic~> of Qx we obtain a 1-contravariant,
tensor
~
tor on Qx is a 1-covariant,
- !7~
2-cord-
in M. It follows that
2-contravariant,
a
covec-
symmetric tensor in M.
Similarly as in the preceding Sections the phase bundle is a Whitney sum
where S~ is the bundle of 1-covariant, tensors.
The following objects
~9.3
ot~,
=
dx~® ~
~~
~
form a basis in a f i b r e of the bundle the symmetric
tensor product)
2-contravariant,
® ( - - ~~ d x
symmetric
° .. . ~ d x 3 )
~# -#\~'~
(by
~
,o denote
. Every element of this fibre may be
uniquely expanded with respect to this basis. The expansion coefficients
~#~
dinate system
together with coordinates (x ~, I~ ~
,~m~)
29.4-
O-~'~#~'e ,
At each point x 6 M forms
29.5
29.6
Ox
=
_21
(x~,l~)
in P, Of course
=
in Q form a coor:
9"Era~ ' ~
~x and Co x are given by the formulae
~2~.dr~:
® ~~~-2--jdx°^ x" ~
..
. ^dx3)
Jdx0A.
.
, A d x 31
.
212 q
( the coefficient ~ appears because of the symmetry 29.4) . The infinitesimal configuration bundle is the first jet bundle Qi = jIQ. A
coordinate system {x *, r%L
, ~m I~ ~)
is induced in qi by the coordi-
nate system (x ~) in M. The coefficient
I~
29.7
~
represent derivatives
= ~a~ I/.~
The infinitesimal phase bundle is as usual the quotient bundle pi = = ~lp of the first jet bundle JqP. Coordinates in pi are denoted by x ~, ~
,I~ ~
, ~
, ~
29.8
, where
~ =
~ ~ ~-~'~
The canonical structure in pi is given by forms X
=
Ox
=
29.9
2
~dr~ +~'~dr~
and ~x
i
@i x
=
d
--
z~ ( d ~
Mdiv~
=
~dx °...~dx3
[~ ( d ~
x
~ d ~D
]~ dxO''-^dx:
29.1o
A dr~t + d ~ g ~
The gravitational momentum
gU~~
dl ~ @ ® d x °... A dx3
has a priori too many components
for describing the metric tensor g. However, similarly as in electrodynamics
(formula 28.36)
we may expect the existence of hamiltonian
constraints reducing the number of independent components of gg . The occurence of such constraints is always equivalent to the fact that not all velocities are involved in the field equations. Indeed, Einstein equations contain only the curvature tensor 29.1q
R ~
213
and less than that, namely the Ricci tensor 29.12
R~
=
R ~
Especially important for our purposes is the symmetric part of the Ricci tensor
Kfl~
=
RC~ )
=
R~S + R ~ )
=
29.13
since we don't know a priori wheather R~S is symmetric or not
(the
symmetry of the Ricci tensor will be a consequence of the dynamics in the same way as the Lorentz
condition is the consequence of the dyna~
mics in the case of the Proca field) . We assume in the sequel that the Lagrangian depends only on K j~
and not on all velocities ~
.
We prove that this assumption implies the constraints
29.14
where
~U ~
~
=
$~
~
is a symmetric tensor density.
components of
~
The number of independent
is thus equal to the number of independent
com-
ponents of ~ ' ~ , ie. to the number of independent components of the metric tensor.
We expect that the quantity
~e~ represents the contra-
variant density of the metric.
It will be shown in the sequel that the
correct identification of
with the metric is :
29.15
~
~e~
where k is the gravitational
-
1
k ~
g#~
constant.
Now we prove the formula 29.14. Our assumption about the Lagrangian
214
=
29.16
L dxO~... ^ dx 5
implies the formula
~9.~v where J ~ W
~x(~%~
, ~.)
and
=
I- J j ~ d
I~"
2
+ I_ ~
dK,~
2
are arbitrary coefficients satisfying the symmetry
condition j~
=
J ~ wA~
29.q8
Using the formula 29.Q3 we write
=
2-
~-
I/~.
+
29.19
.~
2
+ ~I ( g . ~
+
(S~c ~'~-
- s~)
d r ~% . .
On the other hand the formula 29.9 and the definition of the Lagrangian imply
2
m
215 Comparing 29.19 with 29.20 we obtain the formula 29.q¢. Moreover
or
29.22
Jm#~
_- ~
~m~ ~
- ~ %~ ( ~ )~) ~
_ 0~ ~ ~ ~
The last term may be rewritten as follows
29.23
We may add to the right-hand
side the expression
The result is
~ g~C~C~)~ =
29.25
=
- ~¢F'/~
_ ~
~6"
I~:
=
- ~ - ~
-urns-/~m~je~
Combining 29.25 and 29.22 we obtain
29.26 =
(we recall that ~
~
GC~/~
is a tensor density)
.
The coefficients
J~ ~
ar e
216
thus components of the covariant divergence of the momentum and may replace
~e~
~%~w as coordinates in pi.
We use the structure described above to formulate the dynamics of the system composed of a matter field field
~
~A
and the gravitational
. Bundles Q, P, Qi, pi are Whitney sums of bundles corres-
ponding to the field
~A
tes are x ~, ? A p~ , ? ~
and bundles constructed above. The coordina-
, ~A' ~
'0~u~'
C~
and
~ ~w . Forms
~x '
i
gO x , @~, CO x are composed of two parts. One part corresponds to the matter field and the other to the gravitational field. For example
=
A+
d
29.27 + - ~ 2 29.28
i CO x
-- d 0 i
die~
®dx O... Adx 3
.
Field equations for such a system are composed of two parts
: equations
of the matter field and equations of the gravitational field. The equations of the matter field are formally the same as in Section 12. They can be derived for example from the Lagrangian of the matter field 25.18 . We denote now this Lagrangian by
29.29
since the symbols
~mat
~
=
Lmat dxOA''" ^ dx3
and L are reserved for the Lagrangian of the
full system composed of the matter and the gravity
~this Lagrangian
will be found later) . Thus the matter equations are :
=
Lmat(
29.30
3
217
The constraint as equations
equation 2 9 . ~
in the space pi -
tions.
enables us to treat the above equations There are also two gravitational
equa-
N °
As the first gravitational
equation
one usually
takes
29.31
This is equivalent
to
29.32
~g~
=
0
V~ CU~
=
0
~
=
0
Or
29.53
Or
29.34
J~
Equation
29.31
~
implies
the Proca field us to express
=
the symmetry
equations
imply the Lorentz
the second gravitational
tion - in terms
of K#~ instead
the general
case which quantity
of Einstein
equations.
the second grangian
stress
the Proca field,
(equal
of R ~ .
condition.
~
to the Hilbert
However,
it is not obvious
the simplest i.e.
in side
case when
the matter La-
. This is the case of the scalar field, field and also hydrodynamics
30. In those cases the first
tensor T ~ )
hand side of the Einstein
equa-
should be used as the right-hand
the electromagnetic
in Section
as
This enables
- the Einstein
tensor of the matter vanishes on
similarly
equation
Let us first consider
does not depend
is discussed
of the Ricci tensor
equations
is commonly :
which
stress tensor T ~
accepted
as the right-
218
4 29.35
The coefficient
V~g 7 is necessary
sor density and not a tensor.
29.36
-
form
in our notation
Contracting
T#~ is a ten-
both sides with g ~
we obtain
=
This enables us to rewrite equivalent
since
the Einstein
equations
in the following,
:
k 29.37
Now we use equation
29.38
25.28
T#~
=
~ g ~ Lma t
and the identity
~g#~
implied by 29.45.
The result
29.~0
Ke~
Equation
9
k
29.39
-
m~~
med that Lma t does not depend
Equations
equations.
j~
29.44,
29.30,
=
~ g~
is
29.34 can also be rewritten
29.41
2
on
&~
in a similar way since we assu:
a
29.40 and 29.44
~
are the complete
set of field
They d e f i n e t h e dynamics DXi ~ P ~ . Using t h e f u n c t i o n ~mat
219
i defined on D x by Lma t we may rewrite the field equations in the following form
29.42
d~_mat = (~Ad ~A+p~ d~ A +~J~/~ d,~%/~-I-K2~ d 0 ~ l I Dix
The above formula can be interpreted as a generating formula for the lagrangian submanifold in the space ~ix described by coordinates p~ , ~ A
, ~@A' ~
(~A,
' gg~ ,K~ ,J~Y) . The generation is meant with respect
to the special symplectic structure given by the form
29.43
~ mat x
The precise definition of ~i Px is the following.
We take the subbundle
a pi defined by the constraint equation 29.14. The symplectic forms COx are degenerate when restricted to fibres With respect to this degeneracy,
cPi.x We reduce
i.e. we pass to the quotient space
~i. Each element of ~i is a class of elements of p i
Two elements of
pi belong to the same class if they have the same value of coordinaX
tes (~A p~ , ~
, ~ A , ~ ,~6~y ,K~,J~ ~) . The
plectic form ~
on the quotient
CO x~i
=
d@matx
=
form CO ix induces a
sym-
:
(d~AAd~ A +
dp~Ad~
A
+
29.44 + 1 - d J ~ A d J-I~~
+ ~-d~A
2
d K ~ ) ~ dx ° ... ^ dx 3
2
The proof of this formula follows from the calculations used in 29.19. The dynamics D xi defines a lagrangian submanifold D~ c
x described by
equations 29.30, 29.40 and 29.41. The function Lma t is the generating ~i function of D x with respect to the special symplectic structure 29.43. This structure gives a mixed picture of the dynamics: picture for the matter field and
I!
quasi-hamiltonian
11
the lagrangian picture for the
220
gravitational
field
(we use "quasi" since the genuine hamiltenian i picture refers to the space Px rather then to the reduced space ~i ) X
Variables and
(~A
~)
"
are the lagrangian variables for the matter field
(P~v ,gg~w) are the hamiltonian variables for the gravitational
field. Equations 29.30 are of lagrangian type. Equations 29.40 and 29.44 are of hamiltonian type. The pure lagrangian description of D i x ( or Dx) is obtained from the above one by the Legendre transformation applied to the gravitational degrees of freedom• The Lagrangian of the full system
(matter + gravity)
is defined by the formula
--
2
X
"
Using the identity
which we proved earlier we obtain
--
2
x
Hence
29.
- _La t )
= d(~i gg~ K#~) Dxi and
29•¢9
L
=
_Lmat + ~ ~ # ' K ~
Di X
•
The Lagrangian 29.49 is scaler in such a way that it vanishes for the
221 vacuum. The vacuum is defined by the matter vacuum the flat space-time geometry
29.50
~K~
=
~
R~
where R is the scalar curvature.
29.5~
_L
=
= O ) . Using
and
29.q5 we write
g
~,~
R~
=
-
1 k-
~-/~R
,
Thus
1
-Lmat
1 ~'~
- -k
=
~mat = 0
- ~
~-g R k
/
cf.
[23]) • The second term of the right-hand side should not be
interpreted as the gravitational Lagrangian according to the "recipe": matter Lagrangian + gravitational Lagrangian = the complete Lagrangian. This term is obtained from the Legendre transformation.
It is analo-
gous to the term pq in the formula L = p~ - H. Both ~mat and ~ are generating functions of the dynamics of the full system composed of matter and gravity with respect to two different special symplectie structures.
The notion "matter Lagrangian"
is also misleading in this
context. The difference between L and Lma t is analogous to the difference between two potentials
in two different control modes
(e.g.
the
internal energy and the enthalpy in thermostatics ) . The function L depends
on
, J~
function Lma t depends on
and
I~
~A
~
or on , itt
~A
~
,~
and K~,. The
• In order to calculaI te L from Lma t it is not sufficient to add the term - ~ ~ - ~ R but it is necessary to eleiminate the variable
field equations.
and ~
~
(the metric)
using the
Similarly the Lagrangian in particle mechanics can
be obtained from the formula L = p~ - H if we eliminate the variable p from the right-hand side.
Example The Lagrangian for the linear scalar field theory is given by the formula 16.52 where F = 0 :
222
29.52
Lma t
=
(gm~O;~?/~
I ~
- m2~ 2 )
Using the formula
29.53
q
k4
det O~ ~
g
2
det g ~
q
k4
-
g
we write
q ~k~U~ ~
+ k 2 ~_detTg~'
m2~2)
The field equations in this case are 16.53, 29.q4, 29.40 and 29.41. It follows from 19.76 that t ~
m%~
29.55
=
g~
= O. Thus T~/~ = t~/c. We conclude that
p~O~
- g%/~Lma t I
= ~
^
i
2
2)
Einstein equation 29.40 can thus be rewritten in the following equivalent form
~
29.56
Ig m 2 2~
This implies
I ~:~K 29.57
and
=_ 1
2m2~
223 I
29.58
2
2
Combining 29.57 with 29.49 we obtain
I~
=
(Lma t + ~
K~/~) D i
=
29.59 1 m2
2 m 2 ~ 2 k 2 V-- detO'd~
2 ~
In order to calculate the determinant g we use again the Einstein equation 29.58 :
29.60
~ 0 t~f~- :~ K%/~
•
Thus
29.6d
g
=
det g%#
=
~ m2~
det ( ? ~ ? # -
~ K~/z
)
Inserting 29.61 in 29.59 we obtain the final formula for the Lagrangian
~:~
q
, is the combination of I~ where K~,
2 -I
and
~
_ I
:
given by formula 29.q3.
The reader may easily check that the Euler-Lagrange equations derived from the variational principle
29.63
~#d.~
=
0
J where as usual
29.6~
~
=
T, d x O . . . ^ d x
3
224
are equivalent to the system composed of equations 16.53, 29.34 and the Einstein equation 29.56
( cf.
~8]).
In the general case the matter Lagrangian depends on
P#~
and
the equation 29.34 can not be written in the form 29.41. It can be shown that equation 29.3~ is incompatible with the matter equations and the Einstein equations in the sense that dynamics defined by them is net lagrangian.
The interaction between matter and gravity would
not be reciprocal,
so we propose to replace equation 29.34 by another
equation which saves the reciprocity of the interaction
(cf.
~8]).
Our equation follows from the following recipe: the matter Lagrangian ~mat
should always be taken as the generating function of the dynamics
in the mixed picture 29.43, i.e. as the Lagrangian for the matter field and the
(quasi) Hamiltonian for the gravitational field. The dynamics
generated this way is described by equations 29.14, 29.30, 29.40 and 29.41. We proved that the equation 29.40 is the Einstein equation
29.65
G#~
=
I K#.~- ~ g # ~ g ~ @ K ~
=
•
k
I
T#~
V--~
It follows from 25.29 that the equation 29.%1 is equivalent to
29.66
J%#~
The condition
r#y
V%g~
=
~
~gW#~ ~
=
- T#~
•
= 0 is no longer satisfied and the connection
is no longer the metric connection
. Hence, the geometry
of the space-time differs from the flat Minkowskian geometry in two aspects
: the curvature and the non-metricity of the connection.
The
source of the curvature is the first stress tensor 29.65. The source of the non-metricity is the second stress tensor 29.66. The formula 29.49 describing the Legendre transformation to the pure lagrangian picture remains valid. Now we formulate the hamiltonian description of the dynamics.
225 The pure hamiltonian picture is not unique since there is no unique connection in the bundle Q. Each coordinate system (x ~) in M defines its own horizontality condition
17'~(x ) =
: the horizontal sections are defined by the
const. The hamiltonian special symplectic struc-
ture defined with respect to this horizontality is given by the form
1 p3~, ~ d ~ l @
ut
29.67
dxO...Adx.3
The Hamiltonian
=
29.68
H d x O A . . . A dx 3
is defined on the constraint manifold
Chap
given by equation 29.14.
The Hamiltonian is a function of variables
,
~ ,fig~ )
and
can be obtained by the Legendre transformation from the Lagrangian
1 ~
/~,~
+ ~1 ( r / ~~,
- I / ~-1~, )
•A =
?A
p~
:
- Lma t - ~1 K~ v@g,,,-~ =
~
29.69
1 =
~1~ e~ -
-
1
.-~t 1~"
-, ~" I~,t
= - Lma t
~,
By ~mat we denote the matter hamiltonian
29.70
=
Hma t
%
~A
PA
- Lmat
"
The Hamiltonian H depends on the choice of the coordinate system (x~). However,
the Cartan form in
~h
226
=
p~ d~°, .. ~ d ~ A . . . ~ d x ~
+
29.7q
+
~
~ ~ dx° . . . .~dl~
^
.
.A d X.~ . Hdx . O
AdX 3
does not depend on the choice of coordinate system as it was proved in Section 20. The form ~
29.72 ~ m a t
=
is composed of two parts. The first part
P~ dxO~ " ' ' ^ d ~ AA'''mdx3 - HmatdxO .... adx3
corresponds to the matter field and the second part
@gr
=
q •# ~ dx0A d ~/~A .Adx 3 + 5~ "'" ~ ""
+ ~ 0~
29.73
=
~
~
-
~
. . . .
~#~ dx ° ....~ d r Z ~ ..~d~3 + A
+
I~
-
~#)
dxO...^dx 3
corresponds to the gravitational field. The form troduced by W. Szczyrba
(see
~7] and
~ g r was first in-
~8] ) who directly proved its
invariance under general coordinate transformations. The construction of the energy E(X,~) is similar to the one given in the preceding Sections. In adapted coordinates the form
@(X,~) is
given by the formula
O(x,~)
~ A d ~ A + pAkd ~ A k - ~AodPA° +
29.74
+ ~I%
~
d I~% + ~1 ~ k
d F~-~
I ~+~od~
~°
227
The corresponding proper function ~(X,~) may be directly obtained by the Legendre transformation,
i.e. by the formula 19.7¢ :
0~
- X~L
=
29.75 1 ~ X I/~ w~
+ -- ~
X~R
2k
where
The Lie derivative of the connection depends on the second jet of the field X :
Using 25.1 f o r the matter energy E~at(X] and the formula 29.77 we obtain
=
1 \FL-g
(
x ~ t~ • + ~
:~
R,$~
-
1 -
E tJ'~
#'~':~
~
R ~,~.
)
+
29.78 + t~ ~ V~ X~ + ~1 ~Z~ ~
X~
~7(/~% )
.
We shall prove later that this formula can be transformed to
29.79
z~(x)
=
~
~(x)
= ~
~(x)
,
where the antisymmetric
tensor density H%~(X) equals
29.8o
k
~,-(x)
-
- x~t~V)
228
29.81
V ~ -- g~# V m
Formula 29.79 implies the conservation law
29.82
w(x)
=
-~z~x)
=
-)~#~#(x)
for any vector field X in space-time.
=
o
This proves that all transfor-
mations of the space.time are symmetry transformations
of the theory.
For physical interpretation of conservation laws 29.82 see and
[30] , [2~
~]. The analysis of the time-evolution picture for General Relativity
is similar to that given in Section 28 for electrodynamics.
If %-
is
described in adapted coordinates by equation x 0 = 0 then the space of Cauchy data P Z is parametrized by sections
29.83
X"
~Xc PZ
The field equations define a subspace compatible with the dynamics.
of those data which are
The symplectic form
M
=
d
is degenerate when restricted to ~ z . with respect to this degeneracy. Elements of ~ z
Odr
The space
To find an appropriate parametrization (cf.
C ~ may be reduced
The result is a quotient space ~ x
(reduced Cauchy data)
serious technical problem
) dx Adx dx3
[2] and
.
are classes of elements of P~. of reduced Cauchy data is a ~5])
which we do not dis-
cuss here. A new solution of this problem is proposed in
We prove now the energy formula 29.79
~
.
( see [28]). ~he formula
29.75 can be rewritten in the following form
229
E~(x)
~,1 - ~ g R
=
29.85
_ ~I R.
~,~
+ I[X~"
_ ~I R % ~ . , ~ , )
~
+
%
+ ~
x ~. _
We use the f o r m u l a
29.86
~-/ \-~. implied
by 25.9
and i n t e g r a t e
1XGR~
X~
- E
by parts
~
the last two terms
of 29.85.
T h e n we a p p l y the f o r m u l a
29.87
The result
~
=
.
is
~(x)
= ~at(X) + X ~21 v -~~ R + + ~,~ ~,~)+
~.~
I x~(p~z~
+
% ( ~ v ~ x,~) +
+ -
2
N o w we use e q u a t i o n s
EJat CX) 29.89
25.1,
25.17
and the E i n s t e i n
equation.
We o b t a i n
230
The non-metricity equation 29.66 implies
29.90
t (x~)~ = 2
~
2
2
and
29.91
The above formulae imply
+
+
~
+
29.92
But q
29.93
=
R[~]~
~ - ~ R~6" ~
=
- k~--~g(R[~]g~%
+ ~
+ R(~)~ )
( r}~= o)
Using an inertial system
=
one oan e a s i l y prove the f o r -
mula
T~ - g ( R F ~ ] g ~ + R ( ~ ) ~ )
The last two formulae show that the second term in 29.92 vanishes. This completes the proof since the formula 29.90 implies 29.95
t(~,~)
=
~3v~-7 ~ ~
231
30. The hydrodynamics To describe
the hydrodynamics
cosity we need a 3-dimensional terpreted
as material
for this theory
points
"material
Q
where M is a space-time see in the sequel
space"
of the fluid.
is the trivial
30.1
of a barotropic
fluid without
Z. Points
bundle
Q
bundle
=
M× Z
equipped
with a fixed geometry
that the dynamics
does not depend
q = (x,z)~ Q tells us that the point
terial
the point x in the space-time.
the union of tensor products
of Z are in-
The configuration
A configuration occupies
vis-
(g, F ) .
We will
on the c o u n e c t i o n z ~ Z of the ma-
The phase bundle P is
13.1 3
30.2
where
=
Pq
q = (x,z).
As a bundle
T z* z ~ A
Tx* M
over M, P has fibres
equal to
3
30.5
Px
If (x ~) is a coordinate
system
in Z, a = 1,2,3,
=
T*Z @ A
T*x M
in M and (z a) is a coordinate
system
then the tensors
3o.4
form a basis
in the vector space Pq. Every element
ce can be uniquely
30.5
It follows
represented
~
=
that the expansion
as a linear
~ 9~ ae~a
~ s P of this spa-
combination
of tensors
30.4:
•
coefficients
~ a% together
with coordi-
232 nares
(x~,z a) in Q form a coordinate system (xl, za, ~ a%) in P. The
canonical forms
~x and
60x are given by the formulae:
30.6
~
m adZ ~ ( ~
3o.7
COx
x
a
=
d~
=
~JdxO
(d ~CaA ~
=
""^
dx3 )
,
dza)® ( ~x~ / d x O A .
" "
Adx3).
The infinitesimal configuration bundle Qi is the first jet bundle J~Q. The coordinate system (x~,z a) in Q induces a coordinate system (x ~, za ,z%) a in Qi . The coordinates zl a represent partial derivatives
a
30.8
z%
=
~
~z
a
.
The infinitesimal phase bundle pi = TriP is
a
quotient bundle of the
first jet bundle jIp. The coordinate system (x ,za, ~ )
in P induces
a coordinate system (x%,za~ ~a, ~ % z ai, ~a) in P where
30.9
ae a
-
_~%1~ ma
"
At each point x ~ M the infinitesimal phase space pi is a symplectic x manifold with the symplectic form
i dO x
=
=
E~ ( d
"~a A d~.a
)I° dx °
dx 3
;3O.lO
(d ~a"dza+ d ~
dzt)~dxO..., dx~ .
The lagrangian special symplectic structure is given by the lagrangian l-form
o~ 30.11
~ ~v0
=
[ ~ ( ~ ] ~ o . . .
~ ~x~
:
233
=
(~adZ a +
The above f o r m a l i s m Each particular space
q ~ a d Z ~ ) ® d x O A ...A dx 3 •
is c o m m o n
t h e o r y needs
Z. In the t h e o r y
to
all t h e o r i e s
an a d d i t i o n a l
of elastic
media
a riemannian
metric
m u c h weaker.
It is a scalar d e n s i t y r
This d e n s i t y
enables
g i v e n volume
in Z. In the t h e o r y
us to m e a s u r e
~ c Z the i n t e g r a l
of the fluid are c o n t a i n e d
of c o n t i n u o u s
structure
of the m a t e r i a l
this a d d i t i o n a l of fluids
this
a differential
the q u a n t i t y
o£ r over
~
media.
structure structure
5-form]
of the fluid.
is is
in Z. For a
tells us how m a n y moles
in (Y.
Given a s e c t i o n
3o.42
M
or,
a mapping
equivalently,
3o.15
we d e f i n e
~
a matter
in M o b t a i n e d
~
:
x
x
current
3o.
current
satisfies
dj
since dr = 0 as a ~ : f o r m the
quantity
~
zCx)
j which
j
5
q(x)
=
=
e
z
e
~
r
q
,
of r t h r o u g h
~
(a
5-form)
:
.
the c o n t i n u i t y
-- d ? * r
(x,z(x))
is a v e c t o r d e n s i t y
by t a k i n g the p u l l - b a c k
30.14
The m a t t e r
~
equation
--
in the 3 - d i m e n s i o n a l
0
space
Z. We c o n s i d e r
234 which describes
the density
moving reference
of matter
a fluid if the Lagrangian
a only via the matter density z2
derivatives
per volume )
in the co-
frame.
Our theory describes
~
depends
on
:
= g(z a, ~z~))
30.17
We call our fluid homogeneuos This happens
if the Lagrangian
when the dynamical
for each ~oint
i i OxIDx
equivalent
properties
z~ Z. The dynamics
3o.18
does not depend
of the fluid are the same
is given by the generating
=
on z a.
formula
dE
to
30.19
where,
~moles
~adZ a + ~adz a
=
dL
as usual,
30.20
~
The field equations
~Ta
v
=
L dx~...
Adx 3
•
are
=
~a
L
=
za 30.21 ~6~
In the simplest
case of
a
=
~ L
az~
homogeneous
fluid the above
duce to ? ~
30.22
--
~e a
--
o
equations
re-
235
~
~L
ba
dL
~
3j ~
1
_
dL
~j~
~z%
where
30.23
u~
1
:
is the unit vector in space-time.
g~J~
This vector is the four-velocity of
the matter. We have used the formula
~
30.24
1
=
--
u,, ,
which the reader may easily check. In order to give the equivalence
an
interpretation of the Lagrangian
~
and to show
of the field equations 30.22 with the Navier-Stokes
equations we pass from the lagrangian picture to the energy ture performing the partial Legendre transformation.
pic-
Since the bundle
Q is trivial the notion of the Lie derivative for sections of Q coincides with the partial derivative.
Dragging the point q = (x,z)~ Q
along the field X is simply moving the fixed material point z along the integral curves of X. Thus
30.25
~X
za
=
X/~ ~/~za
=
X ~a z/~
The formula 19.74 reads
=
za/
a- X~L
=
(z/~a
%a_
~L)X ~ "
It follows from 19.78 that the first energy-momentum tensor is equal to 30.27
t ~/~
=
~ ~a Z ~a -
S ~% L
236
and the second
energy-momentum
tensor vanishes
30.28
t m~
The following
equality
% a ~aZ~
30.29
The simplest
r
The corresponding
j
dL d~
~0 ~ r
~ ( ~/~ - u~'u/~ ) "
unimodular
coordinates
(z a)
£or which we have
=
dz~d~d~
expression
=
:
it is to use
in Z, i.e. the coordinates
3o.3o
0
can be proved
-
way to prove
=
:
3
for the matter
=
~zl
~z2
current
is
~ z3 d f ~ A d f ~ a dx ~b~
3o.54
j~ ( ~ J d x ° . . . ~ d x 3) ~x # where
3o.32
j#-
The index modular
A
means
coordinates
(-~)~
~(z~z2"z3J
(xO ..,x3) A
=
(-1) ~ det
Zo,-..,
z3
a ~ j~ z~ =
j~ %
~
-
j~
~
.
q 2
Using uni-
the reader may easily verify the following
z~
Thus
z3
that the /~-th column has been omitted.
la :
3o.33
q Zo,..-, 2
formu-
237
3o.3~ 1
dL
j~
u,~ j ~')
%
-
~L &% d~(#-
~'u#)
and
t %
~ ~(%-
-
u~u~) - ~ ~
_-
3o.35 _-
On the other hand,
_
_
dL
_
the e n e r g y - m o m e n t u m
%
tensor
u m u~ )
.
in h y d r o d y n a m i c s
is
30.36
where
6
reference In order
is the frame
energy
density
(per unit
volume)
in the c o - m o v i n g
and p is the pressure.
to be able to internret
our t h e o r y
as h y d r o d y n a m i c s
we have
to take
3o.37
The
~,
functions
p(~)
and
--
~(~)
- ~
~(~)
are not independent.
dL or
5o. 59
The
p
function
quantity We have
=
_(5
=
d~ _~)
~ -~)
-~(~ =
We have
S2
dd ~
~
"
e (~) • = Z(2) 9 is the energy density per one mole. The I v - ~ is the volume c o r r e s p o n d i n g to q mole of the fluid.
238
d
~d--~ e(~)
3o.4o
-
and
I
.t2 P
y
f
o
oQ
r
We see that the energy of d mole of the fluid is equal to the energy to the
corresponding
raryfied
state
(density
equal to zero)
plus
the amount of work which is necessary to compress our mole from the raryfied
state to the actual density ~ . Hence, the so called "con-
stitutive
equation"
p = p(~)
uniquely determines
the Lagrangian
and the dynamics. Formula 25.q5 implies that the stress tensor is equal to the energy-momentum tensor~
30.~2
T~
since t ~
=
=
tm/~
,
O. Formula 25.34 is thus equivalent
to the Navier-Stokes
equations
3o.
3
=
where the covariant derivative connection
nations
0
is taken with respect to the metric
P~=
There is an interesting rodynamics.
=
analogy between electrodynamics
and hyd-
In both cases the Lagrangian depends only on a few combi-
of components
of the configuration
mics and j~ in hydrodynamics)
jet
. The independence
nents is connected with a gauge-invariance "gradient gauge"; points without
in hydrodynamics
(f#~
in electrodyna-
of remaining
compo-
(in electrodynamics
a
a change of "names" of material
changing the matter density )
The first pair of Maxwell equations
@
is automatically
satisfied
239
because of the definition of f#~. The continuity equation is automatically satisfied because of the definition of j. The second pair of Maxwell equations is thus an analogue o~ the Navier-Stokes and the material variables les j~.
equations~
(z a) are potentials for the Euler variab-
Appendices
A. Sections
of fibre bundles
Let
A.q
iTg
:
be a differentiable
F
~
mapping of a differentiable
rentiable manifold B. We call ~ ge F b = ~ -q(b)
B
manifold F onto diffe-
a f i b r a t i o n of F if the inverse ima-
of each point b e B is a submanifold
of ~. This subma-
nifold is called the fibre over b. An example of a fibration is the canonical p r o j e c t i o n
A.2
pr B
:
B × F
~
of a product B × F onto the component vial.
B. This fibration
A fibration A.J is said to be locally trivial
has a neighbourhood over
B
~
~
such that the set
is diffeomorphic
to the product
~-IC~) ~×
is called tri-
if each point bEB
of points of F lying
F b in such a way that the
diffeomorphism
A.5
~,
:
~-I(O)
~ 0
× Fb
~
maps fibres of F onto fibres of
satisfies
Formula A.g means that locally trivial
fibration
~
F b. A
is called a fibre bundle and a diffeomorphism
A.3 is called a local trivialization. in
~×
and (yA) is a coordinate
If (x ~) is a coordinate
system
system in F b then a local trivialization
241
cM induces
a
local
coordinate
pression of the mapping
A.5
system
(x~)
=
bundle TB of a manifold
The manifold
in
The coordinate
F.
ex-
9~ is
X (x~,Y A)
The tangent
(x~,yA~
TB is the collection
•
B is an example
of all tangent
of a fibre bundle.
vectors.
The fibra-
tion A.6
assigns
qYB
:
TB
~ B
to each vector the point at which the vector is attached.
2ibre ToB = ~U B1(b) is the vector Given a coordinate
system
space of vectors
(x %) in B we assigne
tangent
Each
to B at b.
to each vector u ~ TB
coordinates (x~,u ~) consisting of coordinates (x~) of % ( u ) and the components
(u z) of u with repect
A.7
u
The coordinate
bundle
A.8
assigns
u~
example
of a fibre bundle
T*B of covectors
~U B
to each covector
~aoh fibre T ~ B = ~ I b ) in B induce
coordinates
:
T*B
p
a local triviali-
is provided
~ B
is the cotangent space at b (x~,p~]
=
by the
in B. The fibration
the point at which the covector
is attached.
Coordinates (xO
in T'B, where p~ are components
a covector p defined by
A. 9
(x ~) :
system (x~,u %) can be used to define
zation of TB. Another cotangent
=
to the system
p%dx %
of
242 I
In general
any tensor bundle
re bundle.
The bundle
A.q0
over a manifold
F
of k-covectors
an example
A section
completely
appearing
s
bundle
:
in a manifold
field
B
is
• F
of b belong to the fibre F b.
B is a section
(a differential
T~B. A differential
mate system
tensors )
in the notes.
such that for each b e B the image s(b)
A covector
antisymmetric
of a fibre bundle A.q is a mapping
A.qq
A vector field
of a fib-
k / ~ T*B
=
(k-covariant,
frequently
B is an example
~form)
of the tangent is a section
k-form is a section of
(x~,y A) of F a section s is described
A
bundle
TB.
of a cotangent
T~B.
by s(x)
In a coordi= (x~,yA),
where yA = fA(x).
B. Tangent mapping Let
B.I
be a differentiable
~
:
mapping
point x e M the derivative at x a vector tangent
M
~
N
of a manifold
of ~
to N at
assigns ~(x).
M on a manifold
N. At each
to each vector u tangent
We denote this vector by
to M
~.u.
The mapping
B.2
~.
:
TM
defined this way is called the tangent
~ TN
mapping
~ induced b y e )
. The
243
restriction Let
of ~ ,
to TxM is thus a linear mapping
(x~) be a coordinate
system
of TxM on T~(x)N.
system in M and let ( y % ) be a coordinate
in N. Let
B. 3
c< Cx ~ )
=
y ~
where
B.~
be
y~
a coordinate
c~(x)
=
expression
for ~
. The value
of ~
on a vector A.7 is
equal to
o~.(u~ ~x~)=
B.5 Hence,
the coordinate
u~ ~
expression
for
~@
~y~ is
B.~
o<~Cx~,u~ = (~¢~) ,u~ %x---7C ~ )
C. Pull-back
of differential
forms
Let
C.I
O<
:
M
~
be a differentiable
mapping
tot p in N attached
at a point
attached
N
of a manifold
M on a manifold
~ ( x ) e N defines
at a point x e M. The definition
of ~ p
a covector
N. A covec~Wp
in M
is given by the con-
dition
c.2 which we assume
< ~,~p> to be fulfilled
= <~u,p> for each vector u ~ TxM.
The mapping
244
C.5
T~(x) N
>
T~x M
defined this way is the adjoint mapping for of ~ ,
~ , TxM
(the restriction
to T x M ) . If
C.~
p
=
p~dy ~
where (y%) is a coordinate
system in N and
C.5
u~
u
=
x~ where (x ~) is a coordinate
system in M and if B.% is a coordinate
ex-
pression for O( then the formula C.2 reads
g<%
U~
~ ~<~dx ~
Hence, the coordinate
expression for
C.7
=
o(~p
o(~ is :
C<~dx ~
p~
or
G.8
--
)
The formula
e n a b l e s us to extend the mapping k
O< '~ to the space of multicovectors k
:
245
Let
0
be a differential
k-form
in N : K
Y
We define
a k-form
The form
c<~O
commutes
c~O
in M by setting
is called the pull-back
with the exterior
c.13
of
~
differentiation
dd,~8
=
from N to M. The pull-back :
o(~d@
D. Jets In order to be able to associate differential
equations
tion about partial
intrinsic
we need geometric
derivatives
meaning with partial
objects
of sections
containing
of fibre bundles.
informaThe par-
tial derivatives
D.I
fA(x)
fA =
~ x .'~
of the section
described ledge
in Appendix A do not define
of these derivatives
in one coordinate
to calculate
such derivatives
set ( f A , f A )
of values
already
sufficient
a geometric
in another
a geometric
since know-
system is insufficient
coordinate
of a section together
to define
object
system.
The joint
with the derivatives
object.
is
If (fA f~)-- are known
246
A I A~ in a coordinate system (x~,y A) then the corresponding values (£ ,f ~ ) in a coordinate system ~x ~',yA~j are given by
D.3
fAI
=
Y
At(fA,x~)
x~
,
[ S
~ x~
]
where x #~
=
x ~ (x~)
yA j
=
i A ,x ~) yA~y
0.5
describes the change of coordinates.
Formulae D.3 and D.4 express
transformation laws of components (fA f A )
of an object called the
first jet of a section s. Let s I and s 2 be two sections having the same jet at a point b ~ B .
0.6
Then
st(b)
=
s2(b
and the two sections have the same partial derivatives at b with respect to any coordinate system(x~).
The geometric meanning of this si-
tuation is that the graphs of the two sections are tangent to each other at the point f = Sl(b ) = sy(b) E F b. We conclude that the class of sections of F tangent to each other at a point f * F b is a suitable abstract representation of a jet. As an example we analyse jets of vector fields. Let
D. 7
X
:
B
~
TB
be a vector field described locally in a coordinate system (xe,u%) ( see Appendix A ) by
247
~.s
X(b)
=
( ~ - , u ~)
=
f~(~)
where
D.9
u~
We denote by f b
the partial derivatives
9f% . Let (x ~') be another x~ coordinate system in B and let (x#,u ~) be the corresponding system in
TB. If
r
= x,(x 0
.Io then
D.lq
u Z~ =
- -xcb~u ~ ~
Transformation laws for components ( f ~ , f ~ )
f%~ =
D.12
D.13
f%'~,
=
9 x ~'
of jets are
~ x %' f% x~
f'%
~
~x %'
8x ~ +
f~
jx ~ ~ x~
I
According to these transformation laws partial derivatives f ~
alone
do not determine partial derivatives f~,. We see that the partial derivatives separately do not define a geometric object although jointly with the values f~ they form the set of components of a geometric object - a jet. A jet is not a tensor. The space of first jects of sections of a fibre bundle F mula A.q)
(for-
is denoted by J~F. The jet of the section D.2 at the point
be B is denoted by jls(b). Coordinates of jqs(b) are thus (x~,fk,fA) . The point s(b)~ F b is called the target of the jet jls(b). The mapping which assignes to each jet jls(b) its target s(b) is called the canonical jet-target projection and is denoted by
D. 14
('F
:
JIF
~
F •
248
We usually jection
treat the space JIF as a fibre bundle
over B with the pro-
~ ~b F.
Let
D.Q5
~
:
be another fibre bundle
G
~
B
over the same basis B. We suppose
that the
mapping D.~6
~
preserves
D.
the fibre
:
F
,
structure,
i.e.
7
=
The mapping
~
induces
D.18
:
defined by the following
E. Bundle
m;
a mapping
J~
The mapping j 1 ~
G
JQF
~
jIG
condition
is called the first jet prolongation
of vertical
of ~
.
vectors
Let
E.I
~U
be a fibre bundle of the tangent
:
F
~
B
(see Appendix A) . By VF we denote
bundle TF composed
of vertical
vectors,
a submanifold i.e.
of vectors
249
tangent to fibres of F. At each point b E B the fibre VbF is equal to the bundle TF b of vectors tangent to the fibre F b. We treat the space VF as a fibre bundle over B
E. 2
~
:
VF
~-
B
where
:3
and
_- : o(:
~FIVF is the restriction
in the tangent bundle TF
IvF)
to V ~ c T F
of a canonical projection
q~F
(see Appendix A) .
If (x~,y A) are coordinates
in F satisfying A.5 then each verti-
cal vector may be written in the form
A
E.4
u
The coefficients te system ( x
=
u
8 ~ yA
u A together with coordinates (x~,y A) form a coordina-
,yA,uA) in VF. The coordinate
expression for the projec-
tion E.2 is
There is a canonical which may be described Coordinates in JqVF
identification
in coordinate
of the bundle VJqF and JqVF
language in the following way.
(xb,yA,u A) in VF induce coordinates ~ x ~ , y A , u A , y A , u A
~ s e e Appendix D ) . If (x~,fA,f A
) are coordinates
each vertical vector in JqF may be represented
E. 6
v
-- v A
B
~fA
+ wA
~ 3 ~A
by
in JqF then
250
This means that coordinates
in VJqF are ( x ~ f A ~ f A , v A ,wA~
tification of the two bundles f~
with y A ,
of
The iden-
is obtained by identifying fA with yA,
v A with u A and w A
F. Tensor product
~ •
with u A
fibre bundles
Let
F.~
0~
:
F
~
B
:
G
~
B
and F.2
be two vector bundles
over B. The notion
"vector bundle"
each fibre F b and G b is equipped with a linear structure.
means that The tensor
product
of the two bundles
is again a vector bundle
over B. The tensor product
is defined by the following formula
This means that fibres of F @ G are tensor products
of fibres of F
and G.
G. The Lie derivative Assuming that F is a tensor bundle over B we construct the Lie derivative
s of a section s of F with respect to a vector field X X in B. A vector field X generates a one-parameter local group of trans-
formations
~
of the manifold B. We denote
Transformations parameter
elements
@t can be applied to tensors. v
local group
~t~
of transformations
of this group by
This results
~t"
in a one-
of the bundle F. The
251
g e n e r a t o r of the group is a v e c t o r f i e l d ~
in 7. The f i e l d ~ is a lift
of the f i e l d X in the sense that
for each f e 7.
G i v e n a s e c t i o n s of the b u n d l e F we can lift the f i e l d
X to a f i e l d sw(X)
of v e c t o r s
and t a n g e n t to the graph. tical vectors ~X
in F d e f i n e d
on the g r a p h of the s e c t i o n
The d i f f e r e n c e ~ -
s~(X)
is a f i e l d of v e r -
d e f i n e d on the g r a p h of s e c t i o n s. The L i e d e r i v a t i v e
s is a s e c t i o n of the b u n d l e VF d e f i n e d by
G.2
A l t h o u g h we a s s u m e d that the b u n d l e F is the t e n s o r b u n d l e
over B the
same c o n s t r u c t i o n will w o r k w h e n e v e r the a c t i o n of d i f f e o m o r p h i s m s B can be l i f t e d to F. In the
case w h e n this can be d o n e
cal w a y we call the b u n d l e F the b u n d l e In the s p e c i a l
case of a t e n s o r bundle
one step f u r t h e r .
Since 7 b is a v e c t o r
of g e o m e t r i c
of
in a c a n o n i -
objects
o v e r B.
the c o n s t r u c t i o n can be c a r r i e d space we can i d e n t i f y TF b w i t h
7b× 7b. With this identification ( ~ s) (b) is a pair (s
G.3
section
B
~
b
~
f(b)
~
7
is a t e n s o r f i e l d of the same type as s. In the case of a t e n s o r b u n dle it is this f i e l d w h i c h
is u s u a l l y c a l l e d the Lie d e r i v a t i v e of s.
List of more important symbols
Whitney sum of vectors bundles
F~B
exterior product of differential forms or covectors
x~
interior product of a vector field X and a differential form
@
tensor product
F×B
cartesian product of differentiable manifolds F and B
av
boundary of a domain V
d~
exterior differential of a differential form covariant derivative
~=
~ x% partial derivative horizontal lift of a unit vector from the basis R q to a vec-
2
~t
tot bundle over RI; see Sections 9 and 23
d__
lift of a unit vector to dynamics; see Sections 9 and 25
dt
s
Lie derivative of a section s with respect to a vector
X field X; see Appendix G []
([]g)wave
TqQ (TQ)
operator
(with respect to a pseudo-riemannian metric
space tangent to a manifold Q at a point q~ Q
(tangent
bundle ) ; see Appendix A T~Q (T~Q) space dual to TqQ
A~M
8qq
(cotangent bundle) ; see Appendix A
space of k-covectors
cotangent to M at x e M
first jet extension of a fibre bundle q
(space of first
jets of sections of Q) ; see Appendix D space of generalized first jets
(divergences)
of sections
of a fibre bundle P; see Section 16 and Appendix D
J qCt) IpCt)
first jet of a section q at a point t; see Appendix D generalized first jet of a section p at a point t; see Section q6
VF
bundle of vertical vectors tangent to a fibre bundle F; see Appendix E
253
the canonical jet-target projection for jets of sections
LF
of bundle F; see Appendix D
canonical projections from a phase bundle to a configuration bundle mapping tangent to a mapping ~ ; see Appendix B
c~p
pull-back of a covector p via ~ ; see Appendix C first energy-momentum tensor;
t
,~
see Section 19
second energy-momentum tensor;
see Section 19
S~
spin angular-momentum tensor; see Section 19
T~
first stress tensor;
/*
see Section 25
second stress tensor;
see Section 25
TH
the Hilbert tensor;
g,~,
metric tensor in space-time
rZ
affine connection in space-time Christoffel symbols + w~
w(~) w,-~m]
:
=
)
. w@~)
~(~
u~p~
see Section 25
(metric
connection
)
symmetric part of a tensor wa@ antisymmetric part of a tensor w~p
value of covector p on vector u
(a contraction of
u and p) contraction of the first factor of the tensor product fl3.1
$
<
,
>2
Q
contraction of the second factor of the tensor product 13.1 restriction of a fibre bundle Q over M to a submanifold ~
sl v ilDi Xl
X
c: M; see Section 19
restriction of a section s to restriction of a form dynamics)
8V; see Section 14
G~ to a lagrangian submanifold D i
; see Section 8
X
REFERENCES
[I]
Abraham R., Marsden J.E.: Foundations of mechanics, N.Y. 1967 Benjamin
[2]
Arnowitt R., Deser S., Misner C.W.: The dynamics of general relativity.
In: Gravitation - an introduction to current research
/ Witten L. ed. / N.Y. 1962, Wiley
[3] [4] [5] [6]
Belinfante F.J.: Physica 6 /1939/ p.887 Belinfante F.J.: Physica 7 /1940/ p.449 Be ssel-Hagen E.: Mathem. Annalen 84 /1921/ p.258 Caratheodory C.: Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Leipzig u. Berlin q935
[7]
Cartan E.: Legons sur les invariants int@graux, Paris 1921, Herman
[8]
Chernoff P.R., Marsden J.E.: Properties of infinite dimensional hamiltonian systems, Springer Lecture Notes in Mathematics, vo!. 425
[9]
Dedecker P.: Calcul des variations, champs g@od@siques;
formes dif£@rentielles
et
In: Coll. internat, du C.N.R.S. Strasbourg
1953
[Io]
Dedecker P.: Calcul des variations et topologie alg@brique, M@m. Soc. Roy. Sci. Liege 19 /1957/
[11]
Dedecker P.: On the generalization of symplectic geometry to multiple integrals in the calculus of variations.
In: Diff.
Geom. Methods in Mathematical Phys., Springer Lecture Notes in Mathematics,
vol. 570
de Donder Th.: Th@orie invariante du calcul des variations, Paris 1935, Gauthier-Villars
[15]
Einstein A.: Preuss. Akad. Wiss. Berlin, Sitzber.
q915, p.884
Einstein A.: Preuss. Akad. Wiss. Berlin, Sitzber.
q925, p.414
Fischer A., Marsden J.E.: Comm. Math. Phys. 28 /q972/ p.1
255
Garcla P.L.: Collect. Math. 19 /1968/ p
[17] [18] bg]
73
Garcia P.L.: Symposia Math. 14 /1974/ p. 219 Garcla P.L., P@rez-Rend$n A.: Comm. Math. Phys. 13 /1969/ p.22 Garcla P.L., P@rez-Rend6n A.: Archly Rat. Mech. and Anal. 43
/ 1971/ p.lOl
[20] [21]
Goldschmidt H., Sternberg S.: Ann. Inst. Fourier 23 /1973/p.203 Hehl F.W., Kerlick G.D., v. der Heyde P.: Z. Naturforsch. 31 /1976/ p.111
[22]
Hermann R.: Lie Algebras and Quantum Mechanics, Benjamin Lecture Notes
[23]
1970
Hilbert D.: KSnigl. Gesel. d. Wiss. GSttingen, Nachr. Math.Phys. KI. 1915, p.395
[24] [25]
Kijowski J.: Comm. Math. Phys. 30 /1973/ P.99 Kijowski J.: Bull. Acad. Polon. Sci. /math., astr., phys./ 22 /1974/ p.1219
[26]
Kijowski J., Szczyrba W.: Comm. Math. Phys. 46 /1976/ p.183 Kijowski J., Sm61ski A.: On the Cauchy problem in general relativity /in preparation/
[28]
Kijowski J.: On a new variational principle in general relativity and the energy of the gravitational field, GRG Journal, 9 /1978/ p. 857
[29]
Kobayashi S., Nomizu K.: Foundations of Differential Geometry, N.Y. 1963
[30] [31]
Komar A.: Phys. Rev. 113 /1959/ p.934 Kumpera A., Spencer D.: Lie equations, Annals of Math. Studies, Number 73, Princeton Univ. Press., Princeton, N.J. 1972
[32]
Landau L., Lifshitz E.: The classical theory of fields, Reading Mass. 1951, Addison-Wesley
[33]
Lang S.: Differential Manifolds, Reading Mass. 1972, AddisonWesley Lepage Th.: Bull. Acad. Roy. Belg., 28 /1942/ P.73 and p.247
256
[35] [36] [37]
Marsden J.E.: Arch. Rat. Mech. Anal. 28 /q968/ p.323 Marsden J.E.: Arch. Rat. Mech. Anal. 28 /q968/ p.362 Menzio M.R., Tulczyjew W.M.: Ann. Inst. H. Poincar$, 28 A
/1978/ p.349
[38]
Minkowski H.: Space and Time, an Address delivered at the 80-th Assembly of German Natural Scientists and Physicians at Cologne, 21 Sept. 1908; In: The principle of relativity, N.Y. 1923, Dodd, Mead and Co
[39]
Noether E.: K~nigl. Gesel. d. Wiss. GSttingen, Nachr. Math.Phys. KI. 1918, p.235
[~o]
Nomizu K.: Lie groups and differential geometry, Tokyo 1956, Math. Society of Japan
[41]
Pirani F.A.E.: Gauss's Theorem and gravitational energy; In: Les Th@ories Relativistes de la Gravitation, Royaumont Conference 1959, C.N.R.S. Paris 1962
[42]
[43] [44]
Rosenfeld L.: Acad. Roy. Belg. 18 /1940/ p i Schouten J.A.: Ricci - calculus, Berlin 1954, Springer Souriau J.M.: Structure des syst~mes dynamiques, Paris 1970 Dunod
[45]
Sniatycki J., Tulczyjew W.M.: Indiana Univ. Math. J. 22 /1972/ p.267
[,~6]
Sternberg S.: Lectures on differential geometry, Englewood Cliffs N.J. 1964, Prentice-Hall
[4?] [48]
Szczyrba W.: Comm. Math. Phys. 51 /1976/ p.163 Szczyrba W.: On the geometric structure of the set of solutions of Einstein equations, Dissertationes Math. Warszawa 1977, PWN
[49]
Trautman A.: Conservation laws in General Relativity; In: Gravitation - an introduction to current research / Witten L. ed./ N.Y. 1962, Wiley
[5o] [51]
Trautman A.: Comm. Math. Phys. 6 /1967/ p.248 Tulczyjew W.M.: Symposia Math. I~ /1974/ p.247
257
[52]
Tulczyjew W.M.: A symplectic formulation of field dynamics; In: Diff. Geom. Methods in Mathematical Phys., Springer Lecture Notes in Mathematics,
[53] [5 1
Tulczyjew W.M.: Ann. Inst. H. Poinear$,
27 A /1977/ p. I01
Tulczyjew W.M.: A generalization of jet theory, Polon
[55]
vol. 570
Sci. (math. p h y s
astr )
26 /1978/
Bull. Acad.
p.611
Weinstein A.: Advances in Math. 6 /1971/ p.329 Weinstein A.: Ann. of Math. 98 /1973/ P.377
[57]
Weinstein A.: Lectures on Symplectic Manifolds, Lectures held at the University of North Carolina, March 8 - 12, 1976 /mimeographed notes/
[58]
Weyl H.: Ann. of Math. 36 /1935/ p.607
