Gauge mechanics

X whose trivialization morphisms 4>a restrict to linear isomorphisms

xPa(x)

:YX^V,

Vz 6 Ua.

Dealing with a vector bundle Y, we always use linear bundle coordinates ( I A , J / ' ) associated with the above-mentioned bundle atlas 9. We have (pr 2 o ipa)(y) = y'a, y = y'e,(x) = yxipa(x)

l

(e<),

where {e,} is a fixed basis for the typical fibre V of Y, while {e,(z)} is the fibre basis (or the frame) for the fibre Yx of Y, which is associated with the bundle atlas 9. By virtue of Theorem 1.1.2, vector bundles have global sections, e.g., the global zero section 0(X). If there is no risk of confusion, we write 0, instead of 0(X). A morphism of vector bundles # : Y —► Y' is defined as a fibred morphism over / : X —► X' whose restriction $ x : Yx —» Y^^ to each fibre of Y is a linear map. It is called a linear bundle morphism over / . The following assertion is a corollary of Theorem 1.1.3. PROPOSITION 1.1.4. If Y —► X and Y' -» X are vector bundles and $ : Y -> V" is a linear bundle morphism of constant rank over X, then the image of $ and the kernel Kerjj$ of $ with respect to the zero section 0 of Y' —> X are vector subbundles of Y' —* X and Y —> X, respectively. Note that a vector subbundle of a vector bundle is a closed imbedded submanifold. □ Unless otherwise stated, by Ker $ of a linear bundle morphism $ is meant its kernel with respect to the zero section 0. There are the following standard constructions of new vector bundles from old. • Let Y —> X be a vector bundle with a typical fibre V. By Y" —► X is meant the dual vector bundle with the typical fibre V, dual of V. The interior product of Y and Y' is defined as a fibred morphism J\:Y®Y"

—>XxR. x

14

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

• Let Y —> X and Y' —» X be vector bundles with typical fibres V and

V,

respectively. Their Whitney sum Y © Y' is a vector bundle over X with the typical fibre V

®V.

• Let V —► X and Y' —» X be vector bundles with typical fibres V and

V,

respectively. Their tensor product Y ® Y' is a vector bundle over X with the typical fibre V ® V". Similarly, the exterior product Y AY is denned. In particular, let $ : Y\ —* Y2 be an injection of a vector bundle iT\ : Y\ —» Xi to a vector bundle 7r2 : Y2 —► X 2 over a diffeomorphism f : X\ —* X2. Then, there is the dual surjection

$ • : y2* — y;,

v^eTrrHr'o^^cy,,

<*•(«),«> =<«,*(«)>, over the diffeomorphism /

u e y2-,

(1.1.3)

'.

AfRne bundles Let W : y —* X be a vector bundle with a typical fibre V. An afSne bundle modelled over the vector bundle Y —» X is a fibre bundle n : Y -+ X whose typical fibre V is an affine space modelled over V, while the following conditions hold. • All the fibres Yx of Y are affine spaces over the corresponding fibres Yx of the vector bundle Y. • There is a bundle atlas * = {(Ua,ipa)} restrict to affine isomorphisms

i M i ) : Y* -» V,

of Y —► X whose local trivializations

Vx e [4.

In particular, every vector bundle has a natural structure of an affine bundle. Dealing with an affine bundle, we use only affine bundle coordinates {xx,y') associated with the above-mentioned bundle atlas * . We have the fibred morphisms YxY x YxY x

—>Y, x —>Y, x

(y\V')^y' i

+ y',

<

(y ,j/' )-2/'-y'i,

where (y1) are linear coordinates on the vector bundle Y.

1.1. FIBRE

BUNDLES

By virtue of Theorem 1.1.2, affine bundles have global A morphism of affine bundles $ ; Y —» Y' is a fibred restriction $z : Yx —► Vi. , to each fibre of Y is an affine bundle morphism over / . Every affine bundle morphism $ : Y —► Y' from an over a vector bundle Y to an affine bundle Y' modelled determines uniquely the linear bundle morphism

15

sections. morphism over / whose map. It is called an affine affine bundle Y modelled over a vector bundle Y

: Y - Y * ,

—

3$'

called the linear derivative of $>. Let Y x Y' be the fibred product of two affine bundles Y —» X and Y' —► X x which are modelled over the vector bundles Y —» X and Y —» X, respectively. This product, called the Whitney sum, is also an affine bundle modelled over Y(BY. Furthermore, let Y' —> X be an affine bundle modelled over a vector bundle Y —► X. Let Y C Y' be an affine subbundle modelled over a vector bundle Y —> X. Assume that Y is the Whitney sum of Y and a complementary vector bundle Z —<■ X. Then one can easily verify that the affine bundle Y' —» X decomposes in the Whitney sum

Y' = Y®Z. x Tangent and cotangent bundles The fibres of the tangent bundle ■Kz : TZ —» Z of a manifold Z are tangent spaces to Z. Given a coordinate atlas

* z = {(%*e)} of a manifold Z, the tangent bundle is provided with the (holonomic) atlas *={(7r;I[/?),^ = T^)},

CHAPTER

16

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

where by Tfy is meant the tangent map to <j>^. The associated linear bundle coor­ dinates (zx) on TZ are called the induced (or holonomic) coordinates with respect to the frames {d\} in tangent spaces TZZ. Their transition functions read dz» Every manifold morphism / : Z —* Z' generates the fibred morphism of the tangent bundles Tf :TZ

—>TZ', I

i*oT/ = §£*. called the tangent map to / . The cotangent bundle of a manifold Z is the dual ■K.Z :T'Z

— Z

of the tangent bundle T Z —> Z. It is equipped with the (holonomic) coordinates (zx, ix) with respect to the coframes {dzx} dual of {d\}. Their transition functions read ., _ dz» . Zx ~ dz^2"Tangent and cotangent bundles of fibre bundles Let TTY : TY —» Y be the tangent bundle of a fibre bundle ir : Y —> X. Given fibred coordinates (x A ,y') on Y, the tangent bundle TY is equipped with the coor­ dinates (xx,y',xx,y'). The tangent bundle TY is a fibre bundle ix o Try : T Y -» X

over X, while the tangent map TIT to 7r defines the fibration Tn:TY

— TX

There is the commutative diagram TY

^TX

I

I

y

—► x Tf

1.1. FIBRE

BUNDLES

17

The tangent bundle TY -> Y of a fibre bundle Y -> X has the vertical tangent subbundle VYd=KerTn of TV, given by the coordinate relation i A = 0. This subbundle consists of the vectors tangent to fibres of Y. The vertical tangent bundle VY is provided with the coordinates {xx,yx,y') with respect to the frames {d,}. Let T $ be the tangent map to a fibred morphism $ : Y —► Y'. Its restriction V = T$\VY

: VY -» VY',

i/"oV$ = ^ $ ' = ^ $ ' ,

(1.1.4)

to VY is a linear bundle morphism of the vertical tangent bundle VY to the vertical tangent bundle VY', called the vertical tangent map to $. Every vector bundle Y —> X admits the canonical vertical splitting of the vertical tangent bundle VY = Y x Y x

(1.1.5)

because the coordinates y' on VY have the same transformation law as the linear coordinates y' on Y. An affine bundle Y —> X modelled over a vector bundle Y —* X also admits the canonical vertical splitting of the vertical tangent bundle VY^YxY

(1.1.6)

x

because the coordinates y' on VY have the same transformation law as the linear coordinates y1 on the vector bundle Y. The cotangent bundle T'Y of a fibre bundle Y —» X is equipped with the coordinates (xx,y1,i\,yi). There is its natural fibration T'Y —» X over X, but not over T'X. The vertical cotangent bundle VY —» V of a fibre bundle Y —» X is defined as the vector bundle dual of the vertical tangent bundle VY —> Y. It should be emphasized that there is no canonical injection of VY into the cotangent bundle T'Y of Y, but we have the canonical projection C : T'Y *

-

VY,

Y x

C : xxdx

+ i/idy* i-> jfidj/*,

(1.1.7)

18

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

where (dy'} are the bases for fibres of VY, which are dual of the frames {9,} in the vertical tangent bundle VY. With VY and V'Y, we have the following two exact sequences of vector bundles over Y: 0 — VY — TY -> Y x TX — 0,

(1.1.8a)

0 -» Y x T'JT — T * r -» v * y -» 0.

(1.1.8b)

For the sake of simplicity, we will denote the pull-backs

YxTX, x

YxT'X x

simply by TX and T ' X . E x a m p l e 1.1.2. Let us consider the tangent bundle TT'X of T'X and the cotan­ gent bundle T'TX of TX. Relative to coordinates (xx, p\ = ±\) on T'X and (xx, xx) on TX, these fibre bundles are provided with the coordinates (xx,p\, xx,p\) and (xx,xx,±\,xx), respectively. By inspection of the coordinate transformation laws, one can show that there is the isomorphism a : TT'X

s

T'TX, P\ <—► Xx,

(1.1.9)

Px *—> i\

of these bundles over TX [43, 96]. Given a fibre bundle Y —► X, there is the similar isomorphism

av : VV'Y S W r , x

t

i

over VY, where (x ,y ,pl,y ,pl) V'VY, respectively. •

ft <—> j/i, x

p, <—► in ,

and (x ,y ,y\yi,y\)

(1.1.10) are coordinates on V V V and

Sheaves There are several equivalent definitions of sheaves [16, 80). We will start from the following. A sheaf on a topological space X is a topological fibre bundle 5 —> X whose fibres, called the stalks, are Abelian groups Sx provided with discrete topology. A presheaf on a topological space X is defined if an Abelian group Sy corresponds to every open subset U C X (Sj = 0) and, for any pair of open subsets V C U, there is the homomorphism rv : Su —> Sy

1.1. FIBRE

BUNDLES

19

such that rv = Id Su, ru _ rv V T w — ~w~v>

W C V C U.

Example 1.1.3. Let X be & topological space, Su the additive Abelian group of all continuous functions on U C X, while the homomorphism rv : Su —* Sy is the restriction of these functions to V C X. Then {Sv, ry} is a presheaf. • Every presheaf {Su, ry} on a topological space X yields a sheaf on X whose stalk Sx at a point x € X is the direct limit of the Abelian groups Su, x e U, with respect to the homomorphisms ry. It means that, for each open neighbourhood U of a point x, every element s e Su determines an element sx € Sx, called the germ of s at x. Two elements s £ Su and s' € S'v define the same germ at x if and only if there is an open neighbourhood W 3 x such that rru s w

. _— rvr sI . w

For instance, two real functions s and s' on X define the same germ sx if they coincide on an open neighbourhood of x. The sheaf generated by the presheaf in Example 1.1.3 is called the sheaf of continuous functions. The sheaf of smooth functions on a manifold X is defined in a similar way. Two different presheaves may generate the same sheaf. Conversely, a sheaf de­ fines a presheaf of Abelian groups T(U, S) of local sections of the sheaf S. This presheaf {T(U,S),ry} is called the canonical presheaf of the sheaf S. It is eas­ ily seen that the sheaf generated by the canonical presheaf {r(U, S),ry} of the sheaf S coincides with S. Therefore, we will further identify sheaves and canonical presheaves. Example 1.1.4. Let Y —* X be a vector bundle. The germs of its sections make up the sheaf S(Y) of sections of Y —» X. The stalk SX(Y) of this sheaf at a point x e X consists of the germs of sections of Y —> X in a neighbourhood of x € X. The stalk SX(Y) is a module over the ring C™(X) of the germs at x G X of smooth functions on X. If we deal with a tangent bundle TX —» X, the stalk SX(TX) is a Lie algebra with respect to the Lie bracket of vector fields. •

20

CHAPTER

1.2

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

Multivector fields and differential forms

Subsections. Vector fields, 20; Vector fields on fibre bundles, 21; Multivector fields, 22; The Schouten-Nijenhuis bracket, 24; Exterior forms, 25; Exterior forms on fibre bundles, 26; Interior products, 28; Bivector fields and 2-forms, 29; The Lie derivative, 31; Tangent-valued forms, 31; Distributions, 32; Foliations, 34. Vector fields A vector field on a manifold Z is defined as a global section of the tangent bundle TZ -» Z. The set 7~(Z) of vector fields on Z is both a module over the ring C°°{Z) of smooth functions on Z and a real Lie algebra with respect to the Lie bracket [v, u] = (vxdxu" -

v^dxv^d^, u = uxd\,

v=

vxd\.

A curve c : () —► Z, () C R, in Z is said to be an integraJ curve of a vector field u on Z if c = u o c, cA(t) = u A (c(t)),

te().

Recall that, for every point z € Z, there exists a unique integral curve

c:(-e,e)-Z,

( >0,

of a vector field u through z = c(0). A vector field u on an imbedded submanifold N c Z is said to be a section of the tangent bundle TZ —► Z over N. It should be emphasized that this is not a vector field on a manifold N since u(N) does not belong to TN C TX in general. A vector field on a submanifold N C Z is called tangent to the submanifold iV if u(N) C TAT. Let U C 2 be an open subset and t > 0. By a JocaJ 1-parameter group of local diffeomorphisms of Z defined on (—c, e) x (/ is meant a mapping G :(-e,()

xU3{t,z)^G,{z)eZ

which possesses the following properties: • for each t e (—e, e), the mapping G ( is a diffeomorphism of £/ onto the open subset Gt(l7) C Z;

1.2. MULTIVECTOR

FIELDS AND DIFFERENTIAL

FORMS

21

• Gt+t,(z) = (Gt o G,)(z) if t + H e (-«, e). If such a mapping G is defined on 1 x Z, it is called a 1-parameter group of diffeomorphisms of Z. THEOREM 1.2.1. [100]. on U C Z defines a local to the curve s(t) = Gt{z) Z. For each z G Z, there local 1-parameter group

Each local 1-parameter group of local diffeomorphisms G vector field u on U by setting u{z) to be the tangent vector at ( = 0. Conversely, let u be a vector field on a manifold exist a number e > 0, a neighbourhood (/ of 2 and a unique of local diffeomorphisms on (—e, e) x U, which determines

u. D

In brief, every vector field u on a manifold Z is the generator of a local 1parameter group of local diffeomorphisms. In particular, every exterior form (p on a manifold Z is invariant under a local 1-parameter group of local diffeomorphisms G u with the generator u, i.e., g'4> = 0,

V 5 e G„,

if and only if its Lie derivative Lu4> along u vanishes. If a vector field u on a manifold Z is induced by a 1-parameter group of diffeo­ morphisms of Z, then u is called a complete vector Held. Vector fields o n fibre b u n d l e s A vector field u on a fibre bundle Y —> X is said to be projectable if it projects over a vector field ux on X, i.e., if the following diagram Y

X

JL^TY —*TX

is commutative. A projectable vector field has the coordinate expression U= TlV)& + « W ) 3 i .

it* = uxdx-

A vector field r = TA&A on a base X of a fibre bundle V —» X can give rise to a vector field on Y, projectable over r, by means of some connection on this fibre bundle (see (1.4.7) below). Nevertheless, a tensor bundle T=(®TX)®(®TmX),

22

CHAPTER

1. INTERLUDE: BUNDLES, JETS,

CONNECTIONS

admits the canonical lift r = T*0„ + [duT^x^;0a-

+ ...

°/3iT xv0i-0k

• ' J o - a , am

(1.2.1)

of any vector field r o n l . In particular, there exist the canonical lift f = T<^ + 3„Ta±"A

(1.2.2)

of r onto the tangent bundle TX, and its canonical lift ri

f = r^dy, -

d0Tvxu-— oxff

(1.2.3)

onto the cotangent bundle T'X. Hereafter, we will use the compact notation (1.2.4)

dx = — dxx

A projectable vector field u = u'<9, on a fibre bundle Y —* X is said to be vertical if it projects over the zero vector field ux = 0 on X. Let Y —» X be a vector bundle. Using the canonical vertical splitting (1.1.5), we obtain the canonical vertical vector field (1.2.5)

uY = y'd,

on Y, called the Liouville vector Held. For instance, the Liouville vector field on the tangent bundle TX reads x

(1.2.6)

UTX = i d\.

Accordingly, any vector field r = rxd\ on a manifold X has the canonical vertical lift TV = TXdX

(1.2.7)

onto the tangent bundle TX. Multivector fields A multivector field d of degree | t? |= r (or simply an r-vector field) on a manifold Z is a section

4= V r!

1

Xr

dXlA---AdXr

1.2. MULTIVECTOR

FIELDS AND DIFFERENTIAL

FORMS

23

of the exterior product l\TZ —> Z. Let us denote by %{Z) the vector space of r-vector fields on Z. In particular, T\{Z) is the space of vector fields on Z (denoted by T(Z) for the sake of simplicity), while TQ(Z) is the vector space Cca{Z) of smooth functions on Z. All multivector fields on a manifold Z make up the real Z-graded vector space T,{Z) which is also a Z-graded exterior algebra with respect to the exterior product of multivector fields. Given a manifold Z, the tangent lift tf onto TZ of an r-vector field d on Z is denned bv the relation d{F,...,<j')=d{a\...^) where: (i) ak = akdxx

(1.2.8)

are arbitrary 1-forms on the manifold Z, (ii) by

ak = x»dvO$dxx + akxdxx are meant their tangent lifts (1.2.24) onto the tangent bundle TZ of Z, and (iii) the right-hand side of the equality (1.2.8) is the tangent lift (1.2.22) onto TZ of the function -d(or,..., a1) on Z [67]. We then have the coordinate expression t? = - r t f A l V d A l A - - A d A r , r: (1.2.9) 7"!

tfAl"*'£&,

A • • • A S A , A •■•A& r ].

In particular, if r is a vector field on a manifold Z, its tangent lift (1.2.9) coincides with the canonical lift (1.2.2). If an r-vector field # is simple, i.e., tf = T ' A - - - A T ' ,

its tangent lift (1.2.9) reads T

■d= ^ T y A - - - A f i - - - A r ( ; , i=l

where T£ is the vertical lift (1.2.7) onto TZ of the vector field r*. E x a m p l e 1.2.1. The tangent lift of a bivector field

w^^w^d^hdv

24

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

is w = - ( i A a A w ^ 4 A 3„ + I O " " ^ A dv + ufdn

A dv)

• Schouten-Nijenhuis bracket The exterior algebra of multivector fields on a manifold Z is provided with the Schouten-Nijenhuis bracket which generalizes the Lie bracket of vector fields as M o w s [13, 181]: [., .]SN : %{M) x TS(M) -» T r + S _,(M),

r!

(1.2.10)

SI

[tf,u]sK = * * t » + ( - l ) " t ; * t f , * *"

=

^j(^

2

A r

^a'

a

* ^ 2 A • • • A 3 Ar A dol A • • • A da.).

There following relations hold:

[ M a , = (-1) WM M]SN.

(1.2.11)

[«/, 0 A V]SH = [u, tf]SN A v + (-1){M-DI«I^ A [v, u] S N , (-ijMCM-D^ (tfi U]SN]SN + ( _ i ) W 0 " l - « ^ [„, „] SN ] SN +

(1.2.12)

(-l)M(l«l-i)[ t , j [ V | t f ] s s ] S N

=

(1.2.13)

0.

Example 1.2.2. Let w = -uiMI/dM A dv be a bivector field. Its Schouten-Nijenhuis bracket reads [U>,W]SN = w^d^w^d^

A dx, A dX3.

The Schouten-Nijenhuis bracket commutes with the tangent lift (1.2.9) of multivectors [67], i.e., [tf,v]SN = [IMSN-

(1.2.14)

1.2. MULTIVECTOR

FIELDS AND DIFFERENTIAL

25

FORMS

Remark 1.2.3. Let us point out another sign convention used in the definition of the Schouten-Nijenhuis bracket [125]. This bracket, denoted by [., .]SN-, is (1.2.15)

[IMSN' = - ( - 1 ) ' " [ 1 M S N . The relation (1.2.11) for this bracket reads

[MSN<

= (-ir'-'X'^MlsN'.

(1.2.16)

The relation (1.2.12) keeps its form, i.e., [v, t? A v]SN, = [u,tf]SN
A

[„,

v]m,t

(1.2.17)

while the relation (1.2.13) is replaced by ( _ 1 ) (M-.)(M-i, [ l / ] [dMsN,]sN,

+ (^M-iXW-Dp,

(_1)(M-i)(W-i)[w[j/^sN,]sN,

=

[Vf v ] s N , ] s N ,

+

(1.2.18)

0

The equalities (1.2.16) and (1.2.18) show that, with the modified Schouten-Nijenhuis bracket (1.2.15), the Z-graded vector space T.{Z) of multivector fields on a manifold Z is a graded Lie algebra, where the Lie degree of a multivector field $ is | d \ — 1. In particular, a&d{v)^[d,v\sw

(1.2.19)

is a graded endomorphism of degree | ■& | — 1 of the graded Lie algebra T,(Z). is a vector field, the endomorphism (1.2.19) is the Lie derivative adtf(u) = L#v

If ■&

(1.2.20)

of the multivector field v along ti. •

Exterior forms An exterior r-form on a manifold Z is a section cj>^-d>x1...xrdzXlA---AdzXr r! of the exterior product A T'Z —» Z. We denote by Dr(Z) the vector space of exterior r-forms on a manifold Z. This is also a module over the ring D°(Z) = C°°{Z).

26

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

From now on we will use the notation C°°{Z) for the ring of smooth functions on a manifold Z, while D°(Z) stands for the vector space of these functions as a rule. All exterior forms on Z constitute the exterior Z-graded algebra 0"{Z) with respect to the exterior product. The exterior differential is the first order differential operator d:Dr(Z)-»Dr+1(Z), d = - S „ 0 A l r! on D'{Z).

Ar dz"

A dzx> A • • • dzx',

It obeys the relations

dod = Q, d{4> A a) = d{4>) A a + (-l) w <£ A d(a), where | <> / | is the degree of cj>. Given a manifold map / : Z —* Z*, by f'(j> is meant the pull-back on Z of an r-form <j> on Z' by / , which is defined by the condition r4>{v\ ■ ■ .y){z)

= (PiTfiv1),...,

Tf(vr))(f(z)),

Vv\ ■•■■«/

eTzZ.

We have the relations /•(^Aa) = / > A / V ,

df'4> = r{d4>). For instance, if in ■ N —» Z is a submanifold, the pull-back i j ^ onto TV is called the restriction of an exterior form <j> to N. Exterior forms on fibre bundles Let n : Y —► X be a fibre bundle with fibred coordinates (i A ,y'). The pull-back on Y of exterior forms on X by n provides the inclusion it* : 0*(X) -» D*(y). Exterior forms

<&: y - A r*x,
1.2. MULTIVECTOR

FIELDS AND DIFFERENTIAL

FORMS

27

on Y such that d\<j> = 0 for arbitrary vertical vector field d on. Y are said to be horizontal forms. A horizontal n-form is called a horizontal density. We will use the notation w = dx 1 A • • • A d z " ,

^A

(1.2.21)

= d\\w.

In the case of the tangent bundle TX —► X, there is a different way, besides the pull-back, to lift onto TX the exterior forms on X [67, 110, 189]. Let / be a function on X. Its tangent lift onto TX is defined as the function

/ = ±XdxJ-

(1.2.22)

Let a be an r-form on X. Its tangent lift onto TX is said to be the r-form a given by the relation (1.2.23)

^ ( n , . . . , f r ) = Cr(Ti,...,T r ),

where r4 are arbitrary vector fields on TX, and ?< are their canonical lifts (1.2.2 onto TX. We have the coordinate expression a = -aXv.\rdxXi r!

A---Adxv,

a = —[x}idliaxl...xrdxXi r!

A • • • A dx Ar +

(1.2.24)

r

5^CV- ArdzAl A • • • A dxA' A • • • A dx Ar ]. t=l

The following equality holds: da = da. Example 1.2.4. Given a 2-form fi = -Sl^dx*

A dx"

on a manifold X, its tangent lift (1.2.24) onto TX reads 0 = hx^xSl^dx11 •

A dx" + il^dx"

A dx" + fi^dx" A dx").

(1.2.25)

28

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

Interior products The interior product (or the contraction) of a vector field u = u^d^ and an exterior r-form <j> is given by the coordinate expression u

\ = £ *=l

{ ~\-uA*
-

A • • • A d / ' A • • • A dz A ' = (1.2.26)

7 7TT«M0^,2...ardza* A • • • A dz Q '. ( r - 1)! It satisfies the relations (1.2.27)

0 ( u i , . . . , U r ) = Ur\ ■■■Ui\4>, W

u\{<j>Aa) = u\4> A a + {-\) (j>

(1.2.28)

Au\a,

[U,U']J0 = u\d{u'\4>) - u'\d{u\4>) - u'\u\d<j>,

0 6O'(Z).

(1.2.29)

The generalization of the interior product (1.2.26) for multivector fields is the left interior product

0\4> = W), l*!
^eD'(Z),

tf e r.(z),

of multivector fields and exterior forms, which is derived from the equality (j>(u{ A--- A Mr) = ^ ( U i , . . . , ^ ) ,

* 6 0'(Z),

u, e T ( Z ) ,

for simple multivector fields. We have the relation ■d\v\<j> = (t)A #)J0 = ( - l ) | u | | % J t f | 0 ,

*eom

i9,w<=T.(Z).

Example 1.2.5. The formula (1.2.29) can be generalized for multivector fields as follows [13]: [ I M S N J 0 = ( - l ) M ( " | - I , 0 J d M 0 ) + (-l)l*l«jd(tfj«) - ujdjd*, where | 0 | = | i? | + | u | - 1 . • The right interior product

*!.* = *(*).

Ul
tfeO'(Z),

i? 6 T.(Z),

1.2. MULTIVECTOR

FIELDS AND DIFFERENTIAL

FORMS

29

of exterior forms and multivector fields is given by the equalities

0(&,...,0r)=0|VL*i.

^

AeO'CZ),

0 e rr(Z),

= ( T Z ^ ^ - ^ - ' ^ A , A • • • A d0r_„ 0

6D'(Z).

It satisfies the relations (tf A u)|> = tf A (v|>) + (-1) M (T?L<W A U,

^efl'iZ),

tf(<M,aeO'(Z). In particular, if | ■d | = | 0 |, we have the natural pairing

(,):T r (Z)xD r (Z)-C°°(Z), (0, 0) = tf|0 = T?[0 = #(0) = « W

(1.2.30)

Bivector fields and 2-forms Each bivector field w = -w^df,. A d„ on a manifold Z defines the linear fibred morphism Wt-.T'Z

w,(a)

z

-*TZ,

= —w(z)[a,

IW'(Q) =

a e r;z,

(1.2.31)

)w

w {z)aiLdl,,

which fulfills the relation to(z)(a,/J)

= w'(a)J/8 = w(z)[/?|a,

zeZ,

a,(3eT'zZ.

One says that a bivector field w is of r a n i r at a point z € Z if the morphism (1.2.31) has rank r at 2. If this morphism is an isomorphism at all the points z € Z, the bivector field w is said to be non-degenerate. Such a bivector field can exist only on an even-dimensional manifold. The morphism (1.2.31) can be generalized to the homomorphism of graded al­ gebras £>m(Z) —► T,(Z) in accordance with the relation W*(^)(
0 eo m

1

aiCO ^).

(1.2.32)

30

CHAPTER

1, INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

This is clearly an isomorphism if the bivector field w is non-degenerate. Each 2-form ft = ^zQ.^dz'1 A dzv on a manifold Z defines the linear fibred morphism ftb : TZ — T'Z, (1.2.33)

n\v) = -n^iz^dz". One says that a 2-form ft is of rank r at a point z 6 Z if the morphism (1.2.33) has rank r at z. This is the maximal number Ik such that A Q(z) ^ 0. The kernel of a 2-form ft is defined as the kernel Kerft d = \J{v e TZZ : ujujft = 0, Vu 6 TZZ) zez

(1.2.34)

of the morphism (1.2.33). Its fibre Ker 2 ft at a point z e Z is a vector subspace of the tangent space TZZ whose codimension equals the rank of ft at z. If a 2-form ft is of constant rank, its kernel (1.2.34) is a subbundle of the tangent bundle TZ in accordance with Proposition 1.1.4. A 2-form ft is called non-degenerate if its rank is equal to dim Z at all points z G Z. A non-degenerate 2-form ft can exist only on a 2m-dimensional manifold. Then A ft is nowhere vanishing, and can play the role of a volume element on Z. On a 2m-dimensional manifold Z, there is one-to-one correspondence between the non-degenerate 2-forms ftm and the non-degenerate bivector fields WQ in accor­ dance with the equalities wn(,°) = n«,(wJi(«?l)),wJi((7)),

(1.2.35a)

nw(-d,v) =

(1.2.35b)

wn(ni(ti),{t(v)),

0,aeD'(2),

tf,i/eT(Z),

where the morphisms WQ (1.2.31) and ftj„ (1.2.33) obey the relations

4 = (Ar\ ftiua/jU'n

=

°3<

i.e.,

wtittm = 0,

<£(«&(*)) = -

1.2. MULTIVECTOR

FIELDS AND DIFFERENTIAL

FORMS

31

T h e Lie derivative The Lie derivative of an exterior form 4> along a vector field u is given by the equality Lu<£ = u\d(j> + d{u\4>).

In particular, if / is a function, then Uf

= u{}) = u\d}.

The relation Lu(<£ A a) = Lucj> A a + <j> A Lucr is fulfilled. Given the tangent lift <j> (1.2.24) of an exterior form 4>, we have L u (0) = u'4> [67, 147]. The Lie derivative (1.2.20) of a multivector field v along a vector field u is LUV = [U,V]SN' = [u,f]sN,

and it obeys the equality L„(i9 Au) = Lutf A « + t)A Luw in accordance with the relation (1.2.17). T a n g e n t - v a l u e d forms Elements of the tensor product OT(Z) ® T(Z) r-forms

are called the

tangent-valued


Xrdz

Xl

A • • • A dzXr ® d„.

There is one-to-one correspondence between the tangent-valued 1-forms <j> on a ma­ nifold Z and the linear bundle endomorphisms over Z: 4>:TZ -> TZ, 0:T2Z3«~?;Jtf>(z)eT2Z,

(1.2.36)

32

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

and $' : T'Z — T'Z, $' : T'ZZ 3 B ' H {z)\v" e T ; Z .

(1.2.37)

In particular, the canonical tangent-valued J-form 9Z = dzx dx

on Z corresponds to the identity morphisms (1.2.36) and (1.2.37). Let Z = TX. There is the fibred endomorphism J of the tangent bundle TTX of T X such that, for every vector field r on X, we have J of =

TV ,

J o TV = 0,

where r is the canonical lift (1.2.2) and Tv is the vertical lift (1.2.7) onto TTX vector field r on T X . This endomorphism reads

JW0 = a*,

J0X) = 0.

of a

(1.2.38)

It is readily observed that J o J = 0, and the rank of J equals n. The endomorphism J (1.2.38), called an almost tangent structure [110, 189], corresponds to the tangentvalued form <j>j = dxx ® dx

(1.2.39)

on the tangent bundle TX. Distributions An n-dimensional smooth distribution on a /c-dimensional manifold Z is an ndimensional subbundle T of the tangent bundle TZ. We will say that a vector field v on Z is subordinate to a distribution T if it is a section of T —► Z. An integral curve of a vector field, subordinate to a distribution T, is called admissible with respect to T. A distribution T is said to be involutive if the Lie bracket [u, v!\ is a section of T —> Z, whenever u and u' are sections of the distribution T —» Z. A connected submanifold TV of a manifold Z is called an integral manifold of a distribution T on Z if the tangent spaces to N belong to the fibres of this distribution at each point of N. Unless otherwise stated, by an integral manifold we mean an

1.2. MULTIVECTOR

FIELDS AND DIFFERENTIAL

FORMS

33

integral manifold of maximal dimension, equal to dimension of the distribution T. An integral manifold N is called maxima] if there is no other integral manifold which contains N. T H E O R E M 1.2.2. [185]. Let T be a smooth involutive distribution on a manifold Z. For any point z £ Z, there exists a unique maximal integral manifold of T passing through z. □ In view of this fact, involutive distributions are also called completely integrable distributions. If a distribution T is not involutive, there are no integral submanifolds of di­ mension equal to the dimension of a distribution. However, integral submanifolds always exist, e.g., the integral curves of vector fields, subordinate to T. We refer the reader to [68] for a detailed exposition of differential and Pfaffian systems. A differential system S on a manifold Z is said to be a subbundle of the sheaf S(TZ) of vector fields on Z whose fibre S z at each point z € Z is a submodule of the Cf> (Z)-module SZ{TZ) (see Example (1.1.4)). The germs of sections of a distribution T obviously make up a differential system S(T). The Bag of a differential system S is the sequence of differential systems S,=S,

S 2 = [S,S],

•••

S, = [Si-i,S].

Here [S,S'] 2 is the Cf (Z)-module generated by [v,u], v € S 2 , u e S^. Let S(T) be a differential system associated with a distribution T, and let S(T) =

S , c S j C -

be its flag. In general, S, is not associated with a distribution. If this is the case for all i, we may define the Sag of a distribution T = T i C T 2 C • • •.

(1.2.40)

A distribution is called regular if its flag (1.2.40) is well defined. The sequence (1.2.40) stabilizes, i.e., there exists an integer r such that T r _i ^ T r = T r + i [184]; moreover T r is involutive. In particular, if r = 1, we are dealing with the integrable case. If T r = TZ, the distribution T is called totally non-holonomic. A codistribution T* on a manifold Z is a subbundle of the cotangent bundle. For instance, the anniMator Ann T of an n-dimensional distribution T is a (k — n)dimensional codistribution.

34

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

THEOREM 1.2.3. [185]. Let T be a distribution and A n n T its annihilator. Let A Ann T be the ideal of the exterior algebra O'(Z) which is generated by elements of Ann T. A distribution T is involutive if and only if the ideal AAnn T is a differential ideal, i.e., d(AAnnT) C AAnnT. D COROLLARY 1.2.4. Let T be a smooth involutive r-dimensional distribution on a fc-dimensional manifold Z. Every point z £ Z has an open neighbourhood U B z which is a domain of a coordinate chart ( z \ . . . , zk) such that the restrictions of the distribution T and its annihilator Ann T to U are generated by the r vector fields dz1'''''

dzr

and the (k - r) 1-forms dzk r + 1 , . . . , dzk, respectively. It follows that integral man­ ifolds of an involutive distribution make up a foliation. □ Example 1.2.6. Every 1-dimensional distribution on a manifold Z is integrable. Its section is a nowhere vanishing vector field u on Z, while its integral manifolds are the integral curves of u. By virtue of Corollary 1.2.4, there exist local coordinates (z ,..., zk) around each point z £ Z such that u is given by

u = d/dz1. • A Pfaffian system S' is asubmodule of the C°°(Z)-module D1(Z). In particular, sections of a codistribution constitute a Pfaffian system. Any Pfaffian system S" defines the ideal AS" of the exterior algebra D'(Z) which is generated by elements of 5*. Given a flag (1.2.40) of a regular distribution, one can introduce the coBag of the codistribution Ann (T) D Ann (T 2 ) D ■■■.

(1.2.41)

The coflag (1.2.41) stabilizes. In particular, a distribution T is totally non-holonomic if and only if its coflag (1.2.41) shrinks to zero. Foliations An r-dimensional (regular) foliation on a k-dimensional manifold Z is said to be a partition of Z into connected leaves FL with the following property. Every point

2.3. JET MANIFOLDS

35

of Z has an open neighbourhood U which is a domain of a coordinate chart (za) such that, for every leaf Ft, the connected components Ft D U are described by the equations z r + 1 = const.,

zk = const.

[90, 150]. Note that leaves of a foliation fail to be imbedded submanifolds, i.e., topological subspaces in general. Example 1.2.7. Submersions n : Y —» X and, in particular, fibre bundles are foliations with the leaves 7r"'(x), x G 7r(V) C X. A foliation is called simple if it is a fibre bundle. Any foliation is locally simple. • Example 1.2.8. Every real function / on a manifold Z with nowhere vanishing differential df is a submersion Z —» R. It defines a 1-codimensional foliation whose leaves are given by the equations f(z) = c,

c e f{Z) c R.

This is the foliation of level surfaces of the function / , called a generating function. Every 1-codimensional foliation is locally a foliation of level surfaces of some function on Z. • The level surfaces of arbitrary function / ^ const, on a manifold Z define a sin­ gular foliation F on Z [90]. Its leaves are not submanifolds in general. Nevertheless if df(z) ^ 0, the restriction of F to some open neighbourhood U of z is a foliation with the generating function f\y-

1.3

J e t manifolds

Subsections: Jet manifolds, 35; Canonical horizontal splittings, 38; Second order jet manifolds, 38; The total derivative, 40; Higher order jet manifolds, 40; Differential operators and differential equations, 41. Jet manifolds Given a fibre bundle Y —» X with bundle coordinates (xx,yl), let us consider the equivalence classes j^s, x € X, of its sections s, which are identified by their

36

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

values s'(z) and the values of their first derivatives 3 M s'(i) at points i e X. The equivalence class j^s is called the first order jet of sections s at the point i £ l The set JXY of first order jets is provided with a manifold structure with respect to the adapted coordinate atlas

(z\
=

{x\s>{x),dxs\x))}

(a + )2/

^ = S " ^ "-

(1.3.1)

It is called the jet manifold of sections of the fibre bundle Y —► X (or simply the jet manifold of the fibre bundle Y —> X). The jet manifold admits the natural fibrations TT1

: JlY

*\;JlY

(1.3.2)

3 jls ■-» x 6 X,

3jls~s(x)eY,

(1.3.3)

where (1.3.3) is an affine bundle modelled over the vector bundle T'X®VY

-» K

Y

For the sake of convenience, the fibration JlY —> X is further called a jet bundie, while the fibration JlY —» Y is an affine jet bundle. There are the following two canonical monomorphisms of the jet manifold JlY over y : A: J 1 Y ' - - > T * X ® 7 T ,

(1.3.4)

A = dxx ® dA = dxx ® (dA + j/^a,), where d\ is called the total derivative, and 0! : J'Y Sl=p®d,

^T'Y®VY, Y l

(1.3.5) x

= (dy - y\dx ) ® d„

where 9' = dy' -

y\dxx

(1.3.6)

37

1.3. JET MANIFOLDS

is called the contact form. In accordance with these monomorphisms, every element of the jet manifold JXY can be represented by the tangent-valued forms dxx ® (dx + y\di)

and

(dy' - y\dxx) ® dx.

Each fibred morphism $ : F —» Y' over a diffeomorphism / is extended to the fibred morphism of the corresponding jet manifolds J^:JXY

— J1?", *

J^:jis^j}{x){^oSof-% 8(f~xY 1 y'\ * '' )) ££ ^^ TJ _- ., M* y'\ o o J J 1 ** = = (d,
(^«)(x)^iia, ((y\y\)oJ 2 / , , 2 / l )l°s- / 1 s= = ((s>(x),d s ' ( x ) , ^ 5 ,x(s>(x)), x)), of the jet bundle JlY —* X. A section s of the jet bundle JlY —> X is said to be hoionomic if this is the jet prolongation of some section of the fibre bundle Y —* X. Any projectable vector field u=

ux(x")dx+u'(x>',y])d,

on a fibre bundle Y —» X admits the jet prolongation to the vector field

u = n o jlu ■. j ' y -»j'ry ->rj'y, G = uAaA + u% + (dxu> -

fidxurffi,

(1.3.7)

on the jet manifold JlY. One can show that the jet prolongation of vector fields u— i ► u is the morphism of Lie algebras, i.e., [u,u'] = [u,u'J = [u.JZ']. [u,lf}. In order to obtain (1.3.7), we have used the canonical fibred morphism r, : JlTY

-

TJlY,

yi°n vi ° J-I = = {y') (y'hx -- yi± y^v x-

38

CHAPTER

1, INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

In particular, there is the canonical isomorphism VJlY

= JXVY,

(1.3.8)

y'x = (y')xCanonical horizontal splittings The canonical morphisms (1.3.4) and (1.3.5) can be viewed as the morphisms A : JlY x TX B dx >-» dx = dx\X € JlY x TY X

(1.3.9)

Y

and 0i : JlY x V'Y 9 dy* H-. 0> = 9x\dy' € JlY x T'Y,

(1.3.10)

where {dy'} are the bases for the fibres of the vertical cotangent bundle V'Y. morphisms determine the canonical horizontal splittings of the pull-backs J'Y

xTY

= X(TX)

Y

0

jiy

These

(1.3.11)

VY,

xx8x + y% = xx(dx + y\dx) + (y* -

xxy\)d„

and JlY

x T'Y = T'X

(1.3.12)

0 0i(V*y),

jiy

Y

xxdxx + y,dy{ = (i A + y,yx)dxx

+ y^dtf -

y\dxx).

Second o r d e r jet manifolds Taking the first order jet manifold of the jet bundle J ' V —* X, we come to the repeated jet manifold JlJlY, provided with the adapted coordinates (*

y'\x

,y\yx,ylM>2/J.A).

dxa = d^(d°+

yfa)di +

yiadj)y'y

There exist two different affine fibrations of J 1 J*Y over

JlY:

1.3. JET

MANIFOLDS

39

• the familiar affine jet bundle (1.3.3) *ii : J'J'Y

-

J'y,

y\°Tn

= y\,

(1.3.13)

modelled over the vector bundle T'X

® VJlY

— JlY,

(1.3.14)

• and the affine bundle J1^

: JxJlY

-> / F ,

y* ° J^l = vU>

(1.3.15)

whose underlying vector bundle Jl(T'X®VY)^J1Y

(1.3.16)

differs from (1.3.14). In general, there is no canonical identification of these fibrations, but it can be made by means of a symmetric linear connection on X [57]. The points q € J1J1Y, where nn(q) = . / ' ^ ( g ) , make up the affine subbundle J*Y —> J1Y of J^J^Y, called the sesquiholonomic jet manifold. This is given by the coordinate conditions V\x) = y\<

and is coordinated by (xx, y\j&,jfj,^). The second order jet manifold J2Y of a fibre bundle Y —> X is the affine subbundle 7!-? : J2Y -» JlY of the fibre bundle pY — J 1 ^ , given by the coordinate conditions

vU = yU and coordinated by (xx,y',y\,y\ll

VT'X

= y'^)- It

is

modelled over the vector bundle

® yy -» J1Y.

The second order jet manifold PY can also be seen as the set of the equivalence classes j2s of sections s of the fibre bundle Y —> X, which are identified by their

40

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

values and the values of their first and second order partial derivatives at points x€X:

y\(fxs) = dxs\x),

VUJIS)

= d^s'ix).

Let s be a section of a fibre bundle Y —♦ X and J*s its jet prolongation to a section of the jet bundle JlY —» X. The latter gives rise to the section JlJls of the repeated jet bundle Jl JlY —> X. This section takes its values into the second order jet manifold f*Y. It is called the second order jet prolongation of the section s, and is denoted by J2s. PROPOSITION 1.3.1. Let s be a section of the jet bundle JlY —> X and Jls its jet prolongation to the section of the repeated jet bundle J1JiY —> X. The following three facts are equivalent: • s = Jls where s is a section of the fibre bundle Y —> X; • ./'s takes its values into J^Y; • J^s takes its values into J2Y. D

The total derivative We will use the total derivative operator

d^d^ + yidi + yifl. It satisfies the equalities d\{4> A a) = d\(4>) A dx{d) =

a

+ A

dx(a),

d(dx(4>)).

Higher order jet manifolds The k-orderjet manifold JkY of a fibre bundle Y ~* X comprises the equivalence classes j * s , x 6 X, of sections s of Y identified by the k + 1 terms of their Tailor

1.3. JET

MANIFOLDS

series at the points x e X. coordinates

41 The jet manifold JkY

is provided with the adapted

(xx,y\yl...,y\k...Xi),

y\l.xlU*s) = dxr--dxlsi(x),

0
Every section s of a fibre bundle Y —> X gives rise to the section Jks of the fibre bundle JkY -> X such that j / A , . A t ° ^ = aA,...aAlSi,

0 < I < k.

Differential operators and differential equations Let JkY be the fc-order jet manifold of a fibre bundle V —► X and E —* X a vector bundle over X. DEFINITION 1.3.2. A fibred morphism £:JkY^E

x

(1.3.17)

is called a fc-order differential operator on the fibre bundle Y —► X. It sends each section s(x) of Y —> X onto the section (£ o Jks)(x) of the vector bundle £ - » !

□ The kernel of a differential operator is the subset Ker£ = £-1(0(X))cJkY,

(1.3.18)

where 0 is the zero section of the vector bundle E —» X, and we assume that Q(X) C £{JkY). DEFINITION 1.3.3. A system of Ar-order partial differential equations (or simply a differential equation) on a fibre bundle V —► X is defined as a closed subbundle € of the jet bundle JkY -» X [20, 57, 104]. D Its (classical) solution is a (local) section s of the fibre bundle Y —> X such that its A;-order jet prolongation Jks lives in £. For instance, if the kernel (1.3.18) of a differential operator £ is a closed subbundle of the fibre bundle JkY -> X, it defines a differential equation £oJks

= 0.

42

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

The following condition is sufficient for a kernel of a differential operator to be a differential equation. PROPOSITION 1.3.4. Let the morphism (1.3.17) be of constant rank. By virtue of Theorem 1.1.3, its kernel (1.3.18) is a closed subbundle of the fibre bundle JkY —> X and, consequently, is a fc-order differential equation. □

1.4

Connections

Subsections: Connections, 42; The curvature of connections, 44; Linear connections, 44; Affine connections, 44; Flat connections, 45. Connections A connection on a fibre bundle Y —» X is defined as a global section r : Y -*

JlY,

x

T = dx ®{dx

+

T\{x»,y*)di),

of the affine jet bundle JlY —» Y. Combining a connection T and the morphisms (1.3.9) and (1.3.10) gives the splittings X o T : TX -» TY, 0i o r : V'Y -

T'Y

of the exact sequences (1.1.8a) and (1.1.8b), respectively. Accordingly, substitution of the section y\ — r\ into the expressions (1.3.11) and (1.3.12) leads to the familiar splittings of the tangent bundle TY = r(TX)

© VY,

(1.4.1)

Y

x

x

± d, + y% = ±\dx + r\d,) + (y< - i r\)dt, and the cotangent bundle T'Y =

T'X(BT(V'Y),

(1.4.2)

Y

xxdxx + iudtf = (&x + T\ili)dxx + yi{dy' -

r\dxx),

1.4.

43

CONNECTIONS

of a fibre bundle Y —» X with respect to the connection T. In an equivalent way, the connection V defines the corresponding projection T:TY3xxdx

+ y% -> (y* - ixr\)di

G VY

(1.4.3)

and the corresponding section T = (dyl - T\dxx) <S> di

(1.4.4)

of the fibre bundle T'Y ® VY -> Y. Y

Connections on a fibre bundle Y —• X constitute an affine space modelled over the linear space of soldering forms a:Y

-> T'X ® VY, Y

(T = a\dxx (8) diAny connection T on a fibre bundle Y —> X defines the first order differential operator on Y Dr:JlY3z^[zDr =

Tiirliz))] G T'X ® VY,

(1.4.5)

x

(y\-r\)dx ®di,

called the covarianfc differential. Its action on sections s of the fibre bundle Y reads V r s = Dr o Jls = [dxs* - ( r o s)\]dxx ® a,.

(1.4.6)

For instance, a section s is said to be an integral section for a connection T, if V r s = 0, i.e., T o s = J J s . For any section s of a fibre bundle Y —> X, there exists a connection r on Y —> X such that s is its integral section. This connection is an extension of the section s(x) i-» Jls(x) of the affine jet bundle J1Y —> Y over the closed submanifold s(X) C V in accordance with Theorem 1.1.2. A connection T on a fibre bundle Y —» X defines the horizontal lift

rr = r\dx + r\d{) onto Y of each vector field T = rxd\ on X.

(1.4.7)

44

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

The curvature of connections The curvature of a connection T on a fibre bundle Y —» X is said to be the 2-form on Y R:Y

->AT'X®VY, Y x

R= -R\tidx

Adx»®di,

R\„ = ^ r ; - d,r\ + r ^ r ; - r ^ r j ,

(1.4.8)

Linear connections Let Y —» X be a vector bundle. A linear connection on Y —> X reads

r = dxA®[aA + r A i J (xya,]. It defines the duai linear connection

r = dxx ® [5A - r^ A (x) % a'] on the dual vector bundle Y* —» X. For instance, a linear connection .ft" on the tangent bundle TX, and the dual linear connection K* on the cotangent bundle T'X are given by the expressions K = dxx <8> (3A + ^ a „ ( i ) i " a Q ) , A

A" = dx ® (5A -

Kf¥(x)*Jr)-

(1.4.9) (1.4.10)

Affine connections Let Y —> X be an affine bundle modelled over a vector bundle Y —» X. affine connection on Y —» X reads

r = dxA ® [dA + (r*'j(*y + r\(x))di}. It defines the linear connection

r = dxx®[dx + rxlJ(x)yi-?-} on the vector bundle Y —» X.

An

1.4.

CONNECTIONS

45

Flat connections Each connection T on a fibre bundle Y —> X, by definition, yields the horizontal distribution T(TX) C TY on Y, generated by the horizontal vector fields (1.4.7). The following assertions are equivalent. • The horizontal distribution is involutive. • The connection T is flat (curvature-free), everywhere.

i.e., its curvature is equal to zero

• There is an integral section for the connection T through any point y e Y. Hence, a fiat connection r on Y —* X yields the integrable horizontal distribution, i.e., the horizontal foliation on Y, transversal to the fibration Y —> X. Its leaf through a point y 6 Y is defined locally by an integral section sy for the connection T through y. Conversely, let a fibre bundle Y —> X admit a horizontal foliation such that, for each point y e Y, the leaf of this foliation through y is locally defined by a section sy of Y —► X through y. Then the map r : Y -» r(?/) =

JlY, jlxsv,

n(y) = x,

introduces a flat connection on Y —> X. Thus, there is one-to-one correspondence between the flat connections and the horizontal foliations on a fibre bundle Y —» X. Given a horizontal foliation on a fibre bundle Y —> X, there exists the associated atlas of bundle coordinates ( x \ y') of Y such that every leaf of this foliation is locally generated by the equations yl = const., and the transition functions yl —> 2/"(j/J) are independent of the base coordinates xA [23, 57]. This is called the atJas of constant local trivializations. Two such atlases are said to be equivalent if their union is also an atlas of constant local trivializations. They are associated with the same horizontal foliation. Thus, we come to the following assertion. PROPOSITION 1.4.1. There is one-to-one correspondence between the flat connec­ tions r on a fibre bundle Y —> X and the equivalence classes of atlases of constant local trivializations of Y such that T\ = 0 relative to these atlases. □

46

CHAPTER

1.5

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

Bundles with symmetries

Subsections: Tangent and cotangent bundles of Lie groups, 46; Principal 48; The linear frame bundle, 52.

bundles,

Tangent and cotangent bundles of Lie groups Let G be a real Lie group with dim G > 0 and flj [flr] its left [right] Lie algebra of ieft-invariant vector fields £i(g) = TLg(£i(e)) [right-invariant vector Selds £r(
TLgoTR-g\vT{g)).

Let {em = e m (e)} [{^m = £m(e)}] denote the basis for the left [right] Lie algebra, and let c^nn be the right structure constants: [£m,en] =

c^nek.

The mapping J H J ' yields the isomorphism (1.5.1)

P ■fll3 (m >-» £m = - « m 6 0r

of left and right Lie algebras. For instance, we have [tm,en} =

-c^nek.

The tangent bundle TTQ : TG —» G of the Lie group G is trivial. There are the isomorphisms ft : TG 3 q -

(g = nG(q),TL-1(q)) i

P r

: T G 3 ? « ( j = nG(q),TR- (q))

6 G x 6 G x

ft> flr.

The left action L 9 of a Lie group G on itself defines its adjoint representation g t-* Adg in the right Lie algebra gT and its identity representation in the left Lie algebra gj. Correspondingly, there is the adjoint representation e' : e — i ► ade'(£) = [e',e], ade m (e n ) =

c^nek,

1.5. BUNDLES

WITH

47

SYMMETRIES

of the right Lie algebra g r in itself. An action G x Z B (g,z)>-> gz e Z of a Lie group G on a manifold Z on the left yields the homomorphism 8, 3 e -» & 6

T(Z)

of the right Lie algebra g r of G into the Lie algebra of vector fields on Z such that l

Udg(s) =Tgo£eog

(1.5.2)

[100]. Vector fields ££m are said to be the generators of a representation of the Lie group G in Z. Let g" = T*G be the vector space dual of the tangent space TeG. It is called the dual Lie algebra (or the Lie coalgebra), and is provided with the basis {e m } dual of the basis {e m } for TeG. The group G and the right Lie algebra g r act on g* by the coadjoint representation (Ad'g(e'),e)^{e\Adg-1(e)),

£* G 9*,

(adV(e*),e) = -( £ *,[ £ ', £ ]), ad* £m (e") =

n

-c

ee0r,

(1.5.3)

e' e ft.,

k

mke

.

R e m a r k 1.5.1. In the literature (see, e.g., [2]), one can meet another definition of the coadjoint representation in accordance with the relation {Ad'g(e'),

£) = <£*, Ad g(e)).

An exterior form on the group G is said to be left-invariant [right-invariant] if 4>{e) = L'((j>{g)) \<j){e) = Rm((p(g))]. The exterior differential of a left-invariant [rightinvariant] form is left-invariant [right-invariant]. In particular, the left-invariant 1-forms satisfy the Maurer-Cartan equations <^(£,6') = - ^ ( M ) ,

e,e'eg,.

There is the canonical Qi-valued left-invariant 1-form er-TcGBe^eegi

48

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

on a Lie group G. The components #p of its decomposition 0; = 0,mem with respect to the basis for the left Lie algebra g( make up the basis for the space of left-invariant exterior 1-forms on G:

tmW = C The Maurer-Cartan equation, written with respect to this basis, reads

<*r = ^M n A0f. Accordingly, the canonical g r -valued right-invariant 1-form 9T:TcGBe^e&QT on the group G is defined. There are the relations el{vg) = el{TL-g\vg))

=

TL-g\vg),

Vg G TgG,

8T(v9) = 8T(TR;\v9)) = TR;>(vg), p{0,(v9)) = -TLgoTR9l9r(v9) = -Adg(er(v9)), where p is the isomorphism (1.5.1). Principal bundles We refer the reader to [100, 170, 192] for the general theory of principal bundles. Let iTP : P —» X be a principal bundle with a real structure Lie group G. There is the canonical free transitive action

Rc-.PxG-^P, x Rg

P-> pg,

(1.5.4)

peP,

geG,

of G on P on the right. A principal bundle P is equipped with a bundle atlas * p = {(Ua,^)} trivialization morphisms

Va ■ T P W -> Ua x G obey the condition pi2oip[oRg

= go pr 2 o t/£,

VpeC.

whose

1.5. BUNDLES

WITH

SYMMETRIES

49

Due to this property, every trivialization morphism tp£ uniquely determines a local section za:Ua-*P such that pr 2 o v £ o za = e. The transformation rules for za read z0(x) =

za{x)pa0(x),

x e Ua n Up,

(1.5.5)

where pa0 are the transition functions of the atlas tfp. Conversely, the family {(Ua,za)} of local sections of P, which obey (1.5.5), uniquely determines a bundle atlas * P of P. A principal bundle P —» X admits the canonical trivial vertical splitting a:VP^

PxQt

such that a~l(em) are fundamental vector fields on P corresponding to the basis elements em for the left Lie algebra gj. Taking the quotient of the tangent bundle TP —* P and the vertical tangent bundle V P of P by the tangent map TRg, we obtain the vector bundles TGP = TP/G

and

VGP = VP/G

(1.5.6)

over X. Sections of TGP —* X are G-invariant vector fields on P, while sections of VQP —» X are G-invariant vertical vector fields on P. Hence, the typical fibre of VQP —• X is the right Lie algebra g r of the right-invariant vector fields on the group G. The group G acts on this typical fibre by the adjoint representation. The Lie bracket of vector fields on P goes to the quotients (1.5.6) and defines the Lie bracket of sections of the vector bundles TQP —» X and VGP —* X. It follows that VGP —> X is a fibre bundle of Lie algebras (the gauge algebra bundle in the terminology of gauge theories) whose fibres are isomorphic to the right Lie algebra g r of the group G. Example 1.5.2. When P = X x G is trivial, we have VGP = X x TG/G

=Xx

9 r

.

Example 1.5.3. Given a local bundle splitting of P, there are the corresponding local bundle splitting of TGP and VGP. Given the basis {ep} for the Lie algebra g r ,

CHAPTER

50

1. INTERLUDE:

we obtain the local fibre bases {d\,ep} the fibre bundle VCP — X. If

BUNDLES, JETS,

CONNECTIONS

for the fibre bundle TGP —► X and {e p } for

t, i;: X -» T G P,

e=^+^%,

77 = r/'d,, + r/%,

are sections, the coordinate expression of their bracket is

&»?] = K"fl^ - V^ A )a A + (^a^ r - vxdxe + C;,£VKLet JlP be the first order jet manifold of a principal bundle P —► X with a structure Lie group G. Bearing in mind that the jet bundle JlP —» P is an affine bundle modelled over the vector bundle T'X®VP^

P,

p

let us consider the quotient of the jet bundle JXP —» P by the jet prolongation of the canonical action (1.5.4). We obtain the affine bundle C =

J1P/G-^X

J lR g

(1.5.7)

modelled over the vector bundle C = T'X ® VGP -> X. Hence, there is the canonical vertical splitting

VC^CxC. x In the case of a principal bundle P —» X, the exact sequence (1.1.8a) reduces to the exact sequence 0 — VGP -^TGP x

— TX -» 0.

(1.5.8)

A principal connection A on a principal bundle P —» X is defined as a section A : P —* JlP which is equivariant under the action (1.5.4) of the group G on P, i.e., J1RgoA

=

AoRg,

VpeG.

(1.5.9)

1.5. BUNDLES

WITH

SYMMETRIES

51

Turning now to the quotients (1.5.6), such a connection defines the splitting of the exact sequence (1.5.8). It is represented by the tangent-valued form A = dxx ® (dx +

Ale,),

(1.5.10)

p

where A x are local functions on X. On the other hand, due to the property (1.5.9), there is obviously one-to-one correspondence between the principal connection on a principal bundle P —> X and the global sections of the fibre bundle C -* X (1.5.7), called the bundie of principal connections. Let a principal connection on the principal bundle P —* X be represented by the vertical-valued form A (1.4.4). Then the form -^T'P®VPl'^$T*P®Ql

A:P

p

is the familiar g r valued connection form on the principal bundle P. Given a local bundle splitting (U^,z^) of P, this form reads A = dp - ~A\dxx ® £„ where Op is the canonical g;-valued 1-form on P, {tp} is the basis of gj, and Apx are local functions on P such that AUpg)cq =

Al(p)Adg-\eq).

The pull-back z^A of A over U^ is the well-known local connection 1-form At = -Aqxdxx

® e„

(1.5.11)

where Aqx = A9xo z^ are local functions on X. It is readily observed that the coefficients A\ of this form are precisely the co­ efficients of the form (1.5.10). Moreover, given a bundle atlas of P, the bundle of principal connections C is equipped with the associated bundle coordinates (xx, ax) such that, for any section A of C —» X, the local functions A\ =

a\oA

are again the coefficients of the local connection 1-form (1.5.11). In gauge theory, these coefficients are treated as gauge potentials. We will use this term to refer to sections A of the fibre bundle C —* X.

52

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

Let now (1.5.12)

Y = ( P x V)/G

be a fibre bundle associated with the principal bundle P —> X whose structure group G acts on the typical fibre V of Y on the left. Let us recall that the quotient in (1.5.12) is defined by identification of the elements (p, v) and (pg,g~lv) for all g 6 G. Briefly, we will say that (1.5.12) is a P-associated Rbre bundle. As is well known, the principal connection A (1.5.10) induces the corresponding connection on the P-associated fibre bundle (1.5.12). If Y is a vector bundle, this connection takes the form A = dxx®(dx

+ AvxI'pdt),

where Ip are generators of the representation of the right Lie algebra g r on V. This is called the associated principal connection or simply a principal connection on

Y-+X. The linear frame bundle Let X be an n-dimensional connected oriented manifold. Let nLX ■ LX —> X be the principal bundle of oriented linear frames {sa} in the tangent spaces to X (or simply the frame bundle). Its structure group is GL+(n,R). The frame bundle is associated with the tangent bundle TX and with the cotangent bundle T'X of X. Given frames {<9M} in the tangent bundle TX, every element {sa} of the frame bundle LX takes the form SQ

— S a C/^i,

where s^a are matrix elements of the group GL+(n, R). These matrix elements constitute the bundle coordinates \X , s

aj,

S

°

dx*

"'

on the frame bundle LX. With respect to these coordinates, the canonical action of GL + (n,R) on LX on the right reads Rg : s*0 H-> s"bgba,

g£GL+{n,R).

1.6. COMPOSITE

FIBRE

BUNDLES

53

The frame bundle LX is equipped with the canonical Rn-valued 1-form OLX

= s^dx"

®
(1.5.13)

where {ta} is a fixed basis for R n , while s"^ are elements of the inverse matrix of o a-

The frame bundle, like tensor bundles, admits the canonical lift of any diffeomorphism / of its base X to the automorphism /:(xA,5A0)^(/A(x),^/Va). These automorphisms and the corresponding automorphisms of associated bundles are called generai covariant transformations or holonomic automorphisms. For in­ stance, in the case of the tangent bundle TX, the holonomic automorphisms f = Tf are the tangent maps to the diffeomorphisms / . Generators of general covariant transformations of tensor bundles are the canonical lifts r (1.2.1) of vector fields r on X.

1.6

Composite fibre bundles

Let us consider the composition of fibre bundles Y -> E -► X,

(1.6.1)

HYT. - Y - » E

(1.6.2)

7T*-> v

(1.6.3)

where

and * ?,

*

X

are fibre bundles. This is called a composite fibre bundle. The composite fibre bundle (1.6.1) is provided with an atlas of fibred coordinates (xx,am, y'), where (x M ,(r m ) are fibred coordinates on the fibre bundle (1.6.3) and the transition functions am —► am(xx,ak) are independent of the coordinates yx. The following two assertions on composite fibre bundles are useful in applications to field theory and mechanics [57, 159].

54

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

PROPOSITION 1.6.1. Given a section h of the fibre bundle E -» X and a section s E of the fibre bundle Y —» E, their composition (1.6.4)

s = S£ o /i

is a section of the composite fibre bundle Y —* X (1.6.1). Conversely, every section s of the fibre bundle Y —» X is the composition (1.6.4) of the section h = nys o s of the fibre bundle E —► X and some section s^ of the fibre bundle Y —► E over the submanifold h(X) C E. □ PROPOSITION 1.6.2. Given a composite fibre bundle (1.6.1), let h be a global section of the fibre bundle E —» X. Then the restriction

Yh = h'Y

(1.6.5)

of the fibre bundle Y —> E to h(X) C E is a subbundle ih ■ YK ^

Y

of the fibre bundle Y -> X. a Let us consider the jet manifolds J ' E , J^Y, and JXY of the fibre bundles E —» X, Y —> E and Y —» X, respectively. They are parameterized by the coordinates

( „A

_.m _.fn\

(x\<7m,2A&,24),

(x^.a"1,?/1,^,^),

respectively. There is the canonical map rp

: J ' E x j ' y —» j ' y , E v

(1.6.6)

2/w = 2/>r+£In particular, let

^ = dxx ® {dx + A\di) + dam ® (9m + 4,3,)

(1.6.7)

be a connection on the fibre bundle Y —* E, and

r = dxx ® {dx + r?dm) a connection on the fibre bundle E —» X. Then the connection

B = p o {A o TT* x r o TT1) = dxx ® \dx + r7<9m + (>i^r^ + A'x)d,}

(1.6.8)

1.6. COMPOSITE

FIBRE

55

BUNDLES

on the composite fibre bundle Y —► X is defined. connection. In brief, we will write

This is called the composite

B = A o r. For instance, let us consider a vector field r on the base X, its horizontal lift IV onto E by means of the connection V and, in turn, the horizontal lift A{Vr) of TT onto Y by means of the connection A. Then A(FT) coincides with the horizontal lift BT of r onto Y by means of the composite connection (1.6.8). Given a composite fibre bundle Y (1.6.1), there are the following exact sequences of vector bundles over Y: 0 — v E y <-> VY -> Y x VE -» 0,

(1.6.9a)

E

o -> y x V*E «-»v*y - > y ? y -»o, E

(1.6.9b)

*-

where Vj;y and V£Y are vertical tangent and cotangent bundles of the fibre bundle y —» E, respectively. Every connection A (1.6.7) on the fibre bundle Y —» E determines the splittings VY = V E y ®A(Y

(1.6.10a)

xKE),

K

E

y-a, + amdm = (y> - A^d™)* VY

+ am(dm +

A^d,), (1.6.10b)

= (Y x V"E) © A(V£Y), m

yidtf + amda

l

= y,(dy ~ A^da™) + [am +

A^da™,

of exact sequences (1.6.9a) and (1.6.9b), respectively. Using the splitting (1.6.10a), one can construct the first order differential operator D: JlY

^T'X®VEY, Y

D = dxx® (y\ - 4 -

4^R

(1.6.11)

called the verticai covariant differential, on the composite fibre bundle 7 - > X . This operator can also be seen as the composition D = pr, o DB : JlY

~* T'X ® VY -> T'X ® Vy E , r Y where DB is the covariant differential (1.4.5) relative to some composite connection (1.6.8), but D does not depend on T.

56

CHAPTER

1. INTERLUDE:

BUNDLES, JETS,

CONNECTIONS

The vertical covariant differential (1.6.11) possesses the following important property. Let ft be a section of the fibre bundle E —► X and V), the subbundle (1.6.5) of the composite fibre bundle Y —► X, which is the restriction of the fibre bundle Y —► £ to h(X). Then the restriction of the vertical covariant differential D (1.6.11) to JHh(JlYh) C JlY coincides with the familiar covariant differential on Yh relative to the connection Ah = dxx ® [ft + (A\ndxhm + (A o h)x)di] on Yh, which is the restriction of the connection A to h(X) in accordance with the commutative diagram j i y , J2% j i y Ak

!

|

Yk

—

AE

Y

E x a m p l e 1.6.1. Let r : Y —» JlY be a connection on a fibre bundle Y —» X. In accordance with the canonical isomorphism V J ' Y = JlVY (1.3.8), the vertical tangent map VT : VY —» VJlY to V defines the connection

VT-.VY ^

JlVY,

vr = dxx ® (dx + r\d, + djTitfdi),

(1.6.12)

on the composite vertical tangent bundle VY —> X. The dual connection on the composite vertical cotangent bundle VY —> X reads VT

: VY

-»j'vy, x

v*r = dx ® (^ + rift - a»rift#).

(1.6.13)

If y —» X is an affine bundle, the connection VT (1.6.12) can be seen as the composite connection (1.6.8) generated by the connection T on Y —► X and the linear connection r = dxx ® (ft + dX\y>dx) + dy* ® ft on the vertical tangent bundle VY —» F . •

(1.6.14)

Chapter 2 Geometry of Poisson manifolds This Chapter is devoted to the basic geometric structures on a manifold, which we meet in mechanics. We start from a Jacobi structure whose particular case is a Poisson structure. Throughout this Chapter, by Z, unless otherwise stated, is meant a Ar-dimensional manifold with coordinates (zx). For the sake of convenience, the Schouten-Nijenhuis bracket [., .]SN is denoted simply by [.,.].

2.1

Jacobi structure

A Jacobi bracket (or a Jacobi structure) map

on a manifold Z is defined as a bilinear

D°(Z) x D°(Z) 9 (f,g) - {/,
CHAPTER 2. GEOMETRY

58

OF POISSON

MANIFOLDS

PROPOSITION 2.1.1. [98, 118, 123]. Every Jacobi bracket on a manifold Z is uniquely defined in accordance with the relation {/. 9} = w(rf/> dg) + u\ {fdg - gdf)

(2.1.1)

by a pair (tu, u) of a bivector field w and a vector field u on Z such that [u, w] = 0,

(2.1.2)

[w, w] — 2u A w.

□ Example 2.1.1. Taking w = 0 in (2.1.1), we find that every vector field u on a manifold Z provides the Jacobi bracket

{f,g}=u\(fdg-gdf). The relations (2.1.2) are obviously satisfied. • Example 2.1.2. The Jacobi bracket (2.1.1) with u = 0 is said to be a Poisson bracket. According to Proposition 2.1.1, a bivector field w on a manifold Z yields a Poisson bracket if it meets the condition [w, w] = w^d^^dx,

A dX2

Adx,=0-

It is called a Poisson bivector Geld. • Let us consider the following examples of Jacobi manifolds which are not Poisson ones. Example 2.1.3. Odd-dimensional contact manifolds are Jacobi manifolds in ac­ cordance with Proposition 2.2.6. • Example 2.1.4. Let fi be a non-degenerate 2-form and cj> a closed 1-form on an even-dimensional manifold Z such that dfi = <j> A fi. The triple (Z, fi, <j>) is called a locally conformally symplectic manifold. This is a Jacobi manifold characterized by the bivector field wn (1.2.35a) and the vector field u = {nW<j>) [98, 118, 123]. If = 0, we have a symplectic manifold. •

2.1. JACOBI

STRUCTURE

59

Let Z\ and Z2 be Jacobi manifolds. A manifold map Q : Z\ —> Z2 is said to be a Jacobi morphism if, for every pair (f,g) of functions on Z 2 , we have {f°Q,g°e}i

=

{f,g}2°e-

In particular, a vector field v on a Jacobi manifold (Z;w,u) is the generator of a local 1-parameter group of Jacobi morphisms (or an infinitesimal Jacobi morphism) if and only if L„w = 0,

L„u = 0.

DEFINITION 2.1.2. Given a function / £ D°{Z) on a Jacobi manifold (Z;w,u), vector field

d,^w\df)

the

+ }u

is called the Hamiltonian vector Held for / . D We have the relation

[*/.».] = w

(2.1.3)

It follows that the map / •-» 1?/ is the Lie algebra homomorphism. The values of all Hamiltonian vector fields at all points of Z constitute the char­ acteristic distribution T on the Jacobi manifold (Z;w,u). A glance at the formula (2.1.3) shows that this distribution is involutive, but it has different dimensions at different points z € Z in general. Therefore, this distribution is not a subbundle of the tangent bundle TZ and, consequently, is not a distribution as that in Section 1.2. A Jacobi manifold Z is said to be transitive if its characteristic distribution coincides with the tangent bundle TZ. Transitive Jacobi manifolds are proved to be either locally conformally symplectic manifolds, if dim Z is even, or the contact ones, if dim Z is odd [98, 118]. T H E O R E M 2.1.3. [44, 98]. The characteristic distribution on a Jacobi manifold is completely integrable in the sense of Sussmann [173] and defines on Z a singular Stefan foliation [171]. Each leaf of this foliation is endowed with a unique Jacobi structure such that its canonical injection into Z is a Jacobi morphism. □ The following definition generalizes for Jacobi manifolds the notion of coisotropic and Lagrangian submanifolds of a Poisson manifold [83] (cf. Definition 2.3.5).

60

CHAPTER

2. GEOMETRY

OF POISSON

MANIFOLDS

DEFINITION 2.1.4. Let (Z\w,u) be a Jacobi manifold, with the characteristic distribution T, and N a submanifold of Z. The submanifold N is said to be: • coisotropic if w«(Ann7yV)C T*N> where AnnTzN

z€N,

C T'Z is the annihilator o(TzN,

and

• Lagrangian if wl(AnnTzN)=TINnTz,

ze

N.

D

Different Jacobi structures can lead to the same characteristic distribution as follows. Let (Z\ w, u) be a Jacobi manifold and a a nowhere vanishing function on Z. Let us consider the bivector field wa and the vector field ua on Z given by ua = wi(da) + au.

wa = aw,

(2.1.4)

Then the pair (wa,ua) (2.1.4) is a Jacobi structure on Z. It is called conformally equivalent to the Jacobi structure {w,u) because of the relation {/,0}a =

a

~{af,ag}.

It is readily observed that the Hamiltonian vector field for a function / with respect to the Jacobi structure (u», u) is also the Hamiltonian vector field for the function i / with respect to the conformally equivalent Jacobi structure (2.1.4). It follows that conformally equivalent Jacobi structures on a manifold Z define the same char­ acteristic distribution on Z. A Jacobi morphism from a Jacobi manifold to a conformally equivalent Jacobi manifold is said to be a conformal Jacobi morphism. Such a morphism preserves the characteristic distribution on a Jacobi manifold. In particular, a vector field v on a Jacobi manifold (Z; w, u) is an infinitesimal conformal Jacobi isomorphism if and only if there exists a function b on Z such that L„w = bw,

hvu = w'(6) + bu.

2.2. CONTACT

2.2

STRUCTURE

61

Contact structure

We will consider contact structures on odd-dimensional manifolds. By a contact structure is meant a strictly contact structure in the sense of [116]. DEFINITION 2.2.1. An exterior 1-form 6 on a (2m + l)-dimensional manifold Z is said to be a contact form if 8 A (d6)m 5* 0

(2.2.1)

everywhere on Z. O If 6 is a contact form, so is ad where a is a nowhere vanishing function on Z. The pair (Z, 8) (or equivalently the pair (Z,a8)) is termed a contact manifold (we refer the reader to [15, 116] for the details of terminology). If a manifold admits a contact structure, this manifold is orientable, and has the volume element (2.2.1). It follows from (2.2.1) that the exterior differential d9 of a contact form 8 is a presymplectic form of rank 2m. There is the following variant of well-known Darboux's theorem [116]. T H E O R E M 2.2.2. Every point z of a (2m + l)-dimensional contact manifold (Z, 8) has an open neighbourhood U which is the domain of a coordinate chart (z°,..., z2m) such that the contact form 8 on U has the local expression m

8 = dz°- 5> m + , dz\

(2.2.2)

These coordinates are called the Darboux coordinates. □ A contact form 8 on a manifold Z defines the isomorphism \>(u)d=u\d8+(u\8)6,

u €

(2.2.3)

T(Z),

of the C°°(Z)-module T(Z) of vector fields on a manifold Z to the C°°(Z)-module £>' (Z) of exterior 1-forms on Z [83]. This isomorphism can be extended to a mapping from the exterior algebra T,(Z) of multivector fields to the exterior algebra D'(Z) of exterior forms by putting b(«i A ••• A u r ) d = b ( u i ) A • • •

Abfa),

u{ 6

T(Z).

(2.2.4)

CHAPTER 2. GEOMETRY

62

OF POISSON

MANIFOLDS

PROPOSITION 2.2.3. Let $ be a contact form on a manifold Z. There exists the unique nowhere vanishing vector field E = b~1{9)

(2.2.5)

on Z, called the Reeb vector field, such that E\6=l,

E\de = Q

[116]. It is readily observed that, relative to the local Darboux coordinates, this vector field reads E = d0. O Example 2.2.1. Let M be a manifold with coordinates (ql) and Z the odddimensional manifold R x T'M, equipped with the coordinates {t,q',Pi = ft). The manifold R x T'M is well known to admit the contact form 6 = p,dq{ - dt.

(2.2.6)

For the sake of convenience, this contact form is chosen to differ in minus sign from the Darboux one (2.2.2). • Example 2.2.2. To generalize Example 2.2.1, let Q —» R be a fibre bundle, with fibred coordinates (t, q'), and Z = V'Q the vertical tangent bundle of Q, equipped with the coordinates (t,q\pl = q,). As mentioned above, V'Q is a phase space of time-dependent mechanics. Given a connection

r = dt g> {dt + Pdi) on the fibre bundle Q —► R, the manifold V'Q is provided with the contact form 6r = pi{dq{ - T'dt) - dt (see Proposition 5.2.11 below). The corresponding Reeb vector field reads

E = -{dt +

Fdi-pjdiFdx).

Since a connection T on a fibre bundle Q —> R is flat, it defines the corresponding atlas of local constant trivializations such that, with respect to this atlas, T = dt®dt and the contact form 9r is brought into the form (2.2.6). •

2.2. CONTACT

STRUCTURE

63

Given a (2m + l)-dimensional contact manifold (Z,6), the tangent bundle TZ of Z admits the splitting TZ =

Kerd6®KeiO,

where KerdO is the 1-dimensional vector subbundle generated by the Reeb vector field E [116]. In particular, every vector field u on Z is decomposed in a unique fashion into

u=

(u\e)E+(u-(u\6)E).

By duality, the cotangent bundle T'Z of the contact manifold Z is found to have the splitting T'Z = 6 e Ker E, where 0 is the 1-dimensional vector subbundle generated by the contact form 6. As a consequence, every 1-form <j) on the contact manifold Z is decomposed into 4>=(E\4>)9 + (4>-(E\4>)e)There is the corresponding splitting of the isomorphism (2.2.3). This sends Kerd6 onto © and Ker 6 onto K e r E . The restriction of b to Ker 8 coincides with the morphism (—d@f (1.2.33) defined by the presymplectic 2-form d8. The 2m-dimensional subbundle Ker0 of the tangent bundle TZ of a contact manifold Z is called the contact distribution on Z. Any contact form a9, where a is a nowhere vanishing function on Z, defines the same contact distribution on Z. As is well known, there exist integral submanifolds of the contact distribution Ker 6 of dimension m, but none of higher dimension [15]. It is readily observed that the contact forms 6 and a6 have the same contact distribution. DEFINITION 2.2.4. A submanifold N oia. (2m + l)-dimensional contact manifold Z is said to be a Legendre submanifold if it is an m-dimensional integral submanifold of the contact distribution on Z. □ Let (Zu0i) and (Z 2 ,0 2 ) b e contact manifolds. A manifold map Q : Zx -» Z2 is said to be a contact transformation if e'02 = o0i

CHAPTER

64

2. GEOMETRY

OF POISSON

MANIFOLDS

where a is a nowhere vanishing function on Z [116]. A contact transformation preserves the contact distribution, but not a contact form in general. We consider especially the contact automorphisms of a contact manifold (Z,9), which keeps the contact form 9. By definition, the Reeb vector field E (2.2.5) is the generator of a local 1parameter group of local contact automorphisms (or an infinitesimal contact au­ tomorphism) of the contact manifold {Z,9), i.e., LE9 = 0. It is readily observed that, if a vector field u on Z is an infinitesimal contact auto­ morphism of the contact manifold (Z, 6), then u is a Hamiltonian vector field with respect to the presymplectic form d6 on Z, i.e., the exterior form u\d0 is exact (see Definition 2.5.2 below). The converse assertion is the following. PROPOSITION 2.2.5. Given a contact manifold (Z, 6), let ds be a Hamiltonian vector field on Z for a function / with respect to the presymplectic form d.9, i.e., d,\d9 =

-df.

Then the vector field

#/ =

[f-{*/m]E+*f

(2.2.7)

is an infinitesimal contact automorphism of the contact manifold Z, i.e.,

Lhe = o.

□ Proof. It follows from direct computation.

QED

Note that the vector field (2.2.7), like dj, is a Hamiltonian vector field for the function / with respect to the presymplectic form d9. A contact manifold is a Jacobi one as follows. PROPOSITION 2.2.6. [123]. Each contact form 9 on a (2m+ l)-dimensional manifold Z yields the associated Jacobi bracket (2.1.2) on Z which is defined by the Reeb vector field E (2.2.5) and the bivector field w such that w>(4>)\9 = 0,

wt()\d9=-()9)

(2.2.8)

2.2. CONTACT

STRUCTURE

65

for any 1-form on Z. □ Relative to the local Darboux coordinates for the contact form 9, the abovementioned Jacobi bracket (2.1.2) reads

»-B£+»-"!■> * &=«.

— - » •

m

{/.»} = E(9m+,ffa,/ - a^ifdtg) + ({9}d0f - [f}dog), where m

i=l m

[g}=^zm^dm+ig~g. i=i

In particular, let (Z,9) be a contact manifold and {w,u) the associated Jacobi structure on Z. A submanifold N of Z is a Legendre submanifold of the contact manifold (Z, 9) if and only if it is the Lagrangian submanifold of the Jacobi manifold (Z;w,u) [83]. Note that, in this case, the characteristic distribution of the Jacobi structure coincides with TZ, and we have w{(AnnTzN)

= TzN.

Let (Z, 0) be a contact manifold, v a vector field on Z, and b a function on Z. One can show [83] that the pair (v, b) is an infinitesimal conformal Jacobi morphism if and only if the pair (t>, —6) is an infinitesimal contact transformation, i.e., Lv9 = -66>. E x a m p l e 2.2.3. Let M be a manifold with coordinates (q') and Z the odddimensional manifold K x T"M, equipped with the holonomic coordinates (£, ql,Pi = Qi). This manifold is provided with the contact form (2.2.6). Hence, it is a Jacobi manifold where w = -{di + p,dt) Ad',

•

E =

-dt.

(2.2.9)

CHAPTER 2. GEOMETRY

66

2.3

OF POISSON

MANIFOLDS

Poisson structure

In accordance with Example 2.1.2, a bivector field w = -w^dp

A dv

on a manifold Z defines a Poisson bracket (or a Poisson structure) {f,g}=w(df,dg)

=

(2.3.1)

w^dllfdl/g

on Z if and only if [w, w] = 0,

w"k,d^wX2X3

+ w^dpW*3*

+ w^d^w^2

= 0.

Besides the conditions (Al - A3), the Poisson bracket (2.3.1) satisfies the Leibniz rule {h,fg}

= {h,f}g

+

f{h,g}.

(2.3.2)

A manifold Z endowed with a Poisson bivector field w is termed a Poisson manifold, while a real vector space D°(Z) of functions on Z forms a Poisson algebra, i.e., O0(Z) is a real associative and commutative algebra with unit with respect to pointwise multiplication, a real Lie algebra with respect to the Poisson bracket, and these two operations intertwine via the Leibniz rule (2.3.2). Example 2.3.1. Each manifold admits a zero Poisson structure characterized by the zero bivector field w = 0. • Example 2.3.2. Let u and v be vector fields on a manifold Z such that [u,v\ = 0 everywhere on Z. It is readily observed from the relation (1.2.12) that w = u A v is a Poisson bivector field. • The Poisson structure (2.3.1) defined by a Poisson bivector field w is said to be regular if the associated morphism w' : T'Z —* TZ (1.2.31) is of constant rank. Hereafter, by a Poisson structure we mean only a regular Poisson structure. The regular Poisson structure (2.3.1) defined by a Poisson bivector field w is said to be non-degenerate if the associated morphism w' : T'Z —» TZ (1.2.31) is of maximal rank. A non-degenerate Poisson structure can exist only on an evendimensional manifold.

2.3.

POISSON

67

STRUCTURE

E x a m p l e 2.3.3. A non-degenerate Poisson manifold (Z, w) is a symplectic manifold with the symplectic form fi w (1.2.35b) (see Proposition 2.4.4 below). • Two functions / and g on a Poisson manifold (Z, w) are said to be in involution with each other if their Poisson bracket {/, g} equals zero. A function C on a Poisson manifold (Z, w) is called a Casimir function if it is in involution with any function on Zy i.e., {
V / 6 0°(Z).

Casimir functions make up the centre of the Poisson algebra (D°(Z),u/). For in­ stance, if a Poisson structure is non-degenerate, all the Casimir functions (on a connected manifold Z) are constant. Let {Z\,wi) and {Z Z2 is said to be a Poisson morphism if {f°Q,g°Q}i

=

{f,g}2°Q,

Vf,geQ°(Z2),

or W2 =

TQOWI,

where Tg is the tangent map to g. If g is a Poisson morphism, the rank of W\{z) is grater or equal to that of W2{g{z)). If a Poisson morphism g is an immersion, tui(z) and w2{g{z)) are of equal rank. A vector field u on a Poisson manifold [Z, w) is the generator of a local 1parameter group of local Poisson automorphisms (or an infinitesimal Poisson auto­ morphism) of (Z, w) if and only if \JUW

= [u, w] = 0.

(2.3.3)

Such a vector field is called canonical Note that there are no pull-back or push-forward operations of Poisson structures by manifold maps in general. The following assertion deals with Poisson projections [181]. PROPOSITION 2.3.1. Let (Z, w) be a Poisson manifold and i : Z - » V a projection. The following properties are equivalent: • for every pair of functions (/, g) on Y and for each point j i e V , the restriction of the function {/ o n,g o n} to the fibre TT~1(y) is constant;

68

CHAPTER 2. GEOMETRY

OF POISSON

MANIFOLDS

• there exists a Poisson structure w' on Y for which 7r is a Poisson morphism, i.e., {fon,goTr}

= {f,g}' OTT.

□ One says that the Poisson structure w' in Proposition 2.3.1 is coinduced by the projection IT. The direct product Z x Z' of Poisson manifolds (Z, w) and {Z',w') has the Poisson structure defined by the bivector field w + w' on Z x Z'. It is called the direct product of Poisson structures. Obviously, the projections p ^ and pr 2 are Poisson morphisms. One can speak of a Poisson submanifold (Z1, w') of a Poisson manifold (Z, w) if Z' is a submanifold of Z and the natural injection Z' —> Z is a Poisson morphism. DEFINITION 2.3.2. Given a real function / on a Poisson manifold (Z, w), the image (2.3.4)

0/ = «*'(#),

0, = wTdJdu, of its exterior differential d/ by the morphism wl (1.2.31) is called the Hamiitom'arj vector Geld for / with respect to the Poisson structure w. O E x a m p l e 2.3.4. The Hamiltonian vector field for a Casimir function equals zero.

It is readily observed that each Hamiltonian vector field is also a canonical vector field. The Hamiltonian vector field $/ for a function / , by definition, obeys the rela­ tions

#f\dg = {f,g}, [0/.*«] =

V
D°(Z),

W

Then it follows from (2.3.5) that / — i »• i?y is the Lie algebra homomorphism. In particular, we have

tf/Jd/ = 0

(2.3.5)

2.3.

POISSON

STRUCTURE

69

and, consequently, dj is tangent to the leaves of the singular foliation of the level surfaces of the function / at its regular points, where df ^ 0. R e m a r k 2.3.5. It is clear that not every a Hamiltonian vector field. Given a vector construct a Poisson bracket and to find a Hamiltonian vector field for this function. problem in Hamiltonian mechanics, which system of first order dynamic equations zx = ux{z)

vector field on a Poisson manifold Z is field u on a manifold Z, one can try to function on Z such that u would be a A closely related subject is the inverse consists in trying to represent a given

(2.3.6)

on a manifold Z as the Hamilton equation with respect to some Poisson structure, called the generating Poisson structure on Z. In Section 3.2, we will present a general technique of constructing a generating Poisson structure for any dynamic equation (2.3.6) at least locally [59, 82]. • The values of all Hamiltonian vector fields at all points of a Poisson manifold Z constitute an even-dimensional characteristic distribution T on the Poisson ma­ nifold (Z,w). By virtue of the relation (2.3.5), this distribution is involutive and, consequently, completely integrable. T H E O R E M 2.3.3. [181]. A Poisson structure induces a symplectic structure on leaves of the characteristic foliation on Z, called a symplectic foliation. □ In particular, if a Poisson structure is non-degenerate, its characteristic distri­ bution T coincides with TZ, and Z is a symplectic manifold. R e m a r k 2.3.6. It should be recalled that we are considering a regular Poisson structure, and the corresponding symplectic foliation is non-singular. • A 2m-dimensional symplectic foliation on a /c-dimensional Poisson manifold Z admits the adapted coordinates (... , z 2 m + 1 , . . . , zk) described in Corollary 1.2.4. Moreover, one can choose these coordinates in such a way to bring the Poisson structure into the following canonical form [181, 186]. PROPOSITION 2.3.4. For any point 2 of a Poisson manifold, there exists a coordinate system

(
(2.3.7)

70

CHAPTER

2. GEOMETRY

OF POISSON

MANIFOLDS

in a neighbourhood of z such that {Pi,9>}=Sh

{q\
= {Pi,zA} = {Az8}

= 0.(2.3.8)

D

The coordinates (2.3.7) are called canonical coordinates. Given canonical coor­ dinates, one says that the coordinates q' and p< are canonicaiiy conjugate. In canonical coordinates (2.3.7), the Poisson bracket (2.3.1) takes the form w = d' A di, {f,9}=d'fd,g-djd\ while the Casimir functions are arbitrary functions of the coordinates zA. Accord ingly, the Hamiltonian vector field for a function / reads ■df = <97d, -

djd'.

Let (Z, w) be a Poisson manifold with the characteristic distribution T, and N a submanifold of Z. The familiar definitions of coisotropic and Lagrangian submanifolds of a symplectic manifold are generalized for a Poisson manifold as follows [181, 187]. DEFINITION 2.3.5. The submanifold N is said to be • coisotropic if w»(AnnTA0CTAf, • Lagrangian if ™"(Ann7W) = TATnT.

a The following theorems generalize for Poisson manifolds the corresponding re­ sults on symplectic isomorphisms (see [178]). THEOREM 2.3.6. [187]. Let $ : Z\ —» Z 2 be a manifold morphism between Poisson manifolds (Z\,w{) and (Z2,11)2). This is a Poisson morphism if and only if its graph A* = { ( z , $ ( z ) ) z e Zi} C Z, x Z2

2.3.

POISSON

STRUCTURE

71

is a coisotropic submanifold of the Poisson manifold [Z\ x Z^, W\ — w-i). □

T H E O R E M 2.3.7. [67]. Let u be a vector field on a Poisson manifold (Z, w). (i) In accordance with the equality (1.2.14), the tangent lift w (1.2.9) of the Poisson bivector field w is a Poisson bivector field on the tangent bundle TZ of Z. (ii) A vector field v. is an infinitesimal Poisson automorphism of the Poisson manifold (Z, w) if and only if u(Z) is a Lagrangian submanifold of the Poisson manifold (TZ,w). □ A Poisson structure on a manifold Z can be defined entirely by its symplectic foliation instead of by a Poisson bivector field [181]. T H E O R E M 2.3.8. Let Z be a manifold and F a foliation on Z such that every leaf Ft of F is endowed with a symplectic structure. Given a function / 6 Q°(Z) on Z, let ■d'j be a Hamiltonian vector field on a leaf F t for the function / \pL with respect to the symplectic structure on this leaf. Put

0f(')=rf{z),

ze FL.

If -df is a differentiable vector field on the manifold Z for arbitrary function / , thei Z has a unique Poisson structure given by the Poisson bracket

{/,
(2.3.9)

Obviously, this is the product of the zero Poisson structure on K and that defined by the canonical symplectic structure on the cotangent bundle T'M. •

72

CHAPTER

2. GEOMETRY

OF POISSON

MANIFOLDS

Example 2.3.8. To generalize Example 2.3.7, let us consider a fibre bundle IT : Z —► X, parameterized by fibred coordinates (xx;q',pl), whose typical fibre V is a symplectic manifold with the Poisson bracket

{f,g} =

d'fdtg-dxfd\

(2.3.10)

Each trivialization chart V{ : rr-^Uf) - U( x V is endowed with the Poisson structure described in Example 2.3.7 with the Poisson bracket (2.3.10). To generalize construction of symplectic vector bundles [116], we will say that Z —» X is a symplectic bundle if it admits a bundle atlas $ whose transition functions pK(x) : {x} x V - {x} x V,

x e c/f n u(,

provide isomorphisms of the symplectic manifold V. Then the fibre bundle 2 - > X is equipped with the Poisson structure given by the Poisson bracket (2.3.10). For instance, let Z = V'Q be the vertical cotangent bundle of a fibre bundle Q —► K. The typical fibre of V'Q is the cotangent bundle T'M of the typical fibre M of the fibre bundle Q —> R. The fibre bundle V'Q —> R is a symplectic bundle, provided with the Poisson bracket (2.3.10) written with respect to the coordinates (£, q%, p ( ) on V'Q. Indeed, it is readily observed that this Poisson bracket is invariant under all holonomic coordinate transformations of the vertical cotangent bundle V'Q. This is the canonical Poisson structure on a phase space V'Q of time-dependent mechanics. In Section 5.1, we will obtain it in a different way. The notion of a symplectic bundle is naturally extended to that of a Poisson bundle. One should distinguish this notion from that of Jacobi bundles in the sense of Kirillov [98, 123] where a Jacobi bracket is generalized for sections of a 1-dimensional linear bundle, called a Jacobi bundle. • Example 2.3.9. The Lie coalgebra g* of a Lie group G is provided with the canonical Poisson structure, called the Lie-Poisson structure (see [2, 116, 181]). This Poisson structure is given by the bracket

{f,g}^(e'Adf(e'),dg(e')}),

/.seOV).

(2.3.11)

73

2.4. SYMPLECTIC STRUCTURE

on g*, where df(e'),dg(e') 6 flr since we can regard them as linear mappings from Te'2' = Q* to R. The Lie-Poisson bracket (2.3.11) is defined by the Poisson bivector field w

mn

= CkmnZky

where c£,n are the structure constants of the Lie algebra 0 r and zk are coordinates on g* with respect to a basis {ek}. We have the coordinate expression

{f,9}=ckmnzkV»fdng. There is the following well-known theorem. T H E O R E M 2.3.9. [186]. The symplectic leaves of the Lie-Poisson structure on the coalgebra g* of a connected Lie group G are the orbits of the coadjoint representation (1.5.3) of G o n g * . D •

2.4

Symplectic structure

Non-degenerate Poisson manifolds are symplectic manifolds (see Proposition 2.4.4 below). DEFINITION 2.4.1. A non-degenerate exterior 2-form Q on a manifold Z is said to be symplectic if it is closed, i.e., dQ. = 0. A manifold equipped with a symplectic form is called a symplectic manifold. □ Every symplectic manifold (Z, £l) is 2m-dimensional. It is orientable, and 1 m V=—rAfi m!

(2.4.1)

is the volume element on Z. DEFINITION 2.4.2. A morphism (, : Z -> Z' of a. symplectic manifold (Z, Q) to a symplectic manifold (Z', fi') is called a symplectic morphism if Q = C*ft'. □ By very definition, a symplectic morphism is an immersion.

74

CHAPTER

2. GEOMETRY

OF POISSON

MANIFOLDS

R e m a r k 2.4.1. One should distinguish a symplectic morphism from a symplectomorphism in the terminology of [116]. The latter is a diffeomorphism. • A vector field u on a symplectic manifold (Z, Q) is said to be a generator of a local 1-parameter group of symplectic automorphisms (or an infinitesimal symplectic automorphism) of a symplectic manifold (Z, fi) if and only if L u n = 0. Such a vector field is called canonical Example 2.4.2. Let M be a manifold with coordinates (ql) and TT.M : T'M —> M its cotangent bundle, equipped with the holonomic coordinates (q',Pi). The cotangent bundle T'M is endowed with the canonical Liouville form 9 = p,dq\ This form is defined by the condition v\6{p)

=Tn.M(v)\p,

Vu € TPT'M,

p€

T'M.

Its exterior differential is the canonical symplectic form (2.4.2)

n = dd = dpxA dq*

on T'M. All holonomic coordinates on the cotangent bundle T'M are canonical for this form. It should be emphasized that the canonical symplectic form (2.4.2) is not a unique symplectic form on the cotangent bundle T'M. For every closed 2-form on a manifold M, the form

n* = n + ir;M<j> is also a symplectic form on T'M

(2.4.3) [116]. •

The canonical symplectic form (2.4.2) plays the fundamental role in view of Darboux's theorem [116]. T H E O R E M 2.4.3. Let (Z, Q) be a symplectic manifold. Each point of Z has an open neighbourhood U which is the domain of a canonical coordinate chart (q\...,qm,pl,...,pm)

2.4. SYMPLECTIC

75

STRUCTURE

such that the symplectic form fi on U is given by the coordinate expression (2.4.2). D This theorem is an immediate consequence of Proposition 2.3.4 and Proposition 2.4.4 below. We have the following relationship between the non-degenerate Poisson struc­ tures and the symplectic ones. PROPOSITION 2.4.4. [117]. On a 2m-dimensional manifold Z, there is one-to-one correspondence between the symplectic forms SI and the non-degenerate Poisson bivector fields w given by the equalities (1.2.35a) - (1.2.35b). □ The canonical coordinates for a symplectic form Q are also canonical for the corresponding Poisson bivector field w, and we have fi = dp, A dql,

w^&Adi.

With respect to these coordinates, the bundle isomorphism fr (1.2.33) reads 0 k : v'di + Vidx — i ► -Vidq' + v'dplt while the isomorphism wl = (£r)

(1.2.31) is

wl : Vidq' + vldpi — i » vxdx — Vid\ In view of Proposition 2.4.4, a Poisson structure is termed sometimes a cosymplectic structure (one should distinguish this notion from the cosymplectic structure in Remark 4.8.8). A local structure of an arbitrary Poisson manifold in relation to a symplectic one is described by the following two theorems [181, 186]. THEOREM 2.4.5. Any point of a Poisson manifold has an open neighbourhood which is Poisson equivalent to a product of a manifold with the zero Poisson structure and a symplectic manifold. □ Let (Z,w) be a Poisson manifold. Its symplectic realization is said to be a symplectic manifold (Z1, £3) together with a projection Z' —> Z such that the Poisson structure w on Z is coinduced from the Poisson structure wn on Z'. T H E O R E M 2.4.6. Any point of a Poisson manifold has an open neighbourhood which is realizable by a symplectic manifold. □

76

CHAPTER

2. GEOMETRY

OF POISSON

MANIFOLDS

In local Darboux coordinates, this symplectic realization is seen as follows. The Poisson structure given by the Poisson bracket (2.3.8) with respect to the canonical coordinates is coinduced from the symplectic structure given by the symplectic form Q = dpi A dq' + dlfy A dzA with respect to the coordinates (q , - - , q

,P\,--,Pm,Z

, •••,-Z ,Z

. •••>2 )

by the projection Cn1

nm T)

(q,...,q

,pi,,,.

l
n

72m+l

,Pm,Z

k ?2m+l

,...,z,z

i P l , - ■ -,Pm,Z

yk\

, ...,z/t—» ,■ ■ ■ ,Z )■

Example 2.4.3. Let V'Q be the vertical cotangent bundle of a fibre bundle Q —» R as in Example 2.3.8. It is provided with the Poisson structure given by the Poisson bracket

{f,9} =

d'fd,g-d,fd'g.

Its symplectic realization is the cotangent bundle T'Q of Q endowed with the canon­ ical symplectic form (2.4.4)

dE = dp Adt + dp, A dq',

written with respect to the holonomic coordinates (t, q',p,Pi) (see Section 5.1). • Let (Zi,fti) and (Z2, £h) be symplectic manifolds equipped with the associated Poisson structures w-i and w2, respectively. The map g : Z\ —> Z2 is a Poisson isomorphism if it is a symplectic isomorphism. Accordingly, a vector field on a symplectic manifold is canonical if and only if it is canonical for the associated Poisson structure. However, if dim Z\ > dim Z2 and g is a Poisson morphism, then this is not a symplectic morphism, i.e., w7=Tgowi,

Qi ^

e'n2.

The notion of a Hamiltonian vector field on a Poisson manifold is restated for a symplectic one as follows. DEFINITION 2.4.7. A vector field d on a symplectic manifold (Z, Q) is said to be locally Hamiltonian [Hamiltonian] if the exterior form tfjft is closed [exact]. □ As an immediate consequence of Definition 2.4.7, we find the following.

2.4. SYMPLECTIC

77

STRUCTURE

• A vector field ■d on a symplectic manifold (Z, il) is locally Hamiltonian if and only if it is an infinitesimal symplectic automorphism, i.e., USl = d(ti\n) = 0. • A vector field i ) o n a symplectic manifold (Z, fi) is Hamiltonian if and only if it is a Hamiltonian vector field in accordance with Definition 2.3.2 for some function / on Z, and then d;\U=

-df.

(2.4.5)

• The Poisson bracket defined by a symplectic form $1 reads {/.g} = < W j n In the literature (see, e.g., [2]), one may meet a definition of Hamiltonian vector fields, which differs in the minus sign from (2.4.5). E x a m p l e 2.4.4. Let us consider the cotangent bundle T'M equipped with the canonical symplectic form fl (2.4.2). Let u = u'dx be a vector field on M. Then its canonical lift u = uldi — PjdtvSd'

(1.2.3) onto T'M

is a Hamiltonian vector field. We have

u]Q = —d(u'pi).

Let us turn now to submanifolds of a symplectic manifold. Let (Z, Q) be a 2m-dimensional symplectic manifold and w the corresponding Poisson bivector field on Z. Let ./V be an n-dimensional submanifold of Z. The set O r t h n T A r = (J {v

e

TZZ : v\u\Q = 0, Vu € TZN},

is called the orthogonal of TN relative to the symplectic orthogonal space ofTN.

(2.4.6) form Q. or simply Q-

CHAPTER

78

2. GEOMETRY

OF POISSON

MANIFOLDS

Recall the useful formulas Orth n (Orth n 7W) = TN, Ann TAT =

ft^OrthnTAT),

iu»(AnnTAr) = Orth n TAr, Ann(Orth n TAr) = n''(TAr). The inclusion TNt C TN2 is equivalent to OrthnTATi D OithnTN2. Orth n (TAf! nTJV 2 ) = OrthnTAr, +

We have also

OrthnTN2

and, in particular, TN D Orth n TAr = Orth n (TN + OrthnTAT). Note that, in general, TN n OrthnTAf ^ 0, while TZ \N is not the sum TN +

OrthnTN.

Henceforth, when there is no risk of confusion, we will write OrthTN instead of OrthnTA^. Let fijv = 'Jv^ D e 'he restriction of the symplectic form ft to the submanifold AT. This is a presymplectic form on A^. Its kernel (1.2.34) is Ker QN = TNH

Orth^TN.

The following special submanifolds of a symplectic manifold (Z, ft) are usually considered. The presymplectic form QN on any of such special manifolds is always of constant rank. DEFINITION 2.4.8. A submanifold N C Z of a 2m-dimensional symplectic manifold (Z, ft) is said to be (cf. Definition 2.3.5): • coisotropic if OrthnAf C TN, n > m; • isotropic if TN C Orth^A^,

n<m;

• Lagrangian if OrthnAT = TN, n = m; • symplectic if ft# is a symplectic form on N.

2.4. SYMPLECTIC

STRUCTURE

79

□ As an immediate consequence of this definition, a symplectic form restricted to isotropic or Lagrangian submanifolds is equal to zero. One can classify the germs of special submanifolds of a symplectic manifold [48]. Recall that a germ of a submanifold N at a point z G Z is the equivalence class (N; z) of submanifolds of the manifold Z which pass through z and coincide with N in an open neighbourhood of z. With respect to local canonical coordinates (q\pt), a Lagrangian submanifold of a symplectic manifold is given by the equations q" = dbS,

Pa =

-daS,

(2-4.7)

where S(qa,p0) is a function, called the generating function, of the m variable: {<7°,P(,; a 6 A, b E B} for some partition [A, B) of the set ( 1 , . . . , m). Then the germ of a Lagrangian submanifold (N; z) is symplectically equivalent to the gern of the subspace {(q,p)€R2m:

p, = 0,t = l , . . . , m }

(2.4.8)

at the point 0 6 R 2 m of the symplectic space M2m. The expression (2.4.8) results from (2.4.7) by means of the symplectic automorphisms qb >-* ~Pb,

Pb<->qb

and Pi^Pi

+

d,S(qa,q").

Example 2.4.5. Let the cotangent bundle T'M of a manifold M be equipped with the canonical symplectic form Ci. Then the image 0(M) of the zero section 6 of T'M —» M is a Lagrangian submanifold {pi = 0} of the symplectic manifold (T'M,Q). • The germ of a (2m — r)-dimensional coisotropic submanifold (N; z) is symplec­ tically isomorphic to the germ of the subspace { ( g , p ) € R 2 m : gi = 0 , t = l , . . . , r } at the point 0 € R 2 m of the symplectic space R 2m .

CHAPTER 2. GEOMETRY

80

OF POISSON

MANIFOLDS

The germ of a 2(m - r)-dimensional symplectic submanifold (N; z) is symplectically isomorphic to the germ of the subspace { ( g , p ) e R 2 m : q<=Pl =

0,l=l,...,r}

at the point 0 e R 2m of the symplectic space R 2m . We will need the following two constructions in the sequel [178]. Example 2.4.6. Let {ZuQi)

and (Z 2 ,n 2 ) be symplectic manifolds. The 2-form

fii 0 &2 = pr'fii — prjf22 is clearly a symplectic form on the product Z\ x Zi- Then the graph of a sym­ plectic diffeomorphism of (ZuVl{) onto (Z 2 ,^2) i s a Lagrangian submanifold of the symplectic manifold (Z\ x Z 2 , Jli 0 f22). • Example 2.4.7. Let Q be the canonical symplectic form (2.4.2) on the cotangent bundle T'M of an m-dimensional manifold M. Then its tangent lift ft (1.2.23) is the symplectic form fi = dpi A dxf + dp, A dqx

(2.4.9)

on the tangent bundle TT'M of T'M, written with respect to the holonomic co­ ordinates (<7',g',p„Pi) on TT'M. Due to the isomorphism a : TT'M = T'TM (1.1.9), the form (2.4.9) is also the pull-back O'^T-TM of the canonical symplectic form J I T T M o n t n e cotangent bundle T'TM of the manifold TM. Let H b e a function on the cotangent bundle T'M. One can think of H as being a Hamiltonian of conservative mechanics. The Hamiltonian vector field dn = VUdi - diHff for H defines the closed 2m-dimensional submanifold q' = d"H,

P, =

-diH

(2.4.10)

of the 4m-dimensional symplectic manifold (TT'M, fi). The restriction of the sym­ plectic form (2.4.9) to the submanifold (2.4.10) is equal to 0. Hence, this is a Lagrangian submanifold with the generating function H(q',Pi).

2.5. PRESYMPLECTIC

STRUCTURE

81

Let £ be a function on the tangent bundle TM. On can think of £ as being a Lagrangian of conservative mechanics. The exterior differential rf£ of £ yields the morphism a - 1 odC : TM -» T'TM

-» TT*M.

Its image, given by the coordinate relations Pi = di£,

p, = 4 £ ,

(2.4.11)

is a Lagrangian submanifold of the symplectic manifold (7T*M,ft). Its generating function is £( bb ea nxterior r-form oo nhe eangent tundle TM, treated aa a generalized Lagrangian [10]. Then we have the morphism

a-'o4>:TM.T'TM—TT'M and the relation (c-lo4>yh

= d4>.

It follows that (a-'o<j>)(TM) is a Lagrangian submanifold of the symplectic manifold (TT'M, ft) if and only if the 1-form <j> on TM is closed. •

2.5

Presymplectic structure

As in the case of Poisson structures, there are no pull-back or push-forward operations of symplectic structures by manifold maps in general. Pull-backs of symplectic forms are the presymplectic ones. DEFINITION 2.5.1. An exterior 2-form ftona manifold Z is said to be presymplectic if it is closed, but not necessarily non-degenerate. A manifold equipped with a presymplectic form is called a presymplectic manifold. □ Using the formula (1.2.29), one can justify the fact that the kernel Ker ft (1.2.34) of a presymplectic form n of constant rank is an involutive distribution, called the characteristic distribution [116]. It defines the characteristic foliation on a presymplectic manifold Z. The restriction of the presymplectic form O to the leaves of this foliation is equal to zero.

82

CHAPTER

2. GEOMETRY

OF POISSON

MANIFOLDS

Remark 2.5.1. In contrast with a symplectic structure, a presymplectic one is not associated with any Poisson structure in a canonical way. Nevertheless, there is a construction [49] which may make a Poisson bivector field w correspond to a presymplectic form Cl on a manifold Z in accordance with the relations w o f t o w = ur, Kerw'nlmn^O.

• The notion of a Hamiltonian vector field on a symplectic manifold is extended in a straightforward manner to a presymplectic manifold. DEFINITION 2.5.2. A vector field ■d on a presymplectic manifold (Z, ft) is said to be locally Hamiltonian [Hamiitonian] if the exterior form tf J Cl is closed [exact]. □ As in a symplectic case, we find the following. • A vector field ■d on a presymplectic manifold (Z, Q) is locally Hamiltonian if and only if it is an infinitesimal automorphism of this presymplectic manifold, i.e., L,,fi = d(ti\Q) = 0. • A Hamiltonian vector field tij for a function / on a presymplectic manifold obeys the relation tij\df = 0 and, consequently, is tangent to the leaves of the singular foliation of the level surfaces of the function / at regular points, where df ^ 0. Note however that, in contrast with Poisson manifolds, not each function on a presymplectic manifold admits an associated Hamiltonian vector field. The pull-back of a symplectic form by a manifold map is obviously a presym­ plectic form. The converse is the following. PROPOSITION 2.5.3. Any presymplectic form 0 on a manifold M can be represented as the pull-back of some symplectic form. □

2.5. PRESYMPLECTIC

STRUCTURE

83

Proof. Let be a presymplectic form on a manifold M. Then it is the pull-back 0 = 6*(ft + 7r; M ^) of the symplectic form ft0 (2.4.3) on the cotangent bundle T'M of M by the global zero section 0 of T'M —> M. QED Moreover, it is readily observed that O r t h ^ T O ( M ) = (T0)(Kertf>) C T0(M). It follows that the imbedding 6 of the presymplectic manifold (M, <j>) into the sym­ plectic manifold (T'M, Q^) is coisotropic. If a presymplectic form Q on a manifold M is of constant rank, the stronger result holds [62, 72]. PROPOSITION 2.5.4. Given a presymplectic manifold (M, fi) where fl is of con­ stant rank, there exists a symplectic form on a tubular neighbourhood of the zero section of the dual bundle to the characteristic distribution Ker 0 , where M can be coisotropically imbedded. □ There is another well-known pull-back construction, where a presymplectic form is seen as a pull-back of a symplectic form by a surjective submersion. PROPOSITION 2.5.5. Let a presymplectic form Q. on manifold M be of constant rank and its characteristic foliation be simple, i.e., its leaves are fibres of a fibre bundle 7r : M —> P. Then the base P of this fibre bundle is equipped with a symplectic form fi/> such that fl is the pull-back of Qp by n. □ Proof. The symplectic form Up on P is defined by the relation Q p („y) d != f n(i7,tf) for any v G T-n~l(v), vf 6 T T T " 1 ^ ' ) .

QED

This pull-back construction is the key point of what is called a reduction of a symplectic structure.

84

CHAPTER

2.6

2. GEOMETRY

OF POISSON

MANIFOLDS

Reduction of symplectic and Poisson structures

A submanifold N of a symplectic manifold (Z, Q) fails to be a symplectic one in general. At the same time, if the presymplectic form QN = i%& o n ^Y ' s °f constant rank and the characteristic foliation of fi/v is simple, i.e., its leaves are fibres of a fibre bundle N —> P, the base P of this fibre bundle is a symplectic manifold in accordance with Proposition 2.5.5. This construction is called a reduction of a symplectic structure. It is general­ ized for Poisson structures. Important reduction processes appear in connection, e.g., with constraint Hamiltonian systems and Lie group actions on symplectic and Poisson manifolds. DEFINITION 2.6.1. [2, 116]. A symplectic reduction of a symplectic manifold (Z, fi) is said to be a surjective submersion 7r : N —> P of a submanifold N C Z onto a symplectic manifold (P,QP), which satisfies w'Qp

= i*NQ.

One says that (P, fi/>) is the reduced symplectic manifold of the symplectic manifold (Z, Q) via the submanifold N. □ The above speculations show that, given a submanifold N of a symplectic ma­ nifold (Z, Q), there exists a symplectic reduction 7r : N —» P with connected fibres if and only if • the presymplectic form ils = ?)v^ o n N ' s of constant rank, • the characteristic foliation of Qfj is simple. PROPOSITION 2.6.2. [116]. Let (Z, f2) be a symplectic manifold and N a submani­ fold of Z. (i) If n : N —> P is a symplectic reduction, then the rank of the pull-back presymplectic form £lN on N is constant and equal to d i m P , Kerfi/y = VN, and the connected components of fibres of 7r are the leaves of the characteristic foliation for fi/v. Furthermore, if n has connected fibres, the characteristic foliation of Qjv is simple, and is exactly the fibration N —> P. (ii) Conversely, if the rank of the presymplectic form fijv on a submanifold N of a symplectic manifold (Z, fl) is constant and the characteristic foliation for f2^ is

2.6. REDUCTION

OF SYMPLECTIC

AND POISSON

STRUCTURES

85

simple, the base P of the corresponding fibration N —> P has a unique symplectic form such that this fibration is a symplectic reduction of (Z, fi) via N. This is a unique symplectic reduction via N with connected fibres. Furthermore, if N —» P' is another symplectic reduction, there exists a surjection P —> P' which is a local symplectic isomorphism. □ A coisotropic submanifold N provides the most interesting case when the presymplectic form fijv is of constant rank which is neither dim N nor 0. Example 2.6.1. Let [M,<j>) be a presymplectic manifold, where a presymplectic form 4> is of constant rank and its characteristic foliation is simple. Combining the pull-back constructions in Propositions 2.5.3 and 2.5.5 gives the reduction of the symplectic structure 0.$ (2.4.3) on the cotangent bundle T*M via the coisotropic submanifold 0(M). • The reduction procedure is extended to Poisson manifolds as follows. DEFINITION 2.6.3. [129]. By a reductive structure of a Poisson manifold (Z, w) is meant a triple (Z, N, E) of a submanifold N of M and a vector subbundle E C TZ \N together with a submersion 7r : N —* P if the following conditions are satisfied: (i) E fl TN is tangent to the fibres of the submersion N —> P; (ii) if df and dg, where / , g € D°(Z), belong to Ann E, so is d{f, g}w. A reductive structure is said to be a Poisson reduction if: (i) P above is a Poisson manifold with a Poisson bivector field W and (ii) for any local functions / , g on P and for any local extensions f,g onto Z of the pull-backs / o ir,g o ix such that df,dg C AnaE, the relation {7,9~}w°iN =

{f,g}w°n

holds good. One says that (F, W) is the reduced Poisson manifold of (Z, w) via (N,E).

a

The following assertion furnishes the necessary and sufficient conditions for a reductive structure of a Poisson manifold to be a Poisson reduction. PROPOSITION 2.6.4. [129, 181]. Let (Z, N, E) be a reductive structure of a Poisson manifold (Z, w). This is a Poisson reduction if and only if io»(Ann£) CTN

+ E.

(2.6.1)

86

CHAPTER

2. GEOMETRY

OF POISSON

MANIFOLDS

a The functorality property of the Poisson reduction is given by the following. PROPOSITION 2.6.5. [129]. Let (Z,N,E) and (Z',N',E') be Poisson reductions, and let $ : Z — Z' be a Poisson map such that ${N) C N', T$(E) C E', and $ sends the leaves of JV to the leaves of N'. Then $ induces a unique Poisson map $ : P —» P', called the reduction of $, such that n' o $ = $ o n. □ Example 2.6.2. Let Z = T'G, where G is a connected Lie group, N = Z, and £ comprise the tangent space to the right G-orbits. Then (Z, E) is a Poisson reduction, where ix(Z) = T'G/G = g' is the Lie coalgebra of G provided with the Lie-Poisson structure (2.3.11) [129, 130]. • Example 2.6.3. To compare a Poisson reduction with a symplectic one, let (Z, fl) be a symplectic manifold and w the corresponding Poisson bivector field on Z. Let N be a submanifold of Z such that the presymplectic form QN = i%^ is of constant rank. Put E = OithTN. Then E l~l TN = Kerft N is tangent to the characteristic foliation of QN- If there exists a symplectic reduction ir : N —» P of (Z, Q) to (P, ftp), one can show that (Z, N, OrthTTV) is a reductive structure of the Poisson manifold (Z, w). The condition (i) of Definition 2.6.3 holds since the integral manifolds of the distribution Ker fijv are connected components of fibres of N —» P. Furthermore, since we are on a symplectic manifold, df G Ann (OrthTW) if and only if the Hamiltonian vector field ■&; for / belongs to iu , (Ann(OrthTAf))=TAf. Then, dhdge

(2.6.2)

TN implies

{#f,tis}=#{j,9)eTN. Hence, condition (ii) of Definition 2.6.3 also holds. Moreover, the inclusion (2.6.1) obviously takes place because of (2.6.2). It follows that (Z, N, OrthTN) is a Poisson reduction and the corresponding reduced structure W on P is associated with the symplectic form fip. On the other hand, if we take E = Kerfi/v, the inclusion (2.6.1) requires N to be coisotropic, since tu^Ann E)=TN

+ OrthTN,

2.6. REDUCTION

OF SYMPLECTIC

AND POISSON

STRUCTURES

87

and one may apply the previous result. • One can extend this Example to the Poisson manifold case as follows. LEMMA 2.6.6. [181]. Let N be a. submanifold of a Poisson manifold (Z, w) such that C(N)d^wl(AnnTN)C\TN

(2.6.3)

is a distribution, i.e., is of constant rank. Then C(N) is involutive. Furthermore, if N is transversal to the leaves of the symplectic foliation F for w on Z, the distribution C(N) defines a subfoliation of F D TN which is transversally symplectic along each leaf of the latter. D If the foliation C(N) is simple and corresponds to a fibration n : N —> P, then F n TN projects to a symplectic foliation n(F) on P, and P has a welldefined Poisson structure given by this foliation. This is called the leafwise reduction of (Z,w) via N. It follows in the same way as for the symplectic case discussed earlier that, if wl(AnnTN) has a constant dimension, then (Z, N, w^AnnTN)) is a reductive structure. Because of w^Anniw^AnnTN)))

C TN

and the relation (2.6.1), the reductive triple (Z, N, wt(Ann TN)) is also a Poisson reduction, while the reduced Poisson manifold P is exactly the one defined by the leafwise reduction. Given a Poisson manifold (Z,w), let (P,W) be a reduced Poisson manifold of (Z, w) via a submanifold N. Then one can say that the Poisson algebra Q°(P) of smooth functions on P is the reduction of the Poisson algebra of Q°(Z) of smooth functions on Z. If N is a closed imbedded submanifold of a Poisson manifold Z, this reduction is described in an intrinsic way, i.e., in terms of ideals of the Poisson algebra D°(Z) as follows [97]. Let N be a closed imbedded submanifold of a Poisson manifold (Z,w). Let us consider the set / of real functions f on Z which vanish on N, i.e., i'Nf = 0. This set is the kernel / = Ker i'N of the linear morphism t'N : 0°(Z)

-

Q°(N),

CHAPTER 2. GEOMETRY OF POISSON MANIFOLDS

88

and it is an ideal of the associative commutative algebra Q°(Z). Since N is a closed and imbedded submanifold of Z, we have the isomorphism (2.6.4)

Q°{Z)/I =* £>°(N)

of associative commutative algebras. Let us consider the space of all vector fields on Z which restrict to vector fields on N. It is given by TN d^{u g T{Z) : u\df 6 /, V/ € / } .

(2.6.5)

It is clear that 7]v ^ T(Z) | w . Suppose that -d; is a Hamiltonian vector field which restricts to a vector field on N. By the very definition (2.6.5), this means that 4f\dg = {f,g)zl,

\fg€l.

Hence, the functions whose Hamiltonian vector fields restrict to vector fields on N are those functions in the normalizer of /, denoted by

7(/v) d J= f {/e0°(Z): {/,
(2.6.6)

It follows from the Jacobi identity that the normalizer (2.6.6) is a Poisson subalgebra of Q°(Z). Put I'(N)d=I{N)nl.

(2.6.7)

It is naturally a Poisson subalgebra of I{N). This subalgebra is generally non-zero since I2 C I'(N) due to the Leibniz rule. The following Theorem leads us back to the reduction procedure. THEOREM 2.6.7. [97]. Let N be a closed imbedded submanifold of a Poisson manifold Z. Let us assume that w^AnnTN) and C (2.6.3) are distributions of constant rank, and the foliation C is simple. Then (Z, N, u)"(Ann TN)) is a reductive structure such that there is the associative commutative algebra isomorphism D°(F) S

I(N)/I'(N).

(2.6.8)

Since the right-hand side is naturally a Poisson algebra, this isomorphism defines a Poisson structure on P. O R e m a r k 2.6.4. The result is based on the fact that all sections of wi(AnnTN) are restrictions to N of Hamiltonian vector fields for elements in / , while all sections of

2.7. APPENDIX. POISSON HOMOLOGY

AND

COHOMOLOGY

89

C are Hamiltonian vector fields for elements in I'(N) restricted to N. In particular, if N is coisotropic, / C I(N), i.e., / = I'{N) is a Poisson subalgebra of D°(Z). • Theorem 2.6.7 shows that a Poisson reduction procedure can be seen as an algebraic one; that leads to the following purely algebraic definition [97]. DEFINITION 2.6.8. Let V be a Poisson algebra, J be an ideal of V, J{N) its normalizer (2.6.6), and J'(N) = J{N) D J. One says that the Poisson algebra J(N)/J'(N) is the reduction of the Poisson algebra V via the ideal J. □ In particular, an ideal J of a Poisson algebra V is said to be coisotropic if J is a Poisson subalgebra of V. The above algebraic reduction procedure has been generalized for Jacobi mani­ folds [87]. We meet the Poisson reduction in the sense of Definition 2.6.8 in connection with the BRST scheme [97, 169].

2.7

Appendix. Poisson homology and cohomology

This Section is concerned with the Koszul-Brylinski-Poisson homology and the Lichnerowicz-Poisson cohomology of a Poisson manifold, which find their applica­ tion in the quantization procedure. Remark 2.7.1. Let us recall briefly the basic notions of homology and cohomology of complexes [17, 122]. A sequence n J?° n U *

3

£>o *

i

D

*

"1 *

h_ n aZ±L ' '' *

a

v

'' '

(2.7.1)

of Abelian groups Bp and homomorphisms dp is said to be a chain complex if dpodp+1=0,

VpeN,

i.e., Imdp+i C Ker3 p . The quotient Hp(B.)d^Keidp/lmdp+i is called the pth homology group of the chain complex B, (2.7.1). The chain complex (2.7.1) is said to be exact at an element Bp if HP(B.) = 0. It is called an exact sequence if it is exact at each element.

CHAPTER 2. GEOMETRY

90

OF POISSON

MANIFOLDS

A sequence 0^B°

^ B

l

.*-,... £ ^ B P

(2.7.2)

.*♦...

of Abelian groups Bv and homomorphisms 6P is said to be a cochain complex if «S" o 5*- 1 = 0,

VpeN.

The pth cohomology group of the cochain complex S" (2.7.2) is the quotient Hp(B')d=Ker&'/lm6p-1. The De Rham compiex of exterior forms on a manifold Z ... J - » 0 ' - i ( z ) -*-.0»(Z) - ^ D r + 1 ( Z ) - * - . . . . exemplifies a cochain complex. Its cohomology group HP(Z), called the De Rham pth cohomology group, is the quotient of the space of closed p-forms by the subspace of exact p-forms. • Given a Poisson manifold (Z, w), let us consider the operator 6w:Qr(Z)^Dr-\Z), (2.7.3)

6W = w\ o d — d o w\,

on the exterior algebra D*(Z), called the Poisson codifferential [19, 102, 181]. It is given by the coordinate expression SModfi

A • • • A dfT) = i > l ) i=i

£

, + 1

{/o, h}dh A • • • A df, A • • • A dfT +

(-l)* + Vod{/i, h) A dfx A • • ■ A dfi A • • • A dfi A - • • A dfr.

l
Henceforth, when there is no risk of confusion, we shall write 5 instead of 5W. The operator 6 (2.7.3) obeys the equalities 6 o 6 = 0, do6 +

(2.7.4) 6od=0.

(2.7.5)

Due to the nilpotency property (2.7.4), we have the chain complex . . . J-£)P-i(z)

J-DP{Z)

^ - D P + 1 ( Z ) J-

,

2.7. APPENDIX.

POISSON HOMOLOGY

AND

COHOMOLOGY

91

called the canonical complex of the Poisson manifold (Z, w). The homology H™n(Z) of this complex is termed the canonical or the Koszul-Brylinski-Poisson homology. Furthermore, due to the property (2.7.5), the periodic double complex ET,t(Z) =

Ol~T(Z),

6 : Er,t(Z) -»

Er^(Z),

d:Eri{Z)^ET^{Z),

can also be defined. Let (Z, fi) be a 2m-dimensional symplectic manifold, provided with the volume form V (2.4.1), and IUQ the corresponding Poisson bivector field. Imitating the well-known star isomorphism for Riemannian manifolds, one introduces the star operation

* : QT(Z) - D 2m - r (Z),

»0=u>?,(fljy, where wl is the homomorphism (1.2.32). The (2m — r)-form *0 is called the adjoint of the r-form relative to the symplectic form Q. The following equalities

*{**) = <£,

0 6 D'(Z),

*(cj>Aa) = 4W)J<**) = (-l)m°K(°)\(*) are fulfilled [116]. LEMMA 2.7.1. [19]. In the case of a symplectic manifold, the codifferential S (2.7.3) on £>T{Z) is given by the formula 6=

{-iy+l*d*.

(2.7.6)

□ The relation (2.7.6) establishes an isomorphism of the canonical homology group H^{Z) with the De Rham cohomology group H2m~'(Z) of a symplectic manifold as follows. T H E O R E M 2.7.2. [19, 181]. If (Z, 0 ) is a 2m-dimensional symplectic manifold, then

Hf(Z) =

H^-^Z).

(2.7.7)

CHAPTER 2. GEOMETRY

92

OF POISSON

MANIFOLDS

a Given a Poisson manifold (Z, w), let us now introduce the operation

w : %{Z) -* Tr+1(Z),

gWg-K4

(2.7.8)

tfer.(Z),

on the exterior algebra 7;(Z) of multivector fields on Z, where [.,.] is the SchoutenNijenhuis bracket (1.2.10). This operation satisfies the rules (2.7.9)

w o w = 0, w

(2.7.10)

w{ti Av) = ui(i?) A D + ( - l ) t f A w(v),

and is sometimes called a contravariant exterior differential [181]. Its relation to the familiar exterior differential is iB(w>(<j>)) = -wl(d<j>),

eO'(Z),

(2.7.11)

while that to the codifferential 6W (2.7.3) is given by the formula w{-d)\ = ti\6m{4>) +

{-\)w6w{d\<j>),

|i?H0]-l.

(2.7.12)

Example 2.7.2. It is readily observed that

-w(f) = [w, / ] = * , ,

/ e To(Z),

is the Hamiltonian vector field for a function / on Z. • Due to the nilpotency rule (2.7.9), the operation w (2.7.8) provides the cochain complex ••• ^%.,{Z)

^%{Z)

^Tr+l(Z)

-^••,

called the Lichnerowicz-Poisson cochain complex. Cohomology groups HlP(Z,w) of this complex are said to be Lichnerowicz-Poisson or LP-cohomology groups of the Poisson manifold {Z,w). Example 2.7.3. The LP-homology group //£ p (Z,w) is the centre of the Poisson algebra (D°(Z),w). It consists of functions / € T0(Z) modulo constant functions whose Hamiltonian vector fields dj = —w(f) € KeriD vanish. •

2.7. APPENDIX.

POISSON HOMOLOGY

AND

COHOMOLOGY

93

Example 2.7.4. The first LP-homology group HlP(Z,w) is the space of canonical vector fields u for the Poisson bivector field w (i.e., huw = -w(u) = 0) modulo Hamiltonian vector fields -w(f), f e TQ{Z). • Example 2.7.5. The second LP-homology group HlP{Z,w) has the distinguished element {w} whose representative is the Poisson bivector field w itself. We have {w} = 0 if there is a vector field u on Z such that w = w(u) = —huw. • It is readily observed that, due to the property (2.7.10), HlP(Z,w) graded commutative algebra with respect to the product

makes up a

{i>}A{i;}d%At;}. It is also provided with the bracket

[W,M]"{fM) in accordance with the relation (1.2.13). THEOREM 2.7.3. graded algebras H'(Z)

-

[181]. The relation (2.7.11) induces a homomorphism of the

H'h¥{Z)

(2.7.13)

of the De Rham cohomology and the LP-cohomology. This is an isomorphism if the Poisson bivector field w comes from a symplectic structure on Z. D In case of a symplectic manifold, the isomorphism (2.7.13) and the isomorphism (2.7.7) lead to the isomorphism

HT(Z)

=

Hlr\Z).

In general, we have the following relationship between the LP-cohomology and the canonical homology of a Poisson manifold. PROPOSITION 2.7.4. By virtue of the relation (2.7.12), the natural pairing (1.2.30) induces the corresponding pairing

(,) : HUZ) x Hr(Z)

(m,{
-

H^{Z),

94

CHAPTER

2. GEOMETRY

OF POISSON

MANIFOLDS

of vector spaces of the LP-cohomology and the canonical homology of a Poisson manifold. □ The constructions of the LP-cohomology and the canonical homology for Poisson manifolds can be extended to the Jacobi ones [115]. Given a Jacobi manifold (Z;w,u), let us define the operator vm:Tr(Z)->TT+1{Z), wud=-[w,#]+ruAi),

d€Tr(Z),

which generalizes the contravariant exterior differential (2.7.8). This operator sat­ isfies the rules L u o wu = wu o L u , 55 2 (i)) = - M A i o ,

(2.7.14)

i)eT,(Z),

wu(-d A v) = wu{ti) Av+

1

( - l ) * ^ A wu(v).

Its connection with the exterior differential is given by the relations L u (ti/'(0)) = w>(Lu),

(2.7.15)

wu(w*() + u/^uj^) A w.

(2.7.16)

A glance at the expression (2.7.14) shows that the operator wu becomes nilpotent on the subspaces of u-invariant multivectors Tr{Z)

={tf € %{Z) : Lutf = [u,■&] = 0}.

This allows us to introduce the cochain complex IUU

3=

/ rj\

M1U 7r

I ry\

WU

~rf-

j ly\

VJU

The cohomology groups H[^{Z,w) of this complex are called Lichnerowicz-Jacobi or LJ-cohomology groups of the Jacobi manifold (Z; w, u). It is clear that, if u = 0 and (Z, w) is a Poisson manifold, the LJ-cohomology is precisely the LP-cohomology. Example 2.7.6. The first LJ-cohomology group H[p(Z,w) has the distinguished element whose representative is the vector field u, while the second LJ-cohomology group H{jp(Z,w) has the distinguished element whose representative is w. •

2.7. APPENDIX.

POISSON HOMOLOGY

AND

COHOMOLOGY

95

In the same way that the LP-cohomology groups are connected with the De Rham cohomology groups, the LJ-cohomology groups are connected with the cohomology groups of the subcomplex of basic forms of the De Rham complex. An r-form on a Jacobi manifold (Z;w,u) is said to be basic if uJ0 = O,

L U 0 = O.

It is easily seen that, if
-^OT{Z)B

J^Dr+1(Z)B -*-..-

(2.7.17)

of the De Rham complex. On subspaces £>TB{Z) C DT(Z) of basic forms, the relation (2.7.16) takes the form similar to (2.7.11): wu(ui'(0)) = —w>(d(j)),

4> e

(2.7.18)

DB(Z).

Moreover, we deduce from the relation (2.7.15) that, if is a basic r-form, then wl((j>) € Tr(Z). In the same way as in Theorem 2.7.3, the equality (2.7.18) yields a homomorphism H'B(Z) -

H'LJ(Z)

from the cohomology of the complex (2.7.17), called the basic De Rham cohomology, to the LJ-cohomology. To introduce the canonical homology of a Jacobi manifold (Z; w, u), let us con­ sider the codifferential 6W (2.7.3) defined by the bivector field w. It was proved in [38] that, if ^ is a basic form, so is 8W(4>), and 8W o 6W = 0. Then the chain complex

••• h-O'-^Z)

£^QPB{Z)

&-&$l{Z)

^

is defined. The homology of this complex is called the canonicai homology of a Jacobi manifold. If u = 0, this is exactly the canonical homology of a Poisson manifold. Similarly to Proposition 2.7.4, we have the following relationship between the LJ-cohomology and the canonical homology of a Jacobi manifold. T H E O R E M 2.7.5. [115]. There is the pairing (,) : H'LP(Z) x HT(Z)

<{tf},M>=W0>}.

-

m™(Z),

96

CHAPTER 2. GEOMETRY

OF POISSON

MANIFOLDS

of vector spaces of the LJ-cohomology and the canonical homology of a Jacobi manifold. □ The result follows from the fact that, if tf € T.(Z) and 0 is a basic form, then their pairing (d, <j>) (1.2.30) is a basic function, and from the relation wu{-d)\4> = d\6ul{<j>) + {-\)r6w(d\4>),

eOTB(Z),

tfeTr_,(Z).

Note that, bearing additionally in mind the map (2.2.4), one can apply the construction of the LJ-cohomology and the canonical homology of a Jacobi manifold to contact manifolds as the particular case of Jacobi manifolds. We refer to [38, 115] for more details. 2.8

Appendix. More brackets

We will give a brief survey of several extensions of the Poisson bracket to multivectors and differential forms and its generalization to the bracket of n > 2 functions. In this Section, we will return to the notation [., .] S N for the Schouten-Nijenhuis bracket. Let (Z, w) be a Poisson manifold. With the nilpotent operation iD (2.7.8), one can introduce the bracket {d,v}wd=-[w{d),v]sN on an exterior algebra T,(Z) of multivector fields on Z. This bracket has the property [ti,v]w = -(-l)WM[v,i9]w

+ w([#,v}sri),

and is a graded skew-commutative Lie bracket on the quotient where the Lie degree of a multivector field i5 is | ■& | — 1.

T,(Z)/w(T.(Z)),

Remark 2.8.1. Recall that multivector fields on a manifold Z constitute a graded Lie algebra with respect to the Schouten-Nijenhuis bracket [.,.]SN' (1-2.15), where the Lie degree of a multivector field
2.8. APPENDIX.

MORE

97

BRACKETS

and T,(Z). This isomorphism extends the Schouten-Nijenhuis bracket to the algebra of exterior forms [133]:

:£)T(Z)xOs(Z)^Dr+s-1(Z),

{.,.}.

to'Ctoff},,) =[u>'0Vff]sN,

4>,a

(2.8.1) eD'(Z).

(2.8.2)

In the general case of a Poisson manifold, we have the homomorphism w" (1.2.32) which satisfies the property (2.7.11). Therefore, in order to construct the bracket (2.8.1), let us consider the codifferential operator 6W (2.7.3). Then the SchoutenNijenhuis bracket of exterior forms (2.8.1) is defined as {4>, °}w = ( M ) A o + ( - l ) W ^ A {6wo) - 6W{4> A o)

(2.8.3)

[102, 181]. This bracket has the properties

{*,o-}» = (-l) l * l ""{^*}», (_l)W(W-»){0, {a,6}w}w + (-l)M(W-« {ffi {, a)) =

(2.8.4)

l

w \do - w'a^dcj) + d(w((/>,a)). It provides D1(Z) with a Lie algebra structure such that

w* : Q\Z)

-> T{Z)

is a Lie algebra homomorphism (2.8.2). The relationship between the bracket (2.8.4) and the Poisson bracket (2.3.1) is

{df,dg}w = d{f,g}. • Using the nilpotency properties (2.7.4) and (2.7.5) of the codifferential 6W, one can obtain the formula d{<j>,a}w = -{d,o}w -

(-l)w{<j>,da}w.

98

CHAPTER

2. GEOMETRY

OF POISSON

MANIFOLDS

Then the bracket {4>, o}d

= -{dct>,

(2.8.5)

a}w

on an algebra of exterior forms D'{Z) can be introduced [21, 133]. This is a graded skew-commutative Lie bracket {*,*}< =

(-i)w"W}.i

on the quotient D*(Z)/dD*(Z), where the Lie degree of an exterior form is | <j> | —1. Let us turn now to an n-ary generalization of Poisson and Jacobi brackets. DEFINITION 2.8.1. [84, 85). A generalized almost Jacobi bracket of order n on a manifold Z is said to be an n-linear map

{...} :

(2.8.6)

xD°{Z)^0°(Z)

which is • skew-symmetric | . . . , ji,...,

jj,... | —

\..., Jj,..., Ji,... I

with respect to any pair of arguments, • and is a first order linear differential operator on Z with respect to each argu­ ment, i.e., {Si/i, ••■} = 9i{fx, ■••} + /i{ffi, ■■•} - Si/i{l, -..}.

□ If {...} is a generalized almost Jacobi bracket of order n, then there exist an n-vector field w and an (n — 1)-vector field e on a manifold Z such that {/i, - , / « } = w(dfu...,dfn)+

E(-l)'+7.e(rf/i,-,d/„...,d/n)

t=i

and {l,/l,-,/n-l}=e(d/1,...,d/n-l).

(2.8.7)

2.8. APPENDIX.

MORE

BRACKETS

99

Conversely, any pair (w,e) of an n-vector field w and an (n - l)-vector field e defines the generalized almost Jacobi bracket (2.8.6) given by the relation (2.8.7). A manifold Z provided with the bracket (2.8.7) is called a generalized almost Jacobi manifold. If e = 0, we obtain a generalized almost Poisson manifold. In order to reproduce a Jacobi structure in the case of n = 2, one should add the integrability conditions, generalizing the Jacobi identity for the generalized almost Jacobi structure. In fact, two different types of integrability conditions have been suggested. They are • the fundamental identity {/l. • • • i / n - l , { f f l , - . - , 0 n } } = {{fl, {9\,{fl,--,fn-l,92},--,9n}

■ ■ ■ , fn-l,

9l}, 92, ■ ■ ■ , 9n) (2.8.8) +

+

■■■ + { S l , - - , P n - l , { / l , - - , / n - l , 5 n } }

• and the generalized Jacobi identity (2.8.9)

{/l> • • • i/n-li {In,- ■ ■ )/2n-l}} + {/2. • • • ,/n, {fn+l, ■ ■ ■ , / b n - l i / l } } H

=0.

The bracket (2.8.7) satisfying the fundamental identity (2.8.8) is called a NambuJacobi bracket and (Z;w,e) is a Nambu-Jacobi manifold [85, 128]. If e = 0, the bracket (2.8.7) is said to be a Nambu-Poisson bracket, and (Z, w) is a NambuPoisson manifold [9, 84, 175]. The bracket (2.8.7) satisfying the generalized Jacobi identity (2.8.9) is called a generalized Jacobi bracket, and (Z;w,e) is a generalized Jacobi manifold [84, 145]. If e = 0, the bracket (2.8.7) is said to be a generalized Poisson bracket, and (Z, w) is a generalized Poisson manifold [8, 9, 84]. The relationship between Nambu-Jacobi manifolds and generalized Jacobi man­ ifolds can be shown directly by noticing that the generalized Jacobi identity (2.8.9) is the full antisymmetrization of the fundamental identity (2.8.8). It follows that every Nambu-Jacobi manifold is a generalized Jacobi manifold [9, 85]. For n even, the integrability conditions of a generalized Jacobi structure (w, e) are given by the following assertion [145]. PROPOSITION 2.8.2. A generalized almost Jacobi structure (w,e) of even order n = 2p is a generalized Jacobi structure if and only if the relations [w, e]SN = 0,

[w, W]SN = 2(2p - l J m A e ,

CHAPTER 2. GEOMETRY

100

OF POISSON

MANIFOLDS

generalizing the conditions (2.1.2), hold good. □ At the same time, the integrability conditions of a Nambu-Jacobi structure {w, e) in particular imply that • the multivector field w is a Nambu-Poisson structure of order n, • the multivector field e is a Nambu-Poisson structure of order n — 1 [128], • and every Nambu-Poisson multivector w can be written as an exterior product of vector fields [79]. Example 2.8.3. Let us consider the product Z = R x T'M with the holonomic coordinates (*,
e = E Aui2

(see [85] for details). •

2.9

Appendix. Multisymplectic structures

We will give a brief exposition of multisymplectic and vector-valued generalizations of a symplectic structure. Similarly to that a Poisson bivector field is replaced by a multivector one for a generalized Poisson structure or a Nambu-Poisson structure, one can consider an n-ary generalization of a symplectic form. This is a multisymplectic form. Given an m-dimensional manifold M with coordinates (zx), let us consider the fibre bundle C.AT'M^M whose sections are exterior /t-forms on M. This fibre bundle is equipped with the holonomic coordinates (Z A ,PA), where A = (Aj < . . . < \k) are multi-indices of the

2.9. APPENDIX.

MULTISYMPLECTIC

STRUCTURES

101

length | A |= k. The manifold A T ' M is provided with the canonical exterior fc-form 0 defined by the relation u„J...u 1 je(p) = TC(ti„)J...TC(u 1 )Jp 1

p € AT'M,

u, e T p ( A T ' M ) .

Its coordinate expression is

e=

A--- AdzXk,

(2.9.1)

A

where the sum is over all multi-indices A of the length k. The exterior differential dQ of the form (2.9.1) is the (k + l)-symplectic form QM

= de = J^dp\1

\kAdzx>

A---Adzx*

(2.9.2)

A

which belongs to the class of multisymplectic forms in the sense of Martin [131]. If k = 1, the form Q.M (2.9.2) is the familiar symplectic form on the cotangent bundle T'M. E x a m p l e 2.9.1. For instance, let Y —» X be a fibre bundle over a (1 < n)dimensional base X. Let us consider the canonical form 9 (2.9.1) on the exterior product A T'Y. The homogeneous Legendre bundle over Y is said to be the fibre bundle

ZY=T'Y

ACAT'X),

equipped with the holonomic coordinates (xx,yi,pxyp), the transformation law

(2.9.3) where the coordinate p has

'-«(£) ('-$£4

The canonical bundle monomorphism over Y

iz:T'YA{n/\T'X)^AT'Y yields the pull-back form 5 = i"z@ =pcj + pxdyx A uix

(2.9.4)

on the fibre bundle ZY, where we use the notation (1.2.21). Its exterior differential Qz = dp A u + dpx A dy' A u>x

(2.9.5)

102

CHAPTER

2. GEOMETRY

OF POISSON

MANIFOLDS

is also called the multisymplectic form [63]. In particular, if Y —» R, the multisymplectic form (2.9.5) leads to the canonical symplectic form (2.4.4) on the cotangent bundle ZY = T'Y. • Let us touch on another generalization of a symplectic structure when an R n valued 2-form is considered [53, 142]. Let X be an n-dimensional manifold and LX —» X the principal frame bundle coordinated by (xx,s'ia). The frame bundle LX is provided with the canonical Revalued 1-form 6LX (1.5.13) which reads OLX

= s%dz" ta,

where {ta} is a basis for R n . Let us take its exterior differential d0LX = ds% A dx" <8> ta, called the n-symplectic form. It is readily observed that this form is non-degenerate, i.e., given a vector field

u = u*^ + uld» on LX, the equality u\d0Lx = 0 holds if and only if u = 0. By analogy with the symplectic case, one can write Uf\dOLX =

(2.9.6)

-df,

where / is an R n -valued function on the frame bundle LX. There is the class of functions (e.g., which are constant on the fibres of LX) such that vector fields satisfying the equation (2.9.6) exist. Then their Jacobi bracket {f,g} = Uf\dg is defined, and it also belongs to the above-mentioned class of functions. Let now Y —> X be an arbitrary fibre bundle. The Legendre bundle over Y is said to be the fibre bundle

n = AT*X®TX® VY a VY A ( V r x ) , Y

equipped with the holonomic coordinates (xx,y',p*) tions Pi =


(2.9.7)

Y

6t

{£)*

possessing the transition func­

(2.9.8)

2.9. APPENDIX.

MULTISYMPLECTIC

It is provided with the TX-valued

STRUCTURES

103

form

A = dpx A dy{ A u dx,

(2.9.9)

called the polysymplectic form. The Legendre bundle (2.9.7) plays the role of a finite-dimensional phase space in the Hamiltonian formulation of classical field the­ ory [28, 56, 57, 73, 96, 158, 159]. In the case of X = K, the polysymplectic Hamiltonian formalism leads us to the Hamiltonian formulation of time-dependent mechanics (see Chapter 5). The main peculiarity of polysymplectic formalism is that Hamiltonian connec­ tions on the Legendre bundle U —* X play a role similar to Hamiltonian vector fields on a symplectic manifold. A connection

T = da* ®{dx+ -$& + '&%) on the fibre bundle n —♦ X is said to be locally Hamiltonian if the exterior form 7JA is closed (see Section 3.8 for details). Remark 2.9.2. Note that generalization of the Poisson bracket for polysymplectic manifolds meets the difficulty (see [91, 92] and references therein). Nevertheless, in the particular case of an affine bundle Y —> X whose base X is provided with a non-degenerate metric g, the bracket of horizontal 1-forms on n —> X,

u.

i

at) (94>a dap da0d4>a\ >/l7I«fa*. dpf dy{ dp? dy'

9a0

<j> = x{p)dxx,

a =

can be globally defined. •


Chapter 3 Hamiltonian systems The theory of Hamiltonian systems is a vast subject which can be studied from many different viewpoints. This Chapter provides a brief exposition of different types of Hamiltonian systems on Poisson, symplectic and presymplectic manifolds in conser­ vative mechanics. We leave aside many interesting constructions of a Hamiltonian conservative mechanics and its straightforward extension to the direct product [time] x [autonomous phase space] that cannot be applied to time-dependent mechanics because a Hamiltonian is not a function under time-dependent transformations. Nevertheless, any object denned on a phase space V of conservative mechanics or on the product R x Z has the counterpart on a phase space ifr-.W = Rx

Z

(3.0.1)

of time-dependent mechanics, where different trivializations tp (3.0.1) correspond to different reference frames. For instance, there is one-to-one correspondence between the connections on a fibre bundle W —* R (3.0.1) and the time-dependent vector fields 1 x Z

z

-^TZ

because of the diffeomorphism

J V = J*W = J\W

xZ)=RxTZ.

However, this correspondence is not canonical, but depends on a trivialization (3.0.1). 105

106

CHAPTER

3. HAMILTONIAN

SYSTEMS

Note that Hamiltonian relativistic mechanics can be seen as an autonomous Hamiltonian system on the hyperboloid of relativistic momenta, i.e., it exemplifies a Dirac constraint system (see Section 6.3). In this Chapter, we pay special attention to constraint systems, including symplectic Hamiltonian systems, Dirac Hamiltonian systems, and Dirac constraint systems, seen as particular presymplectic Hamiltonian systems.

3.1

Dynamic equations

There are several types of equations in conservative mechanics. These are first and second order dynamic equations, Hamilton equations with respect to Poisson, symplectic or presymplectic structures, and Lagrange equations. The relationship between them is the following. • The dynamic equations, by definition, are differential equations which can be algebraically solved for the highest order derivatives. • The second order dynamic equations on a manifold M, by definition, are particular first order dynamic equations on the tangent bundle TM of M. • The first order dynamic equations on a manifold Z can be represented as Lagrange equations for Lagrangians on the tangent bundle TZ of Z, but they are not Hamilton equations on Z in general. • The Hamilton equations are not necessarily first order dynamic equations. • The Lagrange equations on a manifold M are derived as particular Hamilton equations on the tangent bundle TM, but they are not necessarily second order dynamic equations and differential equations in any strict sense. • The second order dynamic equations on a manifold M fail to be represented as Lagrange equations for Lagrangians on the tangent bundle TM or as Hamilton equations on TM in general. This Section is devoted to the notions of first and second order dynamic equations on a manifold, called autonomous dynamic equations.

3.1. DYNAMIC

EQUATIONS

107

DEFINITION 3.1.1. Let Z, dimZ > 1, be a manifold, coordinated by (zx), and u a vector field on Z. The closed subbundle u(Z) of the tangent bundle TZ, given by the coordinate relations iA =

ux(z),

(3.1.1)

is said to be an (autonomous) .first order dynamic equation on a manifold Z. This is a system of first order differential equations on the fibre bundle K x Z —> R in accordance with Definition 1.3.3 (see Example 5.2.1). □ By solutions of the first order dynamic equation (3.1.1) are meant integral curves of the vector field u. Remark 3.1.1. Generalizing Definition 3.1.1, we will say that an (autonomous) first order differentia] equation on a manifold Z is a closed submanifold <£ of the tangent bundle TZ of Z. A solution of this differential equation is an integral curve of a vector field on a submanifold J V c Z , which takes its values into TN n <£. • In this Chapter, we will deal with the first order dynamic equations (3.1.1) when vector fields u are Hamiltonian vector fields for Poisson, symplectic or presymplectic structure on a manifold Z, treated as a momentum phase space of conservative mechanics. However, they are not equivalent to the Hamilton equations in general. The Hamilton equations with respect to a Poisson structure, by definition, are first order dynamic equations, and so are the Hamilton equations with respect to a symplectic structure. This is not the case of Hamilton equations with respect to a presymplectic structure. Let us bear in mind the well-known inverse problem in conservative Hamiltonian mechanics, which consists in trying to represent a first order dynamic equation on a manifold Z as Hamilton equations with respect to some Poisson structure on Z [32] (see the next Section). The important motivation of the inverse problem is that, with a Poisson structure, one may find integrals of motion of a dynamic equation and also quantize it. Remark 3.1.2. As in the particular case of the general result [76, 81, 156], let us point out that every first order dynamic equation on a manifold Z can be represented as Lagrange equations for a Lagrangian on the tangent bundle TZ, but not in an explicit form. •

108

CHAPTER

3. HAMILTONIAN

SYSTEMS

Let M be a manifold, coordinated by (q'). If it is a configuration space of conservative mechanics, the corresponding velocity phase space is the tangent bundle TM of M, while the cotangent bundle T'M plays the role of the momentum phase space. DEFINITION 3.1.2. An autonomous second order dynamic equation on a manifold M is denned as a first order dynamic equation (3.1.1) on the tangent bundle TM, which is associated with a hoionomic vector field E = q% +

Ei(q^^)dl

(3.1.2)

on TM. This vector field, by definition, obeys the condition •/(•=•) =

UTM,

where J is the endomorphism (1.2.38) and UTM is the Liouville vector field (1.2.6) on TM (see the notation (1.2.4)). □ The vector field (3.1.2) is called a dynamic vector Geld. In the literature, it is often termed a second order dynamic equation. Let the double tangent bundle TTM be provided with the coordinates

(ql,?,$,?)■

(3.1.3)

With respect to these coordinates, the second order dynamic equation defined by the hoionomic vector field 5 (3.1.2) reads

q' = g',

? = =?(?,?).

(3.1.4)

By solutions of the second order dynamic equation (3.1.4) are meant the curves c : () —» M in a manifold M whose tangent prolongations c : () —► TM are integral curves of the hoionomic vector field H or, equivalently, whose second order tangent prolongations c live in the subbundle (3.1.4). They satisfy the second order differential equations

c'(t) = S(c'(t),c'(t)). A particular case of second order dynamic equations on a manifold M is that of geodesic equations on the tangent bundle TM.

3.1. DYNAMIC

109

EQUATIONS

Given a connection K = dq>® (dj + K)di) on the tangent bundle TM —> M, let

K :TMxTM

-» TTM

M

(3.1.5)

be the corresponding linear bundle morphism over T M which splits the exact se­ quence 0 —► VMTM >-+ TTM

—>TM xTM

—* 0.

DEFINITION 3.1.3. A geodesic equation on T M with respect to the connection tf is defined as the image q' =
? = K)q>

(3.1.6)

of the morphism (3.1.5) restricted to the diagonal TM C TM x TM. D By a solution of a geodesic equation on TM is meant a geodesic curve c in M, whose tangent prolongation c is an integral section (a geodesic vector Held) over c C M for the connection K. The geodesic equation (3.1.6) can be written in the form

fdtf = Ktf, where q'{qi) is a geodesic vector field (which exists at least on a geodesic curve), while q'di is the formal operator of differentiation (along a curve). It is readily observed that the morphism K \TM is a holonomic vector field on TM. It follows that any geodesic equation (3.1.5) on TM is a second order equation on M. The converse is not true in general. Nevertheless, we have the following theorem. T H E O R E M 3.1.4. [136]. Every second order dynamic equation (3.1.4) on a manifold M defines a connection K-= on the tangent bundle TM —> M whose components are

K) = \60. D

(3.1.7)

no

CHAPTER

3. HAMILTONIAN

SYSTEMS

However, the second order dynamic equation (3.1.4) fails to be a geodesic equa­ tion with respect to the connection (3.1.7) in general. In particular, the geodesic equation (3.1.6) with respect to a connection K determines the connection (3.1.7) on TM —» M which does not necessarily coincide with K. A second order equation E on M is a geodesic equation for the connection (3.1.7) if and only if E is a spray, i.e., [uTM, E] = =., where UTM is the Liouville vector field (2.4.9) on TM, i.e.,

H* = ayrywy. In Section 4.3, we will improve Theorem 3.1.4 (see Proposition 4.3.3 below). An extensive literature is devoted to the inverse problem for second order dy­ namic equations in Lagrangian mechanics. In Section 4.9, we will touch on the following two aspects of this problem in time-dependent mechanics: • the condition for an Euler-Lagrange type operator to be an Euler-Lagrange one as the particular inverse problem in field theory, • and the corresponding relation between Newtonian and Lagrangian systems. We refer the reader to [136] for a survey on the inverse problem in conservative Lagrangian mechanics. Recall that, in accordance with Remark 3.1.2, a second order dynamic equation on a manifold M, being a first order dynamic equation on TM, can always be thought of as Lagrange equations for a Lagrangian on the double tangent bundle TTM of M. In this Chapter, we will give two examples of the inverse problem for second order dynamic equations in conservative Hamiltonian mechanics. • As is well known, the tangent bundle TM fails to possess any canonical Poisson, symplectic, or presymplectic structures. Nevertheless, there are proce­ dures to study a second order dynamic equation as the Hamilton ones with respect to some Poisson structure on TM (see Remark 3.2.3 below). • Lagrange equations for a Lagrangian £ on the tangent bundle TM can be also seen as Hamilton equations with respect to the presymplectic structure (or the symplectic structure, if £ is regular), defined by this Lagrangian on TM (see Examples 3.3.2 and 3.4.2 below).

3.2. POISSON HAMILTONIAN

3.2

SYSTEMS

111

Poisson Hamiltonian systems

Let (Z, w) be a A:-dimensional Poisson manifold and H a real function on Z. DEFINITION 3.2.1. A Poisson Hamiltonian system (w,H) on a manifold Z for an (autonomous) Hamiltonian H with respect to the Poisson structure w is the set SH=

\J{V€TZZ:V-

w\dH){z) = 0}.

(3.2.1)

26Z

a A solution of the Hamiltonian system (3.2.1) is a vector field tfona submanifold N C Z, which takes its values into TN n Sa­ lt is readily observed that the Poisson Hamiltonian system S w (3.2.1) has a unique solution which is the Hamiltonian vector field &H = wl{dH)

(3.2.2)

for the Hamiltonian H, that passes through any point of Sn- Hence, 5 « is a first order equation in accordance with Definition 3.1.1. It is called the Hamilton equation for the Hamiltonian H with respect to the Poisson structure w. With respect to the local canonical coordinates {q',Pi,zA) (2.3.7) for the Poisson structure w, the Hamilton equation (3.2.1) reads g' = dm,

pj = -djH,

zA = 0,

while the vector field $n (3.2.2) takes the form ■&-H = PHdi - diHdK Its integral curves r : () —► Z satisfy the equations r* = d*n o r,

U = —diH o r,

fA = 0.

(3.2.3)

Recall that, given a dynamic equation (3.1.1) associated with a vector field u on a manifold Z, the Lie derivative L u of a function f on Z along u can be treated as the evolution of / along solutions of this dynamic equation. If

Uf = 0,

(3.2.4)

the function / is constant on any solution of the above-mentioned dynamic equation or equivalently on integral curves of u. A function / ^ const, is called a first integral

112

CHAPTER

3. HAMILTONIAN

SYSTEMS

of motion if it obeys the equation (3.2.4). Prom the geometric viewpoint, this means that the vector field u is tangent to the level surfaces of the function / at its regular points, where df =£ 0. If a dynamic equation is a Hamilton equation with respect to a Poisson structure, one can find its first integrals of motion as functions in involution with a Hamiltonian as follows. Let H be a Hamiltonian of a Poisson Hamiltonian system (w,H). The Lie derivative of an arbitrary real function / on Z along the Hamiltonian vector field tin for H reads

UJ

=

$H\d}={HJ}.

(3.2.5)

The equality (3.2.5) is called the evolution equation. Substituting solutions r of the Hamilton equations (3.2.3) in (3.2.5), we obtain the evolution dt(for)

=

{H,f}or

of a function / along the integral curves of the Hamiltonian vector field ■d-^. In particular, if

{HJ} = 0, the function / is a first integral of motion of the Hamiltonian system

(w,H).

E x a m p l e 3.2.1. A Hamiltonian "H itself is obviously a first integral of motion of the Hamiltonian system (tu,W). • It is easily seen that, if / and / ' are first integrals of motion of a Poisson Ha­ miltonian system, so is their Poisson bracket { / , / ' } . It follows that first integrals of motions constitute a Lie algebra. A vector field v on Z is called an infinitesimal symmetry of a Hamiltonian H if KH = 0. In particular, if / is a first integral of motion of a Poisson Hamiltonian system (w,H), the Hamiltonian vector field $/ for / is an infinitesimal symmetry of the Hamiltonian H, i.e.,

U,n

= 4f\dH =

-{H,f}=0.

3.2. POISSON HAMILTONIAN

SYSTEMS

113

Let us turn now to the inverse problem in conservative Hamiltonian mechanics, mentioned in Remark 2.3.5. It has the following solution. PROPOSITION 3.2.2. For any first order dynamic equation (3.1.1) on a manifold Z, there exists locally a generating Poisson Hamiltonian system (w, H) on Z such that (3.1.1) is locally a Hamilton equation. □ Proof. If u = 0, the statement is obvious. Let u{z) / 0 at a point z € Z. There exists a coordinate system (q1,. .. , qk) on an open neighbourhood of z such that

U=

W

and [u v] = 0 v =

'

'

w

Then u A v is a local Poisson bivector field of rank 2 (see Example 2.3.2). It is readily observed that u is locally a Hamiltonian vector field for the function q2{zx) with respect to the Poisson structure u A » . QED Proposition 3.2.2 leads us to the conditions for the inverse problem of a first order dynamic equation (3.1.1) to have a global solution. PROPOSITION 3.2.3. [59, 82]. Let u be a nowhere vanishing vector field on a manifold Z. If there exist • a nowhere vanishing vector field v on Z such that [u, v] = 0 everywhere on Z, • and a function f on Z such that L u / = 0 and L „ / = 1, then u is the Hamiltonian vector field for the function / with respect to the Poisson structure u A » . □ E x a m p l e 3.2.2. Every first order dynamic equation (3.1.1) on a manifold Z is a Hamilton equation on the product RxZ with coordinates (t, zx) for the Hamiltonian Ti = t and with respect to the Poisson structure w = u A dt. We refer the reader to [59] for the physically interesting examples. • R e m a r k 3.2.3. The method based on Propositions 3.2.2 and 3.2.3 can be applied to study the existence of a generating Poisson structure for a second order dynamic equation (3.1.4) on a manifold M defined by a holonomic vector field E (3.1.2) on the tangent bundle TM. In particular, for any second order dynamic equation

114

CHAPTER

3. HAMILTONIAN

SYSTEMS

(3.1.4) on M, there exists locally a generating Poisson Hamiltonian system {w,H) on TM such that the associated holonomic vector field S is the Hamiltonian vector field for the Hamiltonian H with respect to the Poisson structure w. The necessary condition for a Poisson structure w on the tangent bundle TM to be a generating Poisson structure for a holonomic vector field 5 on TM is L=(w) = [E,tu] = 0. We refer the reader to [182] for a detailed analysis of this condition. •

3.3

Symplectic H a m i l t o n i a n s y s t e m s

Let (Z, fi) be a symplectic manifold. The notion of a symplectic Hamiltonian system is a repetition of Definition 3.2.1 (see [127]). DEFINITION 3.3.1. A symplectic Hamiltonian system (ft.H) on a manifold Z for a Hamiltonian H with respect to the symplectic structure fi is the set

Sn = \J{veTzZ:

v\n + dH(z) = 0}.

(3.3.1)

2€Z

□ As in the general case of Poisson systems, the symplectic Hamiltonian system (fi, H) has a unique solution which is the Hamiltonian vector field tfK such that

V | fi = -dH for the Hamiltonian H, that passes through any point of Sn- Hence, S-H is a first order dynamic equation, called the Hamilton equation. With respect to the local canonical coordinates {q\pi) for the symplectic struc­ ture fi, the Hamilton equation (3.3.1) reads q* = dxH,

pj =

-djTi.

(3.3.2)

In the symplectic case, there is one-to-one correspondence between the Hamil­ tonian systems and the Hamiltonian vector fields on a symplectic manifold (Z, f2). The first integrals of motion of a symplectic Hamiltonian system and infinitesimal symmetries of a Hamiltonian are defined as a repetition of those for a Poisson Hamiltonian system.

3.3. SYMPLECTIC

HAMILTONIAN

115

SYSTEMS

Recall that a Hamiltonian system on a 2m-dimensional symplectic manifold is called completely integrable if there exist m first integrals of motion which are pairwise in involution and whose differentials are linearly independent on a dense open subset of that manifold. E x a m p l e 3.3.1. Let Z = T*M be a symplectic manifold provided with the canon­ ical symplectic form fi (2.4.2). In accordance with Example 2.4.7, any Hamiltonian system (ft, H) on the symplectic manifold (T'M, fi) is the Lagrangian submanifold Sn = $K{T'M) (2.4.10) of the symplectic manifold TT'M equipped with the sym­ plectic form Q (2.4.9). The generating function of this submanifold, given by the coordinate relations

q' = &n,

pi = -aoi,

(3.3.3)

is the Hamiltonian 7i(qx,pi). Any Hamiltonian H on the symplectic manifold T'M defines the fibred morphism over M H d = TT.TM o a o $H : T'M i ; ,

-^TT'M

-> T'TM

-»

TM,

(3.3.4)

q o R=d n,

called the Hamiltonian map. • E x a m p l e 3.3.2. Any non-degenerate autonomous Lagrangian system can be seen as a symplectic Hamiltonian system as follows. An autonomous Lagrangian is defined as a real function C on the tangent bundle TM of an event space M. Throughout this Chapter, we will use the compact notation 7Ti = d{C,

■Kji — djdiC.

Using the tangent-valued form 4>j (1.2.39) on the tangent bundle TM, let us intro­ duce the Poj'ncare-Cartan 1-form He =4>j\d£ = ntdq\ Its differential tie = dnt A dqi = ■K^dc? A dq' + djitidq1 A dq'

(3.3.5)

is a symplectic form on TM if and only if the Lagrangian C is regular, i.e., its Hessian det (TT^) is nowhere vanishing.

116

CHAPTER

3. HAMILTONIAN

SYSTEMS

If a Lagrangian C is regular, one can consider the symplectic Hamiltonian systen (Qci Ec) with respect to the symplectic form Qc for the Hamiltonian (3.3.6)

Ec = v-TMJdC — C = q'lTi — £,

called the energy function, where we make use of the Liouville vector field UTM (1.2.6) on TM. Then we come to the Hamilton equation for the Hamiltonian vector field dc such that (3.3.7)

ticltoL = ~dEc.

With respect to the coordinates (3.1.3) on the double tangent bundle TTM, equation (3.3.7), called the Cartan equation for the Lagrangian £, reads

the

(3.3.8a)

T«(q* - ?') = o, J

J

(3.3.8b)

diC - q'lTji - q 9j7r, + (q - q')dtTX] = 0.

Since the Lagrangian C is regular, the equation (3.3.8a) reduces to the equality (3.3.9)

q' =
while the equation (3.3.8b) comes to the Lagrange equations (3.3.10)

d,C — (pifji — qidj-Ki = 0

for the Lagrangian £. The condition (3.3.9) means that, in the case of a regular Lagrangian £, the Hamiltonian vector field dc is holonomic. It defines the second order dynamic equation on TM which is equivalent to the Lagrange equations (3.3.10). Let us consider the fibre bundles TT'M and T'TM provided with the coor­ dinates (q',Pi,q',Pi) and (q',q', (?,, <ji), respectively, and recall the isomorphism a (1.1.9): a : TT'M

S T'TM,

R

— . ft, ft ^

q,.

The symplectic form fi £ (3.3.5) yields the fibred morphism (1.2.33) over TM:

fij. : 7TM -> T T M , <7i o n b £ = -q>TT:i + q J (3j7rj -

^TT,),

q\ o nb£ = q% y ,

3.3. SYMPLECTIC

HAMILTONIAN

117

SYSTEMS

and the corresponding fibred morphism Q" 1 o ClL : TTM P, O a

_1

~> T'TM

—

TT'M,

f ^ . = -q>TCjx + q'idiTTj - djTTt),

(3.3.11) p, O Q

_1

O £lc = q*7Ty.

We have the relation

cic = -n£n T -™ = -{a~lo n).yn, fi£ = - [ d ( - f Try + q*(flbi - djTT,)) A
Pi = ^TTjn

of the symplectic manifold (TT'M, fi). The generating function of this Lagrangian submanifold is the energy function Ec (3.3.6). Every Lagrangian £ on the configuration space TM yields the fibred morphism over M £d=irT.M

o Q" 1 o dC : TM — T'TM

-> TT'M

-> T'M;

(3.3.12) (3.3.13)

p, = diC,

called the Legendre map. A Lagrangian £ is regular if and only if the corresponding Legendre map £ is a local diffeomorphism. A Lagrangian £ is called hyperregular if the Legendre map £ is a diffeomorphism. The tangent map TC to the Legendre map C is the fibred morphism TC : TTM PioT£

->

= TTi,

TT'M, p , = ix%Jq' + djiTiq>,

over M. Then the image of the Lagrange equations (3.3.10) into TT'M another Lagrangian submanifold Px = TTt,

Pi = di

by TC is

(3.3.14)

CHAPTER 3. HAMILTONIAN

118

SYSTEMS

of the symplectic manifold (TT'M, ft). This is exactly the Lagrangian submanifold (2.4.11), defined by the differential dC (see Example 2.4.7). Its generating function is C. One can find the conditions of a hypen-egular Lagrangian £ on the tangent bundle TM and a Hamiltonian H on the cotangent bundle T'M to define the same Lagrangian submanifolds (3.3.14) and (3.3.3) of the symplectic manifold (TT'M, Q). This takes place if and only if uT-M\dH-H l

pid H-H

= CoH,

(3.3.15)

i

=

C(q\d H),

where UT-M is the Liouville vector field (1.2.5) on T'M —» M, or equivalently uTM\dC

- £ = H o £,

i

q diC-C

=

H(qi,diC),

where UTM is the Liouville vector field (1.2.6) on TM —► M. Taking the differential of the equation (3.3.15), we find that H = C~\ (diC) oH=

-dtH.

Then we have the commutative diagram TM C

\

T'M

-^

TTM \

TC

—>TT'M

• The constructions in Examples 3.3.1, 3.3.2 lead to description of Hamiltonian and Lagrangian dynamics of autonomous systems in terms of Lagrangian submanifolds and to constructing unified Lagrangian-Hamiltonian formalism on the phase space TT'M [111, 176, 177] (see Example 3.4.2 below).

3.4

Presymplectic Hamiltonian systems

The notion of a Hamiltonian system is naturally extended to presymplectic manifolds (see [12, 60, 138, 172]).

3.4. PRESYMPLECTIC

HAMILTONIAN

SYSTEMS

119

DEFINITION 3.4.1. Let Z be a fc-dimensional manifold equipped with a presymplectic form fi, and let H be a real function on Z. A presymplectic Ha.miltonia.ii system for a Hamiltonian H, like (3.3.1), is the set Snd=

\J{ve zez

TZZ : vjfi + dH(z) = 0}.

(3.4.1)

□ A solution of the presymplectic Hamiltonian system (3.4.1) is a Hamiltonian vector field d-u for H which lives in Sn- It satisfies the equality

tfwjfi = -<m.

(3.4.2)

In comparison with the symplectic case, the form £1 is degenerate, the set (3.4.1) fails to be a submanifold in general, and a solution of the equation (3.4.2) does not necessarily exist everywhere on the manifold Z. Note that, for any point z e Z, the fibre {v€TzZ

: v\n + dH(z) = 0 }

(3.4.3)

of the set S-H over z is an affine space modelled over the vector space Ker 2 Q = {v G TZZ : u\v\Q = 0, Vu 6 TZZ} which is the fibre over z of Kerfl (1.2.34) of the presymplectic form fi. The fibre (3.4.3) may be empty. PROPOSITION 3.4.2. The equation v\il + dH{z) = 0,

v G TZZ,

(3.4.4)

has a solution only at the points of the subset N2 = {z G Z : u\<m{z) = 0, Vu G Ker 2 Q} C Z,

(3.4.5)

i.e., where Ker z ft C Ker zdH [60, 138]. □ Proof. Let a vector v G TZZ, satisfying the equation (3.4.4), exist. Then, the contraction of the right-hand side of the equation (3.4.4) with an arbitrary element u G Ker z Q leads to the equality u\dH(z) = 0. In order to prove the converse assertion, it suffices to show that

dH(z) G Imn b ,

CHAPTER

120

3. HAMILTONIAN

SYSTEMS

that results from the inclusions dft(z) £ Ann(Ker dH{z)) C Ann(Ker 2 fi) = Imfi*. QED It follows that, if Z ^ N2a. presymplectic Hamiltonian system is not a differential equation on Z. If N2 = Z, it is a differential equation, but not a dynamic equation. From now on, let us suppose that a presymplectic form f2 is of constant rank, and that N2 (3.4.5) is a submanifold of Z, but not necessarily connected. Then, by virtue of Proposition 1.1.4, the kernel Kerfi is a closed vector subbundle (moreover, an involutive distribution) of the tangent bundle TZ. Accordingly, Sn \N7 is an affine bundle over N2. It admits a global section, but this section does not live necessarily in TN2 C TZ, i.e., is not a vector field on the submanifold N2 in general. To obtain a solution of the presymplectic Hamiltonian system (3.4.1), let us consider a submanifold N C N2 C Z such that

Sn\NnTxN^9,

V z e N,

or equivalently dH(z) 6

Q\TN),

Vze N.

If the morphism 0* \TN is of constant rank, then S-H H TN —> N is an affine bundle modelled over the vector bundle Ker Q n TN —» TV which may be 0. Sections of the affine bundle S-H fl TN —» TV are Hamiltonian vector fields $x (3.4.2) for the Hamiltonian W o n a submanifold N. If such a submanifold exists, it may be found by the following well-known algorithm [12]. Let us take the intersection SH\N2nTN2 and project it to Z. We obtain the subset N3 =

KZ{SH\N,OTN2)CZ.

If N3 is a submanifold, let us take the intersection SH\N3nTN3. Its projection to Z gives a subset N4 C Z, and so on. Since a manifold Z is finitedimensional, the procedure is stopped after a finite number of steps by one of the following results.

3.4. PRESYMPLECTIC

HAMILTONIAN

SYSTEMS

121

• There is i > 2 such that a set TVj is empty. This means that a presymplectic Hamiltonian system has no solution. • A set N,, i > 2, fails to be a submanifold. It follows that a solution may exist, but not everywhere on A/j. • If Ni+i = Nt for some i > 2, this is the desired submanifold N. If the morphism \TN is of constant rank, a solution of the presymplectic Hamiltonian system (3.4.1) exists everywhere on N. Remark 3.4.1. Since N2 ^ Z in general, one can consider another variant of how to solve a presymplectic Hamiltonian system. Let N2 (3.4.5) be a submanifold of Z. It can be provided with the pull-back presymplectic form Q2 = i)v2^- Then the pull-back presymplectic Hamiltonian system SH, =

U iv e TZN2 :v\Q2 zew2

+ dH2(z) = 0}

for the pull-back Hamiltonian H2 = i'^Ti on ./V2 can be examined. As for the initial system S-H, we find that the equation v\Q2 + dH2(z)=0,

veTzN2,

has a solution on the subset N*i = {zeNi:

u\i'N2dn{z)

= 0, Vu € Ker 2 ft 2 C TN2} C N2,

and so on. We obtain the chain of submanifolds N2 D N'3 D N'4 ■ ■ ■

which is finite, and is stopped by one of the following results. • There is i > 2 such that a set N- is empty. • A set AT/, i > 2, fails to be a submanifold. • If N[+l = N? for some t > 2, this is a desired submanifold N' C Z. If the presymplectic form i'N,Q, on N' is of constant rank, then there exists a solution of the presymplectic Hamiltonian system (i*N,£l, i^H) everywhere on N'.

122

CHAPTER

3. HAMILTONIAN

SYSTEMS

It is readily observed that any solution $n C TN of the equation (3.4.2) on a submanifold N C Z also obeys the equation

tiH\i'N{n + dH) = 0 and, consequently, is a solution of the pull-back presymplectic system on N. follows that N C N'. •

It

Solutions of the presymplectic Hamiltonian system 5 « (3.4.1) on a submanifold N constitute an affine space modelled over the linear space of sections of the vector bundle KerCl n TN —» N. If this vector bundle is not 0, different solutions of the Hamilton equation (3.4.1) correspond to different first order dynamic equations on ./V. Therefore, a Hamilton equation with respect to a presymplectic structure on a manifold Z is not equivalent to a first order dynamic equation on Z in general. Sections of the vector bundle Kerfi —» Z are sometimes called gauge fields in order to emphasize that, being solutions of the presymplectic Hamiltonian system (fi,0) for the zero Hamiltonian, they do not contribute to a physical state, and are responsible for the gauge freedom [12, 172]. At the same time, there are physically interesting presymplectic Hamiltonian systems, e.g., in relativistic mechanics when a Hamiltonian is equal to zero (see also Remark 4.8.9 below). In this case, Ker dH = TZ and the Hamilton equation (3.4.4) has a solution everywhere on a manifold Z. Another pull-back construction is connected with the above-mentioned gauge freedom. Let a presymplectic form fi on manifold Z be of constant rank and let its characteristic foliation be simple, i.e., its leaves are fibres of a fibre bundle n : Z —» P over a symplectic manifold (P, $7/>) in accordance with Proposition 2.6.2. Then ft is the pull-back ir'Vlp of a symplectic form ftp on the base P by this fibration. Let a Hamiltonian H be the pull-back n'Hp of a function Tip on P. Then we have Kerfi = VN C Ker dH, and the presymplectic Hamiltonian system (f2, Ti) has solutions everywhere on the manifold Z. Any such solution tin is projected onto a unique solution of the symplec­ tic Hamiltonian system (dp, Tip) on the manifold P, while gauge fields are vertical vector fields on the fibre bundle Z —» P . This is the case of a gauge invariant Hamiltonian system, where P is called a physical phase space. In the case of a presymplectic Hamiltonian system on a manifold Z, this manifold fails to be provided with a Poisson structure in general (see Remark 2.5.1), that is an essential problem for quantization. Nevertheless, by virtue of Propositions 2.5.3 and

3.5. DIRAC HAMILTONIAN

SYSTEMS

123

2.5.4, every presymplectic form on a manifold Z can be represented as a pull-back of a symplectic form by the coisotropic imbedding. It follows that each presymplectic Hamiltonian system can be seen as a Dirac constraint system [26] that we will investigate in Section 3.6. Example 3.4.2. Autonomous Lagrangian systems with constraints are often treated as presymplectic Hamiltonian systems (see [30, 61, 112, 113, 138] and references therein). Every Lagrangian £ on the tangent bundle TM of a configuration mani­ fold M provides TM with the presymplectic form Qc (3.3.5). Let us consider the presymplectic Hamiltonian system (Qc, Ec) on TM for the Hamiltonian Ec (3.3.6). This is exactly the Cartan equation (3.3.8a) - (3.3.8b) for the Lagrangian £. Its restriction to the submanifold q' = q' (3.3.9) of the double tangent bundle TTM of M leads to the Lagrange equations (3.3.10). This means that solutions of the Lagrange equations are solutions of the Cartan equation which are dynamic vector fields on TTM. Conversely, a symplectic Hamiltonian system (fi, H) on the cotangent bundle T'M can be seen as a degenerate Lagrangian system on the velocity phase space TT'M of the momentum phase space T'M as follows. Put the Lagrangian £n = Piq'

-U{q\pi)

(3.4.6)

on this velocity phase space. It is readily observed that the pre-image in TTT'M of the Hamilton equation (3.3.2) on TT'M for the Hamiltonian H by the projection TTT'M -> TT'M is exactly the Lagrange equations (3.3.10) on TTT'M for the Lagrangian (3.4.6). It follows that they have the the same solutions r : () —► T'M. It should be emphasized, that, in general, neither Cartan equations nor Lagrange equations are differential equations in a strict sense. They are differential equations, e.g., when the Hessian matrix 7iy, of £ is of constant rank. •

3.5

Dirac Hamiltonian systems

There is an extensive literature devoted to autonomous mechanical systems with constraints (see, e.g., [121, 127, 137, 166, 172] and references therein). In this and the next Sections, we will deal only with autonomous Hamiltonian systems with con­ straints. We leave the constraints in time-dependent Lagrangian and Hamiltonian mechanics for Chapters 4 and 5.

124

CHAPTER

3. HAMILTONIAN

SYSTEMS

Let (Z, fi) be a 2m-dimensional symplectic manifold and H a Hamiltonian on Z. Let N be a (2m - n)-dimensional closed imbedded submanifold of Z, called the primary constraint space or simply a constraint space. We aim to investigate the following two problems: (A) solutions of the Hamiltonian system (Q, H) on a manifold Z, which live in the constraint space N, (B) and the restriction of the Hamiltonian system (fi, H) to the constraint space N. R e m a r k 3.5.1. The following local relations will be useful in the sequel. Let the constraint space N be given locally by the equations fa{z) = 0,

(3.5.1)

o = l , . ..,n,

where / a (z) are local functions on Z, called the primary constraints. Let us consider the set IN =

(3.5.2)

Ken'NcQ°{Z)

of real functions f on Z which vanish everywhere on N, i.e., i'Nf = 0. This is an ideal of the associative commutative algebra 0 ° ( Z ) such that D°(Z)/IN

=

0°(N)

(cf. (2.6.4)). Its elements / are written locally in the form

/ =

(3.5.3) a=l

where ga are functions on Z. This means that the ideal //v is locally generated by the constraints /„. We will call {/a} a local basis for the ideal 1^. Let dIN be the submodule of the D°(Z)-module O ' ( Z ) , which is generated locally by the exterior differentials df of functions f £ IN. Its elements are the finite sums a =

T.9'df„

/, € IN,

9' €

0°(Z).

i

By virtue of (3.5.3), they are given by local expressions

a=J2(9adfa a=l

+ fa4>a),

(3.5.4)

3.5. DIRAC HAMILTONIAN

SYSTEMS

125

where g" are functions and 4>a are 1-forms on Z. It should be emphasized that the expressions (3.5.3) and (3.5.4) are fulfilled locally on a domain where the constraint space N is given by the equations (3.5.1).

• The solution of problem (A) is obvious. It exists if a Hamiltonian vector field ■d-H, restricted to the constraint space N, is tangent to N, i.e., SH\N = \J{veTzZ:

v\Q + dH{z) = 0} C TN.

(3.5.5)

Then integral curves of the Hamiltonian vector field tin do not leave N. The condition (3.5.5) is satisfied if and only if (3.5.6)

{H, IN} C IN,

i.e., if and only if the Hamiltonian H belongs to the normalizer I(N) (2.6.6) of the ideal IN- With respect to the local basis {fa}, the condition (3.5.6) reads n

#n\dfa = {«,/«} where gc are functions on Z. If the relation (3.5.6) does not hold, let us introduce the secondary {HJa}

(3.5.7)

constraints

= 0.

If the collection of primary and secondary constraints is not closed with respect to the relation (3.5.7), let us add the tertiary constraints {H,{H,fa}}

= 0,

and so on. If a solution exists anywhere on N, the procedure is stopped after a finite number of steps by constructing the complete system of constraints. The complete system of constraints defines the final constraint space, which do not include the points of Z where the Hamiltonian vector field i?M is transversal to the primary constraint space N. From the algebraic viewpoint, we find an ideal 7nn of D°(Z) which is the minimal extension of the ideal IN such that {W, /fin} C /fin-

CHAPTER

126

3. HAMILTONIAN

SYSTEMS

In the framework of the algebraic theory of constraints, problem (A) can be reformulated as follows. Let AT be a closed imbedded submanifold of a symplectic manifold (Z, ft) and IN (3.5.2) the ideal of functions which vanish everywhere on N. All its elements are said to be constraints. We aim to find a Hamiltonian, called an admissible HamiltoniaD, on Z such that the symplectic Hamiltonian system (ft, H) has a solution everywhere on the constraint space N. One can treat N as a final constraint space. Moreover, we may take an ideal IN whether it is an ideal of functions vanishing on some submanifold N C Z or not. In accordance with the condition (3.5.6), any element of the normalizer I(N) (2.6.6) is an admissible Hamiltonian. Let us consider the intersection I'{N) = I(N) n IN (2.6.7). Its elements are called the first-class constraints, while the remaining elements of IN axe the secondclass constraints. The first-class constraints make up a Poisson subalgebra of the normalizer I(N) which, in turn, is a Poisson subalgebra of the Poisson algebra D°{Z) on the symplectic manifold (Z, ft) (see Section 2.6). Recall that IN C I'(N), i.e., the products of second-class constraints are also first-class constraints. E x a m p l e 3.5.2. If N is a coisotropic submanifold of Z, then IN C I{N) I'{N) = IN- It follows that we have only the first-class constraints. •

and

The admissible Hamiltonians which are not first-class constraints are the rep­ resentatives of the non-zero elements of the quotient I(N)/I'(N), which is the re­ duction of the Poisson algebra D°(Z) via the ideal IN (see Definition 2.6.8). In particular, let the presymplectic form i'Nfi on N be of constant rank and its char­ acteristic foliation be simple, i.e., it defines a fibration 7r : N —» P over a symplectic manifold P. In view of the isomorphism (2.6.8), one can think of elements of the quotient I(N)/I'(N) as being the Hamiltonians on the physical phase space P. It follows that the restriction of an admissible Hamiltonian H to the constraint space N coincides with the pull-back onto N of some function / on P , i.e.,

i-NH = *•/,

f e o°(P).

Let us turn now to problem (B). Given a Hamiltonian Wonasymplectic manifold (Z, Q) and a constraint space N C Z, a Dirac Hamiltonian system (ft, H, N) on the constraint space N is defined as the subset S.H = \J{ve

TZZ : i'N(v\n + dri(z)) = 0}

(3.5.8)

3.5. DIRAC HAMILTONIAN

SYSTEMS

127

[127, 137]. Note that SH\N

C

S.n.

Therefore, the set (3.5.8) is not empty. We will say that a Dirac Hamiltonian system on N has a solution if there is a vector field on a neighbourhood of each point z 6 /V, which lives into the subset S.n (3.5.8). For instance, every solution of problem (A) is also a solution of problem (B). E x a m p l e 3.5.3. Problem (B) has a complete solution in the germ terms in the following case. Let Z = T'M be a symplectic manifold equipped with the canonical symplectic form fi, and N its coisotropic submanifold. Then TN is a coisotropic submanifold of the symplectic manifold TT'M equipped with the symplectic form Q (2.4.9). Indeed, let utTTN and v€TTT'M\TTN. There exists a vector field TU on JV such that its canonical lift TU onto TN contains tt. There is also a vector field T„ on T'M such that its canonical lift TV onto TT'M contains v. Let u and v be projected over the same element q £ TN. By the virtue of the relation (1.2.23), one can then write v\u\a

= n&}Jf tt («)Jfi = K v z ) K ( z ) J f t ) ,

z =

xz{q).

In the case of an arbitrary u 6 TqTN, this expression differs from 0 because there exists a vector field T„ on a neighbourhood of z, which does not belong to OrthnTN since N is coisotropic. It follows that Orth-TTN does not include elements of

TTT'M\TTN. Let 7{ be a Hamiltonian on T'M. Then the symplectic Hamiltonian system Sn (3.3.1) is a Lagrangian submanifold of the symplectic manifold (TT'M,fy (see Example 2.4.7). A Dirac Hamiltonian system {Sl,H,N) on a (2m - n)-dimensional coisotropic submanifold TV has a solution if the coisotropic submanifold TN C TT'M and the Lagrangian submanifold Sw are not transversal. The germ of any such pair (TN,S\.) is symplectically equivalent to the pair ({,' = • . • = ,*» = 0}, {qa = 8PS{p)) a = 1 , . . . . 2m}} 0))

(q,p)

eR4m,

where S is the germ of a function such that the rank of the tangent map to the morphism P-»

(d^S,.,.,d2nS)

128

CHAPTER

3. HAMILTONIAN

SYSTEMS

at 0 is not maximal. We refer the reader to [48] for a detailed classification of the germs of such generating functions S. • A Dirac Hamiltonian system (Q, H, N) is called compieteiy integrable if there exists its solution through each point of the set S.n- The obvious necessary condition for a Dirac Hamiltonian system to be completely integrable is the inclusion S.n C TN. In this case, a Dirac Hamiltonian system {Q, H, N) reduces to the symplectic Hamiltonian system (i'NQ, J/yW) on the constraint space N. Let us inspect the above-mentioned necessary condition. Note that, for each point 2 G N, the fibre {v e TZZ : i'N(v\Sl + dH{z)) = 0} of the set S.n over z € N is a non-empty affine space modelled over the vector space {v e TZZ : u\v\Q. = 0, Vu € TZN}

(3.5.9)

which is the fibre of the Q-orthogonal space OrthnTW (2.4.6) of TN. It follows that OrthnT/V C TN. Therefore, a Dirac Hamiltonian system on the constraint space N is completely integrable only if N is a coisotropic submanifold of the symplectic manifold (Z, Q). Then we have u\dH = 0,

Vu e OrthnN.

In order to formulate a sufficient condition for a Dirac Hamiltonian system to be completely integrable, let us assume that the vector spaces (3.5.9) for all z G N are of the same dimension. Then the fi-orthogonal OrthnTTV of TN is a vector bundle over N, while the Dirac Hamiltonian system S.n —► ,/V is an affine bundle over ./V. PROPOSITION 3.5.1. [127], Let a submanifold i V o f a symplectic manifold (Z,f2) be coisotropic. Then OrthnTW is an involutive distribution on N. If the exterior differential dH is constant on the leaves of the corresponding foliation, then the Dirac Hamiltonian system (3.5.8) is completely integrable. □ Proof. Since the set S.n is not empty, it suffices to show that this set belongs to TN. Let v £ S.n and z = TTZ(V). Whenever u € Orth n TAf C TN over z, the equality v\u\Sl = u\dH = 0

3.6. DIRAC CONSTRAINT

SYSTEMS

129

holds. It follows that v € TZN. Since S.« C TN, there is a vector field ■d on N through any point of the Dirac Hamiltonian system S.«. Such a solution is not unique. If ■& is the above-mentioned vector field, then any vector field ■d + v where v lives in OxianTN C TN is also a solution of the Dirac Hamiltonian system. QED Thus, one should consider completely integrable Dirac Hamiltonian systems on coisotropic submanifolds N. In this case, a Dirac Hamiltonian system (fl, H, N) reduces to the presymplectic Hamiltonian system on the constraint space N for the pull-back Hamiltonian i'NH with respect to the pull-back presymplectic form i'NQ. This is the case of a Dirac constraint system.

3.6

Dirac constraint systems

Let (Z, fi) be a 2m-dimensional symplectic manifold, and let Nbea submanifold of Z. Let H be a Hamiltonian on Z.

closed imbedded

DEFINITION 3.6.1. A Dirac constraint system on the constraint space N is the set SN-H™

\J{v

e TZN : v\i%(fl + dH{z)) = 0}.

(3.6.1)

D

In particular, we are dealing with Dirac constraint systems when a Hamiltonian description of degenerate autonomous Lagrangian systems is investigated (see, e.g., [58, 172] and references therein). They are called generalized Hamiltonian systems. In this case, a momentum phase space Z is the cotangent bundle T'M of a config­ uration space M. This phase space is provided with the canonical symplectic form. A primary constraint space iV C Z is an image of the Legendre map L (3.3.12) defined by a degenerate Lagrangian £ on the velocity phase space TM. In fact, one introduces some initial Hamiltonian H on T'M such that the energy function Ec (3.3.6) for C is the pull-back C"H of this Hamiltonian H by the Legendre map C. Thus, we come to a Dirac constraint system on the Lagrangian constraint space N. The goal consists in constructing a Hamiltonian H' on the momentum phase space T'M (but not uniquely in general) such that the solution dw of the symplectic Ha­ miltonian system (fi, H') on T'M provides a solution of the above-mentioned Dirac constraint system on a final constraint space.

130

CHAPTER

3. HAMILTONIAN

SYSTEMS

In fact a Dirac constraint system SN-n is the pull-back presymplectic Hamiltonian system (i'NQ, i'NH) on a submanifold N C Z for the Hamiltonian i'NH and with respect to the presymplectic form i'N£l on N. For instance, Proposition 3.4.2 shows that the equation i'N(v\Q. + dH{z)) = 0,

v g TZN,

has a solution only at the points of the subset N2 = {z e N : u\i'NdH(z)

= 0, Vu £ Ker z fi N }

which is assumed to be a manifold. Such a solution, however, fails to be tangent to N2. Then one should repeat the algorithm for presymplectic Hamiltonian systems in Remark 3.4.1. Nevertheless, one can say more since the presymplectic system SN-H (3.6.1) on TV is the pull-back of the symplectic system (Q, H) on Z. Let a (2m — n)-dimensional closed imbedded submanifold TV of Z be a finai constraint space of the Dirac constraint system (3.6.1). This means that the equation v\QN + dHN(z) = 0, O/v — ^TV^J

v e TZN,

Hpj = i*NH,

has a solution at each point z £ N. By virtue of Proposition 3.4.2, this is equivalent to the condition u\dHN(z)

= 0,

Vu G Ker 2 Q w ,

VzG N,

or Ker nN = TNC\ OrthnTN

C Ker H.

(3.6.2)

We will reformulate these condition in algebraic terms. Let Is be the ideal of functions on Z which vanish everywhere on the final constraint space N. Let I(N) be the normalizer (2.6.6) of 1^ and I'{N) = I{N) H IN. It is readily observed that, restricted to TV, Hamiltonian vector fields dj for elements / of I'{N) with respect to the symplectic form Q on Z take their values into TN nOrth n TJV [97]. Then the condition (3.6.2) can be written in the form {H,I'{N)}CIN,

(3.6.3)

whereas

{H,IN}£IN

(3.6.4)

3.6. DIRAC CONSTRAINT

SYSTEMS

131

in general. A glance at the relations (3.6.3) and (3.6.4) shows that, though the Dirac constraint system on N has a solution, the Hamiltonian vector field dn for the Hamiltonian H with respect to the symplectic form fi on Z is not tangent to N, and its restriction to N is not such a solution in general. Example 3.6.1. Let the final constraint space N be a coisotropic submanifold of the symplectic manifold (Z, fi). Then IN = I'{N), i.e., there are only first-class constraints. In this case, the Hamiltonian vector fields of the Hamiltonian H and of other Hamiltonians H + / , / € IN, provide solutions of the Dirac constraint system on N. Note that, if a primary constraint space is coisotropic, we arrive at a Dirac Hamiltonian system, where the relation (3.6.3) is exactly the condition of Proposition 3.5.1. • The goal of constructing a generalized Hamiltonian system is to find a constraint / € IN such that the modified Hamiltonian H + f would satisfy both the conditions {H +

f,I'(N)}cIN,

{H + / , IN} C IN-

(3.6.5) (3.6.6)

The condition (3.6.5) is fulfilled for any / e IN, while (3.6.6) is an equation for a second-class constraint / . Remark 3.6.2. Since 1^ is the ideal of functions vanishing on a manifold N, the normalizer of IN coincides with the normalizer of J'(JV) [97]. Therefore, the conditions (3.6.5) and (3.6.6) may be combined into {H +

f,I'(N)}cI'(N).

• The equation (3.6.6) is solved in many concrete models, that implies separating first- and second-class constraints. The general problem lies in the fact that the collection of elements which generate If, C I'(N) is necessarily infinite reducible [97]. At the same time, the Hamiltonian vector fields for elements of IN vanish on the constraint space N. Therefore, one can apply the following procedure [137]. Since N is a (2m—n)-dimensional closed imbedded submanifold of Z, the ideal IN is locally generated by a finite basis {/„}, a = 1 , . . . , n, while N is defined locally by the equations (3.5.1). Let the presymplectic form Qn be of constant rank 2m-n — k, i.e., Kerzfl<2 has the dimension k < n at all points z £ Q. It defines a fc-dimensional

CHAPTER 3. HAMILTONIAN

132

SYSTEMS

characteristic foliation on N. Since N C Z is closed, there exist locally k linearly independent vector fields ub on Z which, restricted to N, are tangent to the leaves of this foliation. They read n

« 6 = E &"*/..

6=1,...,*,

o=l

where g% are functions on Z and #/ 0 are Hamiltonian vector fields for constraints fa. Then one can choose the collection of constraints fa, b = 1 , . . . ,n, where the first k functions take the form n

4>b=Y^9bfa<

6=l,...,fc.

Let ■d^ be the Hamiltonian vector fields for these functions. One can easily justify the fact that 0fc|jv = u»|w,

b-l,...,k.

It follows that the constraints fa belong to I'(N) \ I%- Therefore, they are called the first-class constraints, while the remaining fa.+\, ■ ■ ■, fa are the second-class con­ straints. We have the relations n {fa,

fa}

=

J2

C

bca,

b=l,...,k,

c= l , . . . , n ,

where C£. are local functions on Z. It should be emphasized that the constraints fa, • • •, fa do not constitute any local basis for I'{N), though fa,- ■ ■ ,fa make up a local basis for the ideal INNow let us consider a local Hamiltonian on Z n

%' = Ti + 2_, ^ a 0a,

(3.6.7)

where A" are functions on Z. Since Ti obeys the condition (3.6.5), we find {H,fa)=YJBabfa,

6=1...,A,

a=l

where B% are functions on Z. Then, the equation (3.6.6) takes the form {H,fa}+

Y. o=t+l

A " { ^ , 0 c } = Y,D%, 6=1

c=k+l,...n,

(3.6.8)

3.7. HAMILTONIAN

SYSTEMS

WITH

SYMMETRIES

133

where Dbc are functions on Z. This is the system of linear algebraic equations for the coefficients A°, a = k + 1 , . . . n, of second-class constraints. These coefficients are defined uniquely by the equations (3.6.8), while the coefficients \a, a = 1 , . . . , k, of first-class constraints in the Hamiltonian (3.6.7) remain arbitrary. Then, being restricted to the constraint space N, the Hamiltonian vector field of the Hamiltonian H' (3.6.7) on Z provides a local solution of the Dirac constraint system on N. We refer the reader to [137] for a global variant of the above procedure. Example 3.6.3. If AT is a symplectic submanifold of Z, then I'(N) = 1%. Therefore, we have only second-class constraints, and the Hamiltonian (3.6.7) of a generalized Hamiltonian system is defined uniquely. • R e m a r k 3.6.4. In concrete models, the final constraint space N fails to be given in advance. Therefore, a different procedure of separating first- and second-class constraints is usually applied (see, e.g., [39, 58, 172]), but its global treatment is under discussion. •

3.7

Hamiltonian systems with symmetries

In this Section, we summarize the well-known results connected with a group action on symplectic and Poisson manifolds [2, 116, 126, 130, 181]. By G throughout is meant a real connected Lie group. We start from a symplectic manifold case. Let (Z, Q) be a symplectic manifold on which a real connected Lie group G acts on the left so that, whenever g € G, the mapping z *—> gz is a symplectic isomorphism of Z. Such an action of a group G is called a symplectic action. Remark 3.7.1. The classification of transitive symplectic actions of connected Lie groups has been done (see [69, 72, 103]). • Since G is connected, its action on a manifold Z is symplectic if and only if, for any element e of the right Lie algebra gr of a the group G, the corresponding vector field £c on Z is locally Hamiltonian, i.e.,

LCtfi = d(&jn)=0.

CHAPTER

134

3. HAMILTONIAN

SYSTEMS

Let us suppose that it is a Hamiltonian vector field, i.e.,

££Jfi=-d/£, where Jc is a real function on Z. DEFINITION 3.7.1. A symplectic action of a connected Lie group G on a symplectic manifold Z is called a ffamiitonian action if, whenever e 6 QT, there exists a function Jc on Z such that f£ is the Hamiltonian vector field for Jc. □ PROPOSITION 3.7.2. An action of a connected Lie group G on a symplectic manifold Z is Hamiltonian if and only if there exists a mapping, called a momentum mapping, from Z to the Lie coalgebra J ■ Z -

0*

such that, whenever e € 0 r , the function Jt(z)

= (J{z),e),

(3.7.1)

Veeflr,

is that in Definition 3.7.1. □ Proof. If J exists, we obviously have a Hamiltonian action. Conversely, if £c for any e € g r is the Hamiltonian vector field for a function Je on a manifold Z, then the momentum mapping J is defined by the relation (3.7.1) applied to the basis {em} of the Lie algebra gT. QED It is readily observed that, if j and J' are different momentum mapping for the same action of G on Z, then d{J(z)-J'(z),e)=0,

Ve 6 flr,

i.e., J — J' = const, on Z. Example 3.7.2. Let a symplectic form on Z be exact, i.e., ft = dO, and let 6 be G-invariant, i.e.,

Le,0 = d(&J*) + & j n = o,

Ve e a,..

Then the map J : Z —» 0*, given by the relation

<J(z),£) = (&j0)(z),

3.7. HAMILTONIAN

SYSTEMS

WITH

SYMMETRIES

135

is a momentum mapping. •

PROPOSITION 3.7.3. If a Hamiltonian H o n a symplectic manifold Z is invariant under a group G acting on Z, a momentum mapping J defines a family of first integrals of motion of the Hamiltonian system for H. □ Proof. We have LfcW = Ze\dM = 0,

Ve € flr,

and then {H, J£) = 0,

Ve €

flr.

QED Thus, the functions Je{z) (3.7.1) are first integrals of motion. However, they an not pairwise in involution in general. Let us find their Poisson brackets. DEFINITION 3.7.4. A momentum mapping J is called equivariant if J(gz)=

Ad'g(J(z)),

Vp e G,

where Ad'g is the coadjoint representation (1.5.3). □ E x a m p l e 3.7.3. The momentum mapping in Example 3.7.2 is equivariant. In accordance with the relation (1.5.3), it suffices to show that Je{gz) =

JMg-ne)(z),

i.e., (^\8)(gz)

= (£Ad,-i(«)J*)(*).

The latter results follow from the relation (1.5.2). • Example 3.7.4. Let T'M be a symplectic manifold equipped with the canonical symplectic form fi (2.4.2). Let a group G act on M on the left by the generators

U = 4,(9)*-

CHAPTER

136

3. HAMILTONIAN

SYSTEMS

The canonical lift of this action onto T'M has the generators PAe3md\

lm = e'mdi -

(3.7.2)

and preserves the canonical Liouville form 6 on T'M. In accordance with Example 2.4.4, the vector fields (3.7.2) are Hamiltonian vector fields for the functions J

m =

(3.7.3)

e'm(q)pi

and, as follows from Example 3.7.3, there exists the equivariant momentum mapping J = e\n{q)piem. In the case of a G-invariant Hamiltonian on T'M, integrals of motion. •

the functions (3.7.3) are first

In the case of non-equivariant momentum mappings, let us consider the difference a(g) = J(gz) -

Ad'g(J(z)).

(3.7.4)

We refer the reader to [2] for proof that this difference is constant on a symplectic manifold Z, and fulfills the equality o(gg') = o(g)

+Ad'g(o(g')).

(3.7.5)

DEFINITION 3.7.5. A map a : G —> g* is called the cocycle on a group G if the condition (3.7.5) holds. A cocycle a is said to be a coboundary on a group G, if there exists an element p, € g* such that a(g) = p-

Ad'g(p,).

(3.7.6)

The equivalence classes of cocycles modulo coboundaries make up the cohomology group of the group G. □ By this definition, the difference (3.7.4) is the cocycle of the group G associated with the action of G on Z. PROPOSITION 3.7.6. Each symplectic action of a group G on a symplectic manifold Z defines a cohomological class [a] of G. □

3.7. HAMILTONIAN

SYSTEMS

WITH

SYMMETRIES

137

Proof. Let J and J' be different momentum mappings associated with a symplectic action of a group G on a symplectic manifold Z. Since the difference J— J' is constant on Z, then the difference of the corresponding cocycles a — a' is the coboundary (3.7.6) where n= J- J'. QED In particular, an equivariant momentum mapping defines the zero cohomological class of the group G. T H E O R E M 3.7.7. Given a momentum mapping J associated with a symplectic action of a group G on a symplectic manifold Z, the following relation {Je,Jc,}

=

J[c,c>]-{Tea{e'),e)

(3.7.7)

takes place (we refer the reader again to [2], where, however, the left Lie algebra is utilized and Hamiltonian vector fields differ in the minus sign from those we use).

□

In the case of an equivariant momentum mapping as in Example 3.7.4, the relation (3.7.7) leads to the homomorphism {Je ,Je'} = J\t,e'\

(3.7.8)

of the Lie algebra g r to the Lie algebra of functions on a symplectic manifold Z with respect to the Poisson bracket. These are the desired Poisson brackets of first integrals of motion for a G-invariant Hamiltonian system. We will refer to this result in connection with the conservation laws in timedependent mechanics (see Section 5.8). Let now (Z, w) be a Poisson manifold, on which a connected Lie group G acts on the left so that, whenever g e G, the mapping z —> gz is a Poisson automorphism of Z. Such an action of a group G is called a Poisson action. Since G is connected, its action on a manifold Z is a Poisson action if and only if, for any element e of the right Lie algebra g r of a group G, the corresponding vector field £ t on Z is an infinitesimal Poisson automorphism, i.e., the condition (2.3.3) holds. The equivalent conditions are

&({/,»}) = {&(/).$} + {/.&($)}.

&({/,$}) = [&,*/](»)-fc.W). [&.*/] = *«.
CHAPTER

138

3. HAMILTONIAN

SYSTEMS

where / , j £ D°(Z), and by dt is meant the Hamiltonian vector field (2.3.4) for a function / . Note that, by very definition of a Hamiltonian action, generators of a Hamiltonian action of a group G on a Poisson manifold (Z, w) are tangent to the leaves of the symplectic foliation on Z, and there is a Hamiltonian action of G on every symplectic leaf in the sense of symplectic geometry. Definition 3.7.1, Propositions 3.7.2, 3.7.3, and Definition 3.7.4 are naturally ex­ tended to a Poisson action of a connected Lie group G on a Poisson manifold. Let us consider the case of an equivariant momentum mapping. PROPOSITION 3.7.8. [181] An equivariant momentum mapping J : Z —> 0* is a Poisson morphism if the coalgebra a* is provided with the Lie-Poisson structure (2.3.11). a There is the following reduction theorem. THEOREM 3.7.9. [181]. Let (Z,w) be a Poisson manifold endowed with a Hamilto­ nian action of a connected Lie group G and the equivariant momentum map J. Let q be a point of g* such that • q is a non-critical value for all the restrictions of J to the symplectic leaves FL of Z, i.e., the level set Zq = J~l(q) is a submanifold of Z and, for any symplectic leaf Ft, Fuj = ZqV\ FL is a submanifold of FL; • intersections F^ of the submanifold Zq with symplectic leaves of Z are clean, i.e., TFu, = TFL f~l TZq, and so are its intersections with the orbits of G in Z. Let a subgroup Gq C G be the stabilizer of the point q. Then intersections F^ make up a regular foliation on Z,, and its leaves are the orbits of the connected component of the unit in Gq for the action of Gq on Zq. Furthermore, if this foliation is defined by a submersion Zq —> P, the base P has a well-defined Poisson structure whose characteristic distribution is the projection of TC\TZq, where T is the characteristic distribution on (Z, w). This is exactly the reduced Poisson manifold of {Z,w) via (Z„ E = T(orbits G)) (see Definition 2.6.3). a Note that if the momentum mapping J of a Hamiltonian action on a Poisson manifold is not equivariant, we have the relation J(gz)=Ad'g(J(z))+<j(g,z),

(3.7.9)

3.8. APPENDIX.

HAMILTONIAN

FIELD

THEORY

139

similar to (3.7.4) in the symplectic case, where the function a(g, z) is constant on the leaves of the symplectic foliation on Z. Since now a(g, z) depends on z, we have different stabilizers of q for the action given by (3.7.9) on g* for every symplectic leaf. Therefore, the foliation on Zq is no longer regular. We end this Section with the theorem which shows the possibility of reducing a Poisson Hamiltonian system possessing symmetries. T H E O R E M 3.7.10. [129, 181]. Let (Z, N, E) be a Poisson reduction and P a reduced manifold of Z via (N, E). Let H be a Hamiltonian on Z such that the flow $ t of its Hamiltonian vector field dH preserves the submanifold N and the bundle E, and dH is an annihilator of E. Then $ , induces a flow $ £ of Poisson automorphisms of P (see Proposition 2.6.5) along the Hamiltonian vector field dq for the Hamiltonian H on P, defined uniquely by the relation H \N= n'H. Moreover, we have ■dq o 7r = TIT O T3H.

a Thus, one can say that, under the hypotheses of Proposition 3.7.10, the Poisson Hamiltonian system on a Poisson manifold Z reduces to a Poisson system on the reduced Poisson manifold P. Example 3.7.5. Let (Z, Zq, E = T(orbits G)) be a Poisson reduction as in Theorem 3.7.9. If H is a G-invariant Hamiltonian on the Poisson manifold Z, the Poisson Hamiltonian system reduces to a Poisson system on the reduced Poisson manifold of (Z, w) via (Z„ E = T(orbits G)). •

3.8

Appendix. Hamiltonian field theory

As is well-known, applied to a field theory on a fibre bundle Y -+ X, the famil­ iar symplectic techniques take the form of instantaneous Hamiltonian formalism on an infinite-dimensional momentum phase space [64], in contrast with the finitedimensional velocity phase space JlY. The Hamiltonian counterpart of the Lagrangian formulation of field theory is polysymplectic Hamiltonian formalism on the Legendre bundle n (2.9.7) provided with the polysymplectic form A (2.9.9) (see [57, 159] for a survey). Here we aim to give a brief exposition of this formalism

CHAPTER

140

3. HAMILTONIAN

SYSTEMS

because its restriction to fibre bundles Q -> K leads in a straightforward manner to the Hamiltonian formulation of time-dependent mechanics [57, 161). Let Y —> X be a fibre bundle over an n-dimensional manifold X, and n = VY

(3.8.1)

ACA'T'X)

the Legendre bundle (2.9.7). It admits the composite fibration (3.8.2)

T n X = 7T O 1TUY ■ fl —► V —> X.

For the sake of convenience, we will call n —> Y the Legendre the term Legendre bundle stands for the fibration 17 —» X. Given fibred coordinates ( x \ y') on the fibre bundle Y —» X, (3.8.1) is equipped with the holonomic coordinates (xx,y',p*) transition functions (2.9.8). These coordinates are compatible fibration (3.8.2) and are linear bundle coordinates on the vector will call them the canonical coordinates.

vector bundle, while the Legendre bundle together with the with the composite bundle U —> Y. We

There is the canonical bundle monomorphism over Y G : IT

Y

-^n/\T'Y®TX,

(3.8.3)

Y

9 : ( x \ y\p$) ~ -pfdy' Acj®dx, which defines the tangent-valued Liouville form on the Legendre bundle IT Then the polysymplectic form A = dp,A A dy{ A u dx

(3.8.4)

on II is defined as a unique TX-valued (n + 2)-form on II such that the relation A\cf> =-d(G]<j>)

holds for any exterior 1-form 4> on X. Given the vector Legendre bundle II over a fibre bundle Y —» X, we have the exact sequence o —>UXAT"X x

^-> zY

—>n

—>o,

(3.8.5)

where Zy is the homogeneous Legendre bundle (2.9.3) and Tzn : ZY —» II

(3.8.6)

3.8. APPENDIX.

HAMILTONIAN

FIELD

THEORY

141

is an affine bundle over II with 1-dimensional fibres. DEFINITION 3.8.1. Let h be a section of the fibre bundle (3.8.6). Then the pull-back H = h*Z : II-> x

AT*Y,

H = p dy* AUJ\-

(3.8.7)

Hu,

of the canonical form B (2.9.4) on Zy by h is called the polysymplectic form (or simply the Hamiltonian form). □

Hamiltonian

The exterior differential of the Hamiltonian form (3.8.7) is the pull-back dH = h'Qz on II of the multisymplectic form (2.9.5). E x a m p l e 3.8.1. Let r = dxx <8> (dx + T\di) be a connection on the fibre bundle Y —> X. Hence, we have the splitting T : V'Y -» T'Y, T.dtf^

dyl -

T\dxx,

of the exact sequence (1.1.8a). Then T also yields the splitting hr-. n -» ZY, hr : pfdy' ® wA <-> pXdy{ A wx -

pXT\u),

of the exact sequence (3.8.5). It follows that every connection T on the fibre bundle Y —» X defines the Hamiltonian form Hr = h'rE, Hr = pxdy{ ALJX-

p,AI>,

(3.8.8)

on the Legendre bundle n . • PROPOSITION 3.8.2. [57, 159]. Hamiltonian forms on n constitute an affine space modelled over the linear space of horizontal densities H = Hw:Yl^

AT'X

(3.8.9)

142

CHAPTER 3. HAMILTONIAN

SYSTEMS

on the Legendre bundle II —► X. They axe called Hamiltonian densities. □ This means that, if H is a Hamiltonian form and H is a horizontal density (3.8.9), then H - H is also a Hamiltonian form. Conversely, if H and H' are Hamiltonian forms, their difference H - H' is a Hamiltonian density (3.8.9). Example 3.8.1 and Proposition 3.8.2 combine into the following. COROLLARY 3.8.3. Every Hamiltonian form on the Legendre bundle n admits the decomposition Hr= pxdyi A w A - PXT\UJ - Hru>,

H = Hr -

where T is a connection on Y —» X.

(3.8.10)

O

Furthermore, every Hamiltonian form admits a canonical decomposition as fol­ lows. We mean by a Hamiltonian map any fibred morphism

$ : II -> JlY, Y

y[ o $ = &x{q),


(3.8.11)

over Y. Its composition with the canonical morphism A (1.3.4) yields the bundle morphism Ao$:n

Y

^T'X®TY Y

represented by the TY-valued 1-form * = dxx 9 (dx +

$\(q)dt)

(3.8.12)

on the vector Legendre bundle n —> Y. E x a m p l e 3.8.2. Let T be a connection on Y —» X. Then, the composition

f = ro7r ny : n — y -> J1Y, f = dxx ® (dx + rA6ts), is a Hamiltonian map. Conversely, every Hamiltonian map $ : n —» JlY associated connection T* = $ o 0

yields the

3.8. APPENDIX.

HAMILTONIAN

FIELD

THEORY

143

on Y —» X, where 6 is the global zero section of the vector Legendre bundle II —» Y. In particular, we have

r~ = r 1

•

r

1

•

PROPOSITION 3.8.4. [57, 159] Every Hamiltonian form H on the Legendre bundle II defines the associated Hamiltonian map H : U -» J 1 ^ y\oH

= d{H.

(3.8.13)

D

COROLLARY 3.8.5. Every Hamiltonian form H on the Legendre bundle n deter­ mines the associated connection TH = HoO

on Y -> X.

a

In particular, we have T//J. — r , where HY is the Hamiltonian form (3.8.8) associated with the connection T on Y -> X. COROLLARY 3.8.6. Every Hamiltonian form (3.8.10) admits the canonical splitting H = HYH — H.

a The following assertion generalizes Example 3.8.1. PROPOSITION 3.8.7. Every Hamiltonian map (3.8.11) represented by the form (3.8.12) on n defines the associated Hamiltonian form H* = $ j e = pfdy* Aw»-p*$' A w,

144

CHAPTER 3. HAMILTONIAN

SYSTEMS

where 0 is the tangent-valued Liouville form (3.8.3). □ In particular, if HS = H, then H = Hr for some connection r on Y. Hamilton equations in conservative mechanics are the equations of integral curves of Hamiltonian vector fields. Hamilton equations in polysymplectic Hamiltonian for­ malism are the equations of integral sections for Hamiltonian connections as follows. Let J ' n be the first order jet manifold of the Legendre bundle II —> X. It is equipped with the adapted fibred coordinates (x A , 3 / , ,p, A ,^,pV).

We have the commutative diagram j^J'j^Xjly

I

I

n

52. Y

y]t ° J^ar = vlDEFINITION 3.8.8. A connection

y = dx*®(dx + y\d, + rtidl) on the Legendre bundle n —► X is said to be a locally Hamiltonian connection if the exterior form 7J

A = drf A dyl Aux-

( 7 |dp A - y^dy') A u

(3.8.14)

is closed. D Example 3.8.3. Every connection T on a fibre bundle Y —> X gives rise to the connection

f = dxx® [dx + rx(y)di + (-d}rxtf + KX\P] -

(3.8.15)

Kx\f})^\

3.8. APPENDIX.

HAMILTONIAN

FIELD

THEORY

145

on the Legendre bundle II —» X, where A" is a symmetric linear connection (1.4.9) on TX. The connection f (3.8.15) obeys the relation fjA = d ( r j 6 ) . It follows that T is a locally Hamiltonian connection. • Thus, locally Hamiltonian connections always exist on the Legendre bundle n —* X, and every connection r on Y —> X gives rise to a locally Hamiltonian connection on n -> X. It is easily observed that a connection 7 on the fibre bundle n —> X is a locally Hamiltonian connection if and only if 7 fulfills the conditions

dW, ~ dill = 0,

(3.8.16)

d . 7 ^ - %"!& = 0,

(3.8.17)

d/7i +

(3.8.18)

« = 0 .

Using the relation (3.8.18), we find that the second term in the right-hand side of the expression (3.8.14) is a closed form. Then, in accordance with the relative Poincare lemma, this expression is brought locally into the form 7JA = d(pxdy* A u j - Hyu) = dH7,

(3.8.19)

where H-, is a local function on n such that ll = W ,

7AA, =

-dtUr

Given a connection T on the fibre bundle Y —> X, the local form H^ in the expression (3.8.19) can be written as H1 = H r -

HVUJ,

where Hpui is a local horizontal density on n —» X. In accordance with Proposition 3.8.2, it follows that / / 7 defines the local section

h1:(xx,y\p?)~(xx,y\plp=-HJ of the fibre bundle ZY -» n, i.e., H^ is a local Hamiltonian form. Thus, we have proved the following.

CHAPTER 3. HAMILTONIAN

146

SYSTEMS

PROPOSITION 3.8.9. For every locally Hamiltonian connection 7 on the Legendre bundle n —► X, there exists a local Hamiltonian form H in a neighbourhood of each point q € fl such that 7JA = dH.

a Let us formulate the converse assertion. DEFINITION 3.8.10. The Hamilton operator £H for a Hamiltonian form H on the Legendre bundle n —» X is denned to be the first order differential operator EH

:J1n^nA1T'U,

£H = dH-A

= [(y\ - d\H)dp* - (pk + djTi)^]

A u,

(3.8.20)

where A = dpxx A dy' Au A + pxMdy' Au - y\dp* A u is the pull-back of the polysymplectic form A (3.8.4) on J ' n . D A glance at the expression (3.8.20) shows that the Hamilton operator EH is an afEne morphism over n of constant rank. Thereby, its kernel is an affine subbundle (3.8.21)

£H = Ker £H -* n of the affine jet bundle J ' n —> U which is given by the coordinate relations

v\ = d\n, Pi = -diH. This affine bundle is modelled over the vector subbundle of the vector bundle

r*x®vn->n

(3.8.22)

n which is defined by the coordinate relations

yl = o,

(3.8.23)

PA, = 0

with respect to the fibre coordinates (FA.PAI)

on

(3-8.22).

3.8. APPENDIX.

HAMILTONIAN

FIELD

THEORY

147

Since <£« (3.8.21) is an affine subbundle, it is a closed imbedded submanifold of the jet bundle JlU —> X and, therefore, is a system of first order differential equations on II —» X in accordance with Definition 1.3.3. DEFINITION 3.8.11. The first order differential equations € w (3.8.21) are called the Hamilton equations for the Hamiltonian form H on the Legendre bundle II. □ Since the subbundle
(3.8.24)

It follows that every connection on II —» X which takes its values into the Hamilton equations £ # is a locally Hamiltonian connection. DEFINITION 3.8.12. A locally Hamiltonian connection 7 on the Legendre bundle n —> X is said to be a Hamiltonian connection associated with a Hamiltonian form H if 7 obeys the relation (3.8.24). □ Thus, we have proved the following. PROPOSITION 3.8.13. Every Hamiltonian form on the Legendre bundle n has an associated Hamiltonian connection. D We have the equations of a Hamiltonian connection associated with a given Hamiltonian form:

7i = d\H, A

7A i =

- m

(3.8.25) (3.8.26)

By the equation (3.8.25), every Hamiltonian connection 7 for a Hamiltonian form H satisfies the relation Jlirny 0 7 = / / , 7 = dx

x

® (dx + diHdi + T&S*),

(3.8.27)

CHAPTER

148

3. HAMILTONIAN

SYSTEMS

where H is the Hamiltonian map (3.8.13). It projects over the connection YH on Y —> X associated with the Hamiltonian form H. We have the following commuta­ tive diagram

j'n J'*nr

n ■£-> J1Y

o \ / * r„ Y A glance at the equations (3.8.26) shows that there is a set of Hamiltonian connections associated with the same Hamiltonian form H. They differ from each other in soldering forms a on n —» X which obey the equations

5JQ = 0, 31 = 0,

si = o,

and take their values into the constraint subbundle (3.8.23) of the vector bundle (3.8.22). A classical solution of the Hamilton equations (3.8.21), by definition, is a section r of the Legendre bundle n —» X such that its jet prolongation J1r takes its values into the kernel of the Hamilton operator ZH (3.8.20). Then, r satisfies the differential equations

8^ = diU or,

(3.8.28a)

aAr,A = -diH

(3.8.28b)

o r.

Every integral section Jlr = 7 o r of a Hamiltonian connection 7 associated with a Hamiltonian form H is a classical solution of the corresponding Hamilton equations. Conversely, if r is a global solution of the Hamilton equations (3.8.28a) - (3.8.28b) for a Hamiltonian form H, there exists an extension of this solution

Jh-.riX) — j ' n to a Hamiltonian connection which has r as an integral section. Substituting Jlr in (3.8.27), we obtain the identity J'(7Tny or) = H or

3.8. APPENDIX.

HAMILTONIAN

FIELD

THEORY

149

for every classical solution r of the Hamilton equations (3.8.28a - (3.8.28b). Remark 3.8.4. Note that the Hamilton equations (3.8.28a) - (3.8.28b) can be introduced without appealing to the Hamilton operator. They are equivalent to the relation r'{u\dH)

=Q

which is assumed to hold for any vertical vector field u on the Legendre bundle

n-> X. •

Chapter 4 Lagrangian time-dependent mechanics This Chapter is devoted to the formulation of time-dependent mechanics on a phase space of coordinates and velocities. This phase space is the first order jet manifold JlQ of a configuration bundle Q —► R. The familiar case of the direct product Q = R x M corresponds to the choice of a certain reference frame. For the sake of brevity, the above-mentioned formulation is called Lagrangian time-dependent mechanics although equations of motion are not necessarily Lagrange equations. Connections provide the main ingredients in the Lagrangian formulation of timedependent mechanics. These are the reference frames seen as connections on a configuration bundle Q —» R, the dynamic equations represented by connections on the jet bundle JlQ —» R, the dynamic connections on the affine jet bundle JlQ —» Q, the connections on the tangent bundles TQ —> Q and TJlQ —» JlQ, and the Lagrangian connections. For instance, we show that every non-relativistic dynamic equation on a configuration space Q can be seen as a geodesic equation with respect to some connection on the tangent bundle TQ —> Q. Reference frames are necessarily involved when we deal with dynamic equations, forces, accelerations, and conservation laws in time-dependent mechanics. The mass metric is another interesting object of Lagrangian and Newtonian systems, of the inverse problem, and of systems with holonomic and non-holonomic constraints. Throughout this Chapter, TT:Q

(4.0.1)

—R

151

152

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

is a fibre bundle whose typical fibre M is an m-dimensional manifold, while its base R is parameterized by the Cartesian coordinates t possessing the transition functions t' = t+ const. In the universal unit system, the physical dimension of t is equal to [length]. A fibre bundle Q —► R is endowed with bundle coordinates {t,q'}. For the sake of convenience, we also use the compact notation qx where q° = t.

4.1

Fibre bundles over R

In this Section, we point out the most important peculiarities of fibre bundles over R, which we refer to in the sequel. 1. The base R of a fibre bundle (4.0.1) is provided with the standard vector field 3, and the standard 1-form dt which are invariant under the coordinate transformations f = t+ const. The same symbol dt also stands for any pull-back of the standard 1form dt onto a fibre bundle over R. Note on one-to-one correspondence between the vector fields fdt and the real functions / on R. The similar correspondence between the horizontal densities 4>dt and the real functions 4> on a, fibre bundle Q —^ R takes place. Roughly speaking, we may neglect the contribution of TTR and T*R to several expressions. At the same time, one should be careful with such simplification in the framework of the universal unit system. For instance, the coefficient 0 of a horizontal density <j>dt has the physical dimension [length] -1 , whereas a function (j> is physically dimensionless. 2. Since R is contractible, any fibre bundle over R is obviously trivial. Different trivializations rj) : Q = R x M

(4.1.1)

differ from each other in the fibrations Q —> M, while the fibration Q —* R is once for all. Let J ' Q be the first order jet manifold of a fibre bundle Q —» R (4.0.1). It is provided with the adapted coordinates (t,q',q't). Given a direct product Q = Rx M ->1

4.1. FIBRE BUNDLES OVER K

153

coordinated by (f.,if), there is the canonical isomorphism

Jl{RxM) —i

= RxTM,

(4.1.2)

—■

that one can justify by inspection of the transition functions of the coordinates q[ and q , when the transition functions of qx are independent of t. Due to the isomorphism (4.1.2), every trivialization (4.1.1) yields the corresponding trivialization of the jet manifold JlQ = 1 x TM.

(4.1.3)

3. Given the jet manifold JlQ of a fibre bundle Q —> R, the canonical imbedding (1.3.4) takes the form A:

J'Q^TQ,

X:(t,q\q't)^(t,q',i

(4.1.4) = l,q' = ql).

For brevity, we will write A = ck = dt + q\di,

(4.1.5)

where by dt is meant the total derivative. From now on, we will identify the jet manifold JlQ with its image in TQ. Remark 4.1.1. Following precisely the expression (1.3.4), we should write the morphism A (4.1.5) in the form A = dt ® (dt + qldi).

(4.1.6)

With respect to the universal unit system, the physical dimension of A (4.1.5) is [length]" 1 , while A (4.1.6) is dimensionless. • Using the morphism (4.1.4), one can define the contraction

JlQxT'Q Q

->QxR,

Q

(«|; t, 9.) •-» ^J fat + q,dq') = i + q\q„ where (i, g', t, q{) are the coordinates on the cotangent bundle T'Q.

(4.1.7)

154

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

4.

A glance at the expression (4.1.4) shows that the affine jet bundle J ' Q —> Q is modelled over the vertical tangent bundle VQ of the fibre bundle of Q —> R. As a consequence, we have the following canonical splitting (1.1.6) of the vertical tangent bundle VQ J ' Q of the affine jet bundle JXQ -» Q: a : VQJlQ Si JlQ x VQ,

(4.1.8)

"(flf) = ft, VqJlQ

together with the corresponding splitting of the vertical cotangent bundle of JlQ -» Q: a* : V
(4.1.9)

VQ,

i

a*(a 7 j)=3 9 ! where dq\ and dq' are the holonomic bases for VQ JlQ and V'Q, respectively. Then the exact sequence (1.6.9a) of vertical bundles over the composite fibre bundle JXQ -* Q - R

(4.1.10)

reads

I l

0 —*VQJ Q

""'

1

l

l

<^VJ Q

^*J QxVQ

—*0.

Hence, we obtain the following linear endomorphism over J1Q of the vertical tangent bundle V JlQ of the jet bundle JXQ — R:

v^ioa^ony

: VJ'Q^

v(dj) = dl

jjjgj = 0-

VJlQ,

(4.1.11)

This endomorphism obeys the nilpotency rule v o v = 0.

(4.1.12)

Combining the horizontal splitting (1.3.11), the corresponding projection pr 2 : JXQ

xTQ^ Q

dt ~ -q\d\,

JlQ

xVQ^ Q

dx -> d\,

VQJ'Q,

4.1. FIBRE BUNDLES OVER R

155

and the inclusion VJlQ «-» TJlQ, the tangent bundle TJlQ:

one can extend the endomorphism (4.1.11) to

v:TJlQ^TJlQ,

v(dt) = -qldj

vjdi) ■ g,

g(g) = 0.

(4.1.13)

This is called the verticaJ endomorphism. It inherits the nilpotency property (4.1.12). The transpose of the vertical endomorphism v (4.1.13) is v' : T'JlQ

-> T V ' Q ,

v'(dt) = 0,

v'(dq') = 0,

v*(dql) = 6\

(4.1.14)

where 9l = dqi - q\dt are the contact forms (1.3.6). The nilpotency rule v" o v* = 0 is also fulfilled. The homomorphisms v and v' are associated with the tangent-valued 1-form

v = ^ ® a? in accordance with the relations (1.2.36) - (1.2.37). With the endomorphism {?*, one can introduce the vertical exterior differential dy = v* o d,

acting on the algebra D'iJ^) of exterior forms on the jet manifold JlQ. l example, if / is a function on J Q, we have

For

d.f = %f#.

5. In view of the morphism A (4.1.4), any connection

r = dt ® (dt + ra<)

(4.1.15)

156

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

on a fibre bundle Q -» R can be identified with a nowhere vanishing horizontal vector field

T = dt + r<9,

(4.1.16)

on Q which is the horizontal lift rdt (1.4.7) of the standard vector field d, on R by means of the connection (4.1.15). Conversely, any vector field T on Q such that dt\T = 1 defines a connection on Q —► R. Of course, the integral curves c:RD()^Q, i(r) = l,

ci(T) = roc(r)<

re(),

of the vector field (4.1.16) coincide with the integral sections dtc\t)

=

(roc)'(t)

of the connection (4.1.15). Connections on a fibre bundle Q —* R constitute an affine space modelled over the vector space of vertical vector fields on Q —» R. Accordingly, the covariant differential (1.4.5) associated with a connection r on Q —> R takes its values into the vertical tangent bundle VQ of Q —> R: Dr: JlQ q'oDr

-+VQ,

(4.1.17)

= q\ - P .

A connection T on a fibre bundle Q —> R is obviously flat. It yields a horizontal distribution on Q. The integral manifolds of this distribution are integral curves of the vector field (4.1.16) which are transversal to the fibres of the fibre bundle Q~*R. COROLLARY 4.1.1. By virtue of Proposition 1.4.1, each connection T on a fibre bundle Q —» R defines an atlas of local constant trivializations of Q —» R such that the associated bundle coordinates (t,q') on Q possess the transition function q' —> q"(q3) independent of (, and

r = dt

(4.1.18)

with respect to these coordinates. Conversely, every atlas of local constant trivial­ izations of the fibre bundle Q —> R determines a connection on Q —> R which is equal to (4.1.18) relative to this atlas. D

4.1.

FIBRE BUNDLES OVER R

157

A connection T on a fibre bundle Q —» R is said to be complete if the horizontal vector field (4.1.16) is complete. PROPOSITION 4.1.2. Every trivialization of a fibre bundle Q - > R yields a complete connection on this fibre bundle. Conversely, every complete connection r on Q —» R defines its trivialization (4.1.1) such that the vector field (4.1.16) equals dt relative to the bundle coordinates associated with this trivialization. □ Proof. Every trivialization of Q —• R defines the horizontal lift of the standard vector field dt on R to the vector field r = dt on Q which is obviously complete. Then the corresponding connection on Q —» R is also complete. Conversely, let T be a complete connection on a fibre bundle Q —> R. The horizontal vector field T (4.1.16) is the generator of a 1-parameter group Gr acting freely on Q. The orbits of this action are integral curves of the vector field F. Hence, we obtain a projection (4.1.19)

Q -» Q/Gr = M

of Q along the above-mentioned integral curves onto some fibre of Q, e.g., Qt=o — M. Combining the projection (4.1.19) and the projection Q —> R gives a desired QED trivialization (4.1.1) of Q. QED 6.

Let J1J1Q be the repeated jet manifold of a fibre bundle Q —> R, provided with the adapted coordinates

(t,q\qlq\t),qlt)For a fibre bundle Q —* R, we have the canonical isomorphism k between the affine fibrations 7rn (1.3.13) and Jln\ (1.3.15) of JXJ[Q over JlQ, i.e., 7r n ok

= J07T01,

kok^ld^J'Q,

where q't0k = q\t),

q\t) °k = q\,

qtt °k = q'u-

(4.1.20)

In particular, the affine bundle 7ru (1.3.13) is modelled over the vertical tangent bundle V JlQ of JlQ -» R which is canonically isomorphic to the underlying vector bundle JlVQ — JlQ of the affine bundle J 1 ^ (1.3.15).

158

4. LAGRANGIAN

CHAPTER

TIME-DEPENDENT

MECHANICS

By JQJ1Q throughout is meant the first order jet manifold of the affine jet bundle J Q —► Q. The adapted coordinates on JQJlQ are ( g \ q\, q\t). For a fibre bundle Q -> 1 , the sesquiholonomic jet manifold PQ coincides with the second order jet manifold J2Q, coordinated by l

(<.«*. «{,««)■

The affine bundle J2Q —» J ' Q is modelled over the vertical tangent bundle (4.1.21)

VQ J ' Q £ ^ Q x VQ -» J ' Q of the affine jet bundle JXQ —► Q. There are the imbeddings J2Q

^TV'Q

"

* T2Q

VQTQ

C

TTQ,

(4.1.22) >2 : (t, ?', «{, qtt) >-» (*, ?', $> * = 1,9* = ?l = ?«), TA o A2 : (t, q\ q\, g't) w (t,
(i,g\i,g',t,d/,i', R is represented by a horizontal vector field on JlQ such that £\dt = 1. Example 4.1.2. A connection on the jet bundle J*Q —» R is defined as a section of the affine bundle irn (1-3.13). A connection T (4.1.16) on a fibre bundle Q —» R has the jet prolongation to the section J1F of the affine bundle J ' T Q . By virtue of the isomorphism k (4.1.20), every connection r on Q —» R gives rise to the connection JTd=ko

J T : JlQ^

JT = dt + P d , +

J'J'Q, dtrdl

(4.1.24)

on the jet bundle J^Q -* R. • A connection on the jet bundle J1Q —» R is said to be holonomic if it is a section

£ = 4®(A + flfo+fflf)

4.2. DYNAMIC

EQUATIONS

159

of the holonomic subbundle J2Q —> JlQ of J1 JlQ -* JlQ. In view of the morphism (4.1.22), a holonomic connection is represented by a horizontal vector field

(, = a, + q\d{ + g g

(4.1.25) (4.1.25) l

on J ' Q . Conversely, every vector field f on J Q which fulfills the conditions dt\t. = 1,

v(0 = 0,

where v is the vertical endomorphism (4.1.13), is a holonomic connection on the jet bundle JlQ -> R. Holonomic connections (4.1.25) make up an affine space modelled over the linear space of vertical vector fields on the affine jet bundle JlQ —► Q, i.e., which live in VQJ'Q.

A holonomic connection £ defines the corresponding covariant differential (4.1.17) on the jet manifold ^Q: Ds : J ' J ' Q —^VQJlQ q'oDi

=

0,

C

VJlQ,

$ o D( = q\t -

f,

which takes its values into the vertical tangent bundle VQ.J1Q of the jet bundle JlQ —y Q. Then, by virtue of Proposition 1.3.1, any integral section c : () —» JlQ for a holonomic connection £ is holonomic, i.e., c = c where c is a curve in Q.

4.2

Dynamic equations

We start our exposition of Lagrangian time-dependent mechanics from the notion of a second order dynamic equation on a configuration bundle Q —» R. Recall that a dynamic equation, by definition, is a differential equation which can be algebraically solved for the highest order derivatives. D E F I N I T I O N 4 . 2 . 1 . Let

r = dt + F'di be a connection on a fibre bundle Q —> M. The corresponding covariant differential DT (4.1.17) is a first order differential operator on Q. Its kernel, given by the coordinate relations fl* = !*(*,«').

(4.2.1)

160

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

is a closed subbundle of the jet bundle JlQ -» R. Therefore, in accordance with Definition 1.3.3, this is a first order differential equation on the fibre bundle Q —> R, called the first order dynamic equation on Q —► R. □ Solutions of the first order dynamic equation (4.2.1) are integral sections (geodesic curves) for the connection I\ DEFINITION 4.2.2. Let us consider the first order dynamic equation (4.2.1) on the jet bundle J1Q —► R, which is associated with a holonomic connection f (4.1.25) on J ' Q —► R. This is a closed subbundle of the second order jet bundle J2Q —» R, given by the coordinate relations

& = ?«,,
(4.2.2)

Consequently, it is a second order differential equation on the fibre bundle Q —► R in accordance with Definition 1.3.3. This equation is called a second order dynamic equation, or simply a dynamic equation, if there is no danger of confusion. The corresponding horizontal vector field f (4.1.25) is also termed a dynamic equation.

a A solution of the dynamic equation (4.2.2), called a motion, is a curve c in Q whose second order jet prolongation c lives in (4.2.2). It is clear that any integral section c for the holonomic connection £ is the jet prolongation c of a solution c of the dynamic equation (4.2.2), i.e.,

c' = e ° c,

(4.2.3)

and vice versa. Remark 4.2.1. By very definition, the second order dynamic equation (4.2.2) on a fibre bundle Q —> R is equivalent to the system of first order differential equations 9(0 = 9t.

Qlt =

C(t,qj,ql),

(4.2.4)

on the jet bundle JlQ —> R. Any solution c of these equations takes its values into J2Q and, by virtue of Proposition 1.3.1, is holonomic, i.e., c = c. Therefore, the equations (4.2.2) and (4.2.4) are equivalent. The equation (4.2.4) is said to be the first order reduction of the second order dynamic equation (4.2.2) . •

4.2. DYNAMIC

161

EQUATIONS

One can easily find the transformation law of a second order dynamic equation under bundle coordinate transformations ql —> q'^t^g3). This transformation law reads

<& = ?,

e = (?dj + gjgfodk + 24djdt + 9?W% ?)■

(4.2.5)

Example 4.2.2. From the physical viewpoint, the most interesting dynamic equa­ tions are the quadratic dynamic equations, i.e., those which are polynomials in the coordinates q\ of degree < 2. This property is global due to the transformation law (4.2.5). The dynamic equation of a relativistic particle exemplifies a non-polynomial dynamic equation. • A dynamic equation ( o n a fibre bundle Q —» K is said to be conservative if there exists a trivialization (4.1.1) of Q whose corresponding trivialization (4.1.3) of J*Q is such that the vector field f (4.1.25) on JlQ is projectable over M. Then this projection H f = <j,'3 + e(
(4.2.6)

on the fibre bundle K x M —> K in accordance with the isomorphism (4.1.3). We conclude this Section with the following theorem. T H E O R E M 4.2.3. Any dynamic equation on a fibre bundle Q —► K is equivalent to an autonomous second order dynamic equation on a manifold Q. □ Proof. Given a dynamic equation £ on a fibre bundle Q —> R, let us consider the diagram J*Q

'I JIQ

-^T2Q

Is

(4.2.7)

J u TQ

where 5 is a holonomic vector field on the tangent bundle TQ, and we use the morphism (4.1.23). A glance at the expression (4.1.23) shows that the diagram

162

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

(4.2.7) can be commutative only if the component 5° of a vector field S vanishes. Since the transition functions t —> t' are independent of g\ such a vector field may exist on TQ. Now, the diagram (4.2.7) becomes commutative if the dynamic equation f and a vector field E fulfill the relation C = S%q>,i

= l,q>=q>).

(4.2.8)

It is easily seen that this relation holds globally because the substitution of ql = q\ in the transformation law of a vector field H restates the transformation law (4.2.5) of the holonomic connection £. In accordance with the relation (4.2.8), the desired vector field S is an extension of the section TXo\2o£ of the fibre bundle T*Q —> TQ over the closed submanifold JlQ C TQ to a global section. Such an extension always exists by virtue of Theorem 1.1.2, but is not unique. Then the dynamic equation (4.2.2) can be written in the form Itt -

-

(4.2.9)

I(=1I^=?J •

This is equivalent to the autonomous second order dynamic equation t = Q,

f = l,

? = S*',

(4.2.10)

on Q. Being a solution of the equation (4.2.10), a curve c in Q also satisfies the QED equation (4.2.9), and vice versa. It should be emphasized that, written in the bundle coordinates (£, q'), the second order equation (4.2.10) is well defined with respect to any coordinates on Q.

4.3

Dynamic connections

In order to say more than was said in Theorem 4.2.3, we will consider the relationship between the holonomic connections on the jet bundle JlQ —> R and the connections on the affine jet bundle JlQ —> Q (see Propositions 4.3.1 and 4.3.2 below). Let 7 : J'Q -» JlQJlQ be a connection on the affine jet bundle J*Q —» Q. It takes the coordinate form

7 =
(4.3.1)

4.3. DYNAMIC CONNECTIONS

163

with the transformation law

Tj = (VTi + *tf)f£.

(4.3.2)

Remark 4.3.1. In view of the canonical splitting (4.1.8), the curvature (1.4.8) of a connection 7 (4.3.1) reads R: ^Q-^AT'Q R = 2^dqX

® VQ,

JlQ

A dq» ® 5, = {h?kidt}k A dcf + R^dt A dq>) ® du

R\» = 9A7" - d^\ + 7ia j 7 ; -

yid^.

(4.3.3)

Using the contraction (4.1.7), we obtain the soldering form AJ.R = [(«*,«? + R^)dql - R^tfdt] ® a, on the affine jet bundle JlQ — ► Q. Its image by the canonical projection T'Q —► V'Q is the tensor field R:J1Q-*V*Q®VQ, Q

R=(Ry^ + RlJW^di,

(4.3.4)

and then we come to the scalar field R:JlQ^

R, (4.3.5)

R = R-liit + Ron on the jet manifold JlQ. •

4.3.1. Any connection 7 (4.3.1) on the affine jet bundle JlQ -* Q defines the holonomic connection PROPOSITION

£7 = p o 7 : JlQ^

J^JXQ^J2Q,

e, = ^ + q ^ + (7i + qf7j)af. on the jet bundle JlQ -> R. D

(4.3.6)

164

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

Proof. Let us consider the composite fibre bundle (4.1.10) and the morphism p (1.6.6) which reads l l J^Q (q\ q\, - . (q\ql,q\ (q\ q\, q\ = q\,qlt q\, £ = = 4^ + + 4
(4.3.7) (4-3-7)

A connection 7 (4.3.1) and the morphism p (4.3.7) combine into a desired holonomic connection f7 (4.3.6) on the jet bundle JlQ -> R. QED It follows that every connection 7 (4.3.1) on the affine jet bundle J1Q —» Q yields the dynamic equation (4.3.8)

Qu = 7o +
on the configuration bundle Q —> R. This is precisely the restriction to J2Q of the kernel KerD^ of the vertical covariant differential D 7 (1.6.11) defined by the connection 7: Dy : J'J'Q

->

VQJ'Q,

(4.3.9)

q't°D-, = q'tt - 7o -
Therefore, connections on the jet bundle JlQ —> <5 are called the dynamic connec­ tions. The corresponding equation (4.2.3) can be written in the form C1 = p o 7 o c,

where p is the morphism (4.3.7). Of course, different dynamic connections can lead to the same dynamic equation (4.3.8). PROPOSITION 4.3.2. Any holonomic connection £ (4.1.25) on the jet bundle J ' Q —* R defines the dynamic connection

7j = dt ® \dt + (g - yt^)dj]

+ dq>® [d, + jgjgg]

(4.3.10)

on the affine jet bundle JlQ —* Q. □ Proof. Let £ be a holonomic vector field (4.1.25). Given an arbitrary vertical vector field

u = a% + Vd',

4.3. DYNAMIC

CONNECTIONS

165

on the jet bundle J1Q —> R, let us put

h(u)d=[SMu)]-mM), It(u) = -a«a t + (b{ -

aiq&di

where v is the vertical endomorphism (4.1.13). Then, there is the endomorphism

h-VJlQ

^VJ'Q,

h ■ 4% + 9 $ - . -q% + (q\ - cfd]V)d\. It is readily observed that this endomorphism obeys the condition

h°h = hThen, one can construct the projection

J ^ ^

:VJlQ^QVQJlQ,

+ ldVJ'Q)

J< ■?% + &!

"(il-l?%?)%■

Recall that a holonomic connection £ on JlQ —> R defines the projection (1.4.3) which reads £ : TJlQ

3 tdt + q% + q\d\ -> (ql - %)% + (41 ~ i?)$

€

VJlQ.

Let us consider the composition of morphisms

/ ? o £ : TJlQ -» VJ'Q - VQJlQ, idt + q% + gjd? ~

[Q\-m~\4^)-\^d]m-

This corresponds to the connection 7 5 (4.3.10) on the affine jet bundle JlQ —> Q. QED

It is readily observed that the dynamic connection 7^ (4.3.10), denned by a dynamic equation, possesses the property

7* = grj + gjgj which implies the relation

atf = af-tf.

(4.3.11)

166

CHAPTER 4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

Therefore, a dynamic connection 7, obeying the condition (4.3.11), is said to be symmetric. The torsion of a dynamic connection 7 is defined as the tensor field

T:JlQ^V'Q®VQ,

Q

T = J*3j* ® dk, (4.3.12)

Tt = 7* - aho* - d"#tf ■

It follows at once that a dynamic connection is symmetric if and only if its torsion vanishes. Let 7 be a dynamic connection (4.3.1) and £7 the corresponding dynamic equa­ tion (4.3.6). Then the dynamic connection (4.3.10) associated with the dynamic equation £7 takes the form

^! = 5(7? + ahJ + «&•#),

7«,S = £" - ftSc?-

It is readily observed that 7 = 7* if and only if the torsion T (4.3.12) of the dynamic connection 7 vanishes. E x a m p l e 4.3.2. connection

Since the jet bundle J1Q —> Q is affine, it admits an affine

77=dq = dqxx ® ® [9, [d,+ +( 7(y7 yOo ++ 77yyOot
(4.3.13) (4-3.13)

This connection is symmetric if and only if 7^ = f'x. One can easily justify that an affine dynamic connection generates a quadratic dynamic equation, and vice versa. Nevertheless, a non-affine dynamic connection, whose symmetric part is affine, also define a quadratic dynamic equation. An affine connection (4.3.13) on the affine jet bundle J1Q —* Q yields the linear connection

7= = dqX®[d dqX®[d X+lU R. The latter yields the dynamic connection 7 (4.3.10) on the fibre bundle R x M. R x TM - »~*RxM.

4.3.

DYNAMIC

CONNECTIONS

167

Its components 7! are exactly those of the connection (3.1.7) on the tangent bundle TM —> M in Theorem 3.1.4, while 7J make up a vertical vector field

e = T $ = (H1 - i^S^ja,

(4.3.14)

on TM —» M. Thus, we have shown the following. PROPOSITION 4.3.3. Every autonomous second order dynamic equation 5 (3.1.4) on a manifold M admits the decomposition

H1 = Kjq* + e{ where K is the connection (3.1.7) on the tangent bundle TM —» M, and e is the vertical vector field (4.3.14) on TM -» M. □ Remark 4.3.3. With a dynamic connection 75 (4.3.10), we also restate the wellknown linear connection on the tangent bundle TJXQ —> J ' Q , associated with a dynamic equation £ on Q [132], and can write it with respect to holonomic bases. Given a holonomic connection f (4.1.25), we have the corresponding horizontal splitting (1.4.1) of the tangent bundle TJlQ of JlQ: TJlQ

= Hc®

JlQ

VJlQ,

(4.3.15)

tit + q% + $% = tf + (
(4.3.16)

and the corresponding horizontal splitting (1.4.2) of the cotangent bundle T'JlQ

of

T"J1Q = R ® W ' Q ,

tdt + qM + #M = (t + 9t9i + {'$<« +
(4.3.17)

Using these spUttings, one can introduce the non-holonomic bases (£, d„ d\) for the tangent bundle TJXQ, and the non-holonomic bases

(dt,6',dql-Cdt) for the cotangent bundle T'^Q

of JlQ.

168

CHAPTER 4. LAGRANGIAN TIME-DEPENDENT MECHANICS

Furthermore, the dynamic connection 7{ (4.3.10), associated with the dynamic equation £, provides the corresponding splitting (1.6.10a) of the vertical tangent bundle VJlQ of JlQ: VJlQ = VQJlQ © //-,., q% + q\d\ = (q\ - i^fljf 1 )^ + ?'(a, + \&£d[).

(4.3.18)

The splittings (4.3.16) and (4.3.18) combine into the splitting TJ'Q = H,@ Hlt ® VQJXQ

(4.3.19)

together with the non-holonomic bases (4.3.20)

(£, h, = d} + \d)Cdl a?)

for the tangent bundle TJlQ [55]. Note that the fibre bundle H^ in the splitting (4.3.19), like V Q J ' Q , is isomorphic to the pull-back Hlt = JlQ xVQ^

J'Q,

h, <—> dr

(4.3.21)

Accordingly, the dynamic connection 7{ (4.3.10) determines the corresponding splitting (1.6.10b) of the vertical cotangent bundle V'JlQ of JlQ. Combining this splitting with the decomposition (4.3.17) gives the splitting T'JlQ = R ® V'Q ® VAJlQ JlQ

Jl Q

together with the non-holonomic bases

{dt,e\dq\-^dqx) for the cotangent bundle T'JlQ of JlQ. Every connection 7 (4.3.1) on the affine jet bundle JlQ —* Q gives rise to the connection V7 (1.6.12) on the composite vertical tangent bundle VQJlQ - JlQ -» Q. This connection reads Vy = dqx® (dx + 7iaf + ^7i«J'af).

4.3. DYNAMIC

169

CONNECTIONS

Since JlQ —» Q is an affine bundle, the connection Vj, in turn, can be seen as the composite connection (1.6.8) generated by the connection 7 on the affine jet bundle J1Q —> Q, and the linear connection (1.6.14): x k k 7 = dq dq* ®®(d(d d)i\ct dftqid!) dqdq ® dl ® dl x x++ td\) + + 1

(4.3.22) (4.3.22) l

on the vertical tangent bundle VQJ Q —> J Q. Due to the splitting (4.1.8), the connection 7 (4.3.22) in fact is a linear connection (4.3.23) (4.3.23)

x d'-yitfdi) + dqkt ® 6dk 7 = dq dc? ® (dx + d'jxtfdi)

on the pull-back (4.3.21). In particular, if 7 = 75 is the connection (4.3.10), the connection 7^ (4.3.23) yields a connection on the pull-back bundle H-,( —> JlQ in the splitting (4.3.19). Due to the splitting (4.3.19), the tangent bundle TJlQ —* JlQ can be provided with a linear connection A [132] which is the direct sum of • the trivial connection on the linear bundle H^ —> J1Q, • the connection 7^ (4.3.23) on the pull-back bundle H1( —> J ' Q , • and the similar connection (4.3.22) on the vertical tangent bundle VQJ1Q —> JlQ. With respect to the non-holonomic basis (4.3.20), where (4.3.24) (4.3.24)

idt + cfdi + q\d\ = it, + u% + v*dl the above-mentioned linear connection A reads x x ®(d + d^W-^ A =dqdq A = ® (dx x + d ; 7 > J ^

d W ++ f^J^\

^ )

+

d a + ^ 4® ® '■dl

((4.3.25) 4325)

Recall the expression (4.3.10) for the components 7^ of the dynamic connection 7 ? . Substituting qi = iq\+ui,

i = tief + + UkufkkY+ q\ = v* k+v

from (4.3.24) in (4.3.25), we obtain the linear connection A written with respect to the holonomic bases (dx,df) for TJlQ: x

A= A = dq dqx®® (d (dxx + + AAx) +dq\® dq\®(dl(dl+ +A\), A\), x) + Ax + - 4l\^Akfcx + H + <&~&drf Ax = = i[-4dn\d, i[-4dn\d, + (dxC (dxC -- <£drf rf^j "k <7?7^7 ^ \ ~ S*af7i)4f] x - ^ D+S f ] + fld^A + fldrfA + (d (dxlxl)) + + 77 & &7* 7 * -- 7^*7l)9|] I'daDdW ++ # fariM, 9,7^, k A\ = ifa + (d*? - q?dH)dt] + q>dh di.

170

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

The covariant derivatives V relative to the connection A fulfill the conditions VA(0

(4.3.26)

= VKO = o,

V,(5j) = -dtfA, v|(5j) = V|(A,) = o,

VA(^) = - C S T J ^ ,

v*(a3) = -flftflj.

In particular, by virtue of the equality (4.3.26), every solution c of the dynamic equation £ is also a geodesic of the linear connection A. •

4.4

Non-relativistic geodesic equations

In this bundle We bundle

Section, we aim to show that every dynamic equation on a configuration Q —> M is equivalent to a geodesic equation on the tangent bundle TQ —► Q. start from the relation between the dynamic connections 7 on the affine jet JlQ —* Q and the connections

K = dqx® (dx + Kfa)

(4.4.1)

on the tangent bundle TQ —» Q of the configuration space Q. We will use the notation (1.2.4). Let us consider the diagram JlQJlQ

J

, I l

JQ

-±JlQTQ

I .

(4.4.2)

-±-> TQ

where JQTQ is the first order jet manifold of the tangent bundle TQ —» Q, coordi­ nated by

(t,q\i,q\(i)^

(q%).

The jet prolongation over Q of the canonical imbedding A (4.1.4) reads

•/'A : (t,q\ql,qU) - (t,q',i = l,«f = g|, (t)„ = 0,(q% = <£,). Then we have

JJA o 7 ; (t, g', gj) ,_ (<, g\ t = l,g« = q', (*)„ = 0, (g% = -£), X o A : (t, q\ ql) » (t, e?\ « = ! , ? = flj, (t)„ = Kj, («% = /if*).

4.4. NON-RELATIVISTIC

GEODESIC EQUATIONS

171

It follows that the diagram (4.4.2) can be commutative only if the components K° of the connection K (4.4.1) on the tangent bundle TQ —> Q vanish. Since the transition functions t —* t* are independent of q', a connection K = dqx®® (d (3xA + + iq«3i) K\di)

(4.4.3)

with A"° = 0 may exist on the tangent bundle TQ —♦ Q in accordance with the transformation law K"x = (dJq«Ki + d»q*)^-x. K'i-idjfKi + drfl^-y

(4.4.4)

Now the diagram (4.4.2) becomes commutative if the connections 7 and K fulfill the relation ^ = K^o X = Kl(t,q',t

= l,q' = q^).\

(4.4.5)

It is easily seen that this relation holds globally because the substitution of ql = qlt in (4.4.4) restates the transformation law (4.3.2) of a connection on the affine jet bundle JlQ —» Q. In accordance with the relation (4.4.5), the desired connection K is an extension of the section J'A o 7 of the affine jet bundle JQTQ —* TQ over the closed submanifold JlQ C TQ to a global section. Such an extension always exists by virtue of Theorem 1.1.2, but is not unique. Thus, we have proved the following. PROPOSITION 4.4.1. In accordance with the relation (4.4.5), every dynamic equa­ tion on a configuration bundle Q —> R can be written in the form qlt = K'0°\

+ q{K>o\,\

(4.4.6)

where 7f is a connection (4.4.3) on the tangent bundle TQ —» Q. Conversely, each connection K (4.4.3) on TQ —> Q defines the dynamic connection 7 (4.4.5) on the affine jet bundle JXQ —> Q and the dynamic equation (4.4.6) on a configuration bundle Q -» R. □ Then we come to the following theorem. T H E O R E M 4.4.2. Every dynamic equation (4.2.2) on a configuration bundle Q -» R is equivalent to the geodesic equation gif0 = 0,

q° = 1,

i? ? == K\{
(4.4.7)

172

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

on the tangent bundle TQ relative to a connection K with the components K° = 0 and K\ (4.4.5). Its solution is a geodesic curve in Q which also obeys the dynamic equation (4.4.6), and vice versa. □ In accordance with this theorem, the second order equation (4.2.10) in Theorem 4.2.3 can be chosen as a geodesic equation. It should be emphasized that, written in the bundle coordinates (£,
+ b)(qnii +

(4.4.8)

f'(q")-

This property is global due to the transformation law (4.2.5). Then one can use the following two facts. PROPOSITION 4.4.3. There is one-to-one correspondence between the affine con­ nections 7 on the affine jet bundle J ' Q —» Q and the linear connections K (4.4.3) on the tangent bundle TQ —» Q. D Proof. This correspondence is given by the relation (4.4.5), written in the form

< = 7Jo + lU<& = V o ( ? " ) t + KJiWW]^^*

=K;0{q")

+

K*i{
i.e., 7^> —

KJ\. QED

In particular, if an affine dynamic connection 7 is symmetric, so is the corre­ sponding linear connection K. COROLLARY 4.4.4. Every quadratic dynamic equation (4.4.8) on a configuration bundle Q —» R of non-relativistic mechanics gives rise to the geodesic equation

if = 0, 9" =

«7° = 1, t

/V)«V

Jjk(q 'W)(q''W
(4.4.9)

4.4. NON-RELATIVISTIC

GEODESIC

EQUATIONS

173

on the tangent bundle TQ with respect to the symmetric linear connection Kx\

= 0,

^o'o = / ' i

KOJ

on the tangent bundle TQ —» Q.

= -b),

K

kj

= a'kj

(4.4.10)

□

The geodesic equation (4.4.9), however, is not unique for the dynamic equation (4.4.8). PROPOSITION 4.4.5. Any quadratic dynamic equation (4.4.8), being equivalent to the geodesic equation with respect to the symmetric linear connection K (4.4.10), is also equivalent to the geodesic equation with respect to an affine connection K' on TQ —> Q which differs from K (4.4.10) in a soldering form a on TQ —> Q with the components

"S*o,

4 = A* + (s - l)h'kq°,

a'0 = -shlq* - htf + hi

where s and h\ are local functions on Q. D Proof. The proof follows from direct computation.

QED

Proposition 4.4.5 can also be deduced from the following lemma. LEMMA 4.4.6. Every affine vertical vector field a =

(4.4.11)

irm+bWWttf

on the affine jet bundle JlQ —► Q is extended to the soldering form a = {f'dt + b\dqk) ® di on the tangent bundle TQ —> Q.

(4.4.12)

□

Proof. Similarly to Proposition 4.4.3, one can show that there is one-to-one corre­ spondence between the VQ J'Q-valued affine vector fields (4.4.11) on the jet manifold J'<3 and the VgTQ-valued linear vertical vector fields a =

[§{?)# + ftf)?\di

on the tangent bundle TQ. This linear vertical vector field determines the desired soldering form (4.4.12). QED

174

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

In Section 6.4, we will use Theorem 4.4.2, Corollary 4.4.4 and Proposition 4.4.5 in order to study the relationship between non-relativistic and relativistic equations of motion. Now let us extend our inspection of dynamic equations to connections on the tangent bundle TM —» M of the typical fibre M of a configuration bundle Q —> R. In this case, the relationship fails to be canonical, but depends on a trivialization (4.1.1) of Q-*R. Given such a trivialization, let (t.g*) be the associated coordinates on Q, where if are coordinates on M with transition functions independent of t. The corresponding trivialization (4.1.3) of JXQ —» R takes place in the coordinates (t,q',q ), where q are coordinates on TM. With respect to these coordinates, the transformation law (4.3.2) of a dynamic connection 7 on the affine jet bundle JiQ —► Q reads -v"

dq'

10

H

%

(

dq'

ln

+

dq"

at*-

It follows that, given a trivialization of Q —> R, a connection 7 on J ' Q —> Q defines the time-dependent vertical vector field dq

R x TM -* VTM

and the time-dependent connection d RxTM dq' ) on the tangent bundle TM M. Conversely, let us consider a connection dqk ®

K

(W+*

dqh ® (-

(t,
+ Ktf,$)

JXTM C TTM

(4.4.13)

d

d?) on the tangent bundle TM —> M. Given the above-mentioned trivialization of the configuration bundle Q —> R, the connection K defines the connection K (4.4.3), with the components

K

0.

KI

Kk,

on the tangent bundle TQ —> Q. The corresponding dynamic connection 7 on the affine jet bundle J1Q —> Q reads %

0,

7i

rfc.

(4.4.14)

4.5. REFERENCE

FRAMES

175

Using the transformation law (4.3.2), one can extend the expression (4.4.14) to arbitrary bundle coordinates (t, q') on the configuration space Q as follows:

it

^K&(qr),it(Qr,qrt))

%

dtr ±9^4-

P

$«*(*,?)

+ dq^dq^

+

an

dkT,

(4.4.15)

7ir*,

where

is the connection on Q -» R, corresponding to a given trivialization of Q, i.e., P = 0 relative to (t,(f). The dynamic equation on Q defined by the dynamic connection (4.4.15) takes the form
dtr + qid^r + iM - rA).

(4.4.16)

By construction, it is a conservative dynamic equation. Thus, we have proved the following. PROPOSITION 4.4.7. Any connection K on the typical fibre M of a configuration bundle Q —► K yields a conservative dynamic equation (4.4.16) on Q. O

4.5

R e f e r e n c e frames

From the physical viewpoint, a reference frame in non-relativistic mechanics deter­ mines a tangent vector at each point of a configuration space Q, which characterizes the velocity of an "observer" at this point. This speculation leads us to suggest the following mathematical definition of a reference frame in non-relativistic mechanics [57, 132, 161]. DEFINITION 4.5.1. In non-relativistic mechanics, a reference frame is a connection T on the configuration bundle Q —* K. □ In accordance with this definition, one can think of the horizontal vector field (4.1.16), associated with a connection r on Q —> R, as being a family of "observers", while the corresponding covariant differential

q^Dr(qt)

=

ql-r

CHAPTER

176

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

determines the relative velocities with respect to the reference frame I \ In particular, given a motion c : R —» Q, its covariant derivative V r c (1.4.6) with respect to a connection T is the velocity of this motion relative to the reference frame T. For instance, if c is an integral section for the connection T, the velocity of the motion c relative to the reference frame T is equal to 0. Conversely, every motion c : R —> Q, defines a reference frame Tc such that the velocity of c relative to Tc vanishes. This reference frame Tc is an extension of the section c(R) —> JlQ of the affine jet bundle JlQ —> Q over the closed submanifold c(R) G Q to a global section in accordance with Theorem 1.1.2. Remark 4.5.1. It should be emphasized that the vertical tangent bundle VQ of a configuration bundle Q, but not the jet manifold J:Q plays the role of the "space of coordinates and velocities", while elements of J1Q may be termed the absolute velocities. In the universal unit system, elements of VQ, however, have the same physical dimension [q] as elements of Q, whereas absolute velocities are of physical dimension [g] — 1. • By virtue of Corollary 4.1.1, any reference frame T on a configuration bundle Q —> R is associated with an atlas of local constant trivializations, and vice versa. The connection T reduces to

r = dt

(4.5.1)

with respect to the corresponding coordinates (t,? 1 ), whose transition functions q1 —> q" are independent of time. One can think of these coordinates as being also the reference frame, corresponding to the connection (4.5.1). They are called the adapted coordinates to the reference frame I\ Thus, we come to the following definition, equivalent to Definition 4.5.1. DEFINITION 4.5.2. In non-relativistic mechanics, a reference frame is an atlas of local constant trivializations of a configuration bundle Q —> R. □ In particular, with respect to the coordinates if adapted to a reference frame T, the velocities relative to this reference frame are equal to the absolute ones D

r(q't) = fr = 5'c

A reference frame is said to be compJete if the associated connection F is com­ plete. By virtue of Proposition 4.1.2, every complete reference frame defines a trivialization of a bundle Q —> R, and vice versa.

4.5. REFERENCE FRAMES

177

Remark 4.5.2. Given a reference frame T, one should solve the equations

r'M(t,f)) =

a^a^v, +

« , 56

(4.5.2a) (4.5.2b)

in order to find the coordinates {t,qa) adapted to T. Let (i, g") and ((, g2) be the adapted coordinates for reference frames Tj and r 2 , respectively. In accordance with the equality (4.5.2b), the components r \ of the connection 1^ with respect to the coordinates (t, g2) and the components r 2 of the connection T2 with respect to the coordinates (t, g") fulfill the relation

M r - + ra - 0 • Using the relations (4.5.2a) - (4.5.2b), one can rewrite the coordinate transfor­ mation law (4.2.5) of dynamic equations as follows. Let

?u=T

(4.5.3)

be a dynamic equation on a configuration space Q, written with respect to a reference frame (t,if 1 ). Then, relative to arbitrary bundle coordinates (t,ql) on Q —> R, the dynamic equation (4.5.3) takes the form

qlt = dtri +

djr{qi-P)

-g^- r '«- r, >+f^

(4.5.4)

where T is the connection corresponding to the reference frame (t,q"). The dynamic equation (4.5.4) can be expressed in the relative velocities <j{. = q\ — T' with respect to the initial reference frame (t,qa). We have

*t-V4-gij£?**+&M-*>-

(4.5.5)

Accordingly, any dynamic equation (4.2.2) can be expressed in the relative velocities <jf, = q\ — P with respect to an arbitrary reference frame T as follows:

Mi = (£ - Jryt = e- ckr,

(4.5.6)

178

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

where jr is the prolongation (4.1.24) of the connection T onto the jet bundle JlQ —* R. For instance, let us consider the following particular reference frame T for a dynamic equation f. The covariant derivative of a reference frame T with respect to the corresponding dynamic connection 7^ (4.3.10) reads V 7 r = Q -» T'Q x VQJ'Q, k

(4.5.7)

x

wr = 'v}r dq ®dk, vjr* = aAr*-7$or. A connection T is called a geodesic reference frame for the dynamic equation f if

rjv^r

rA(dArk - 7* ° r)

(diC - c o r)di

o.

(4.5.8)

PROPOSITION 4.5.3. Integral sections c of a reference frame T are solutions of a dynamic equation f if and only if T is a geodesic reference frame for f. □ Proof. The proof follows at once from substitution of the equality (4.5.8) in the dynamic equation (4.5.6). QED R e m a r k 4.5.3. The left- and right-hand sides of the equation (4.5.6) separately are not well-behaved objects. This equation will be brought below into the covariant form (4.7.6). • Reference frames play a prominent role in many constructions of time-dependent mechanics. In particular, we obtain a converse of Theorem 4.4.2. THEOREM 4.5.4. Given a reference frame T, any connection K (4.4.1) on the tangent bundle TQ —> Q defines a dynamic equation

e = (K\ - rK°)q* \.0=l.,=q, . D This theorem is a corollary of Proposition 4.4.1 and the following lemma. LEMMA 4.5.5. Given a connection T on the fibre bundle Q - > R and a connection K on the tangent bundle TQ -* Q, there is the connection K on TQ -* Q with the components *A° = 0,

K\ = K\ -

VKl

4.6. FREE MOTION

EQUATIONS

179

□ Proof. The proof follows from the inspection of transition functions.

4.6

QED

Free motion equations

Let us point out the following interesting class of dynamic equations which we agree to call the free motion equations. DEFINITION 4.6.1. We say that the dynamic equation (4.2.2) is a free motion equation if there exists a reference frame ((,?*) on the configuration space Q such that this equation reads (4.6.1)

Tu = 0-

□ With respect to arbitrary bundle coordinates (t,
* = 4P + djr(
- *)<<* - n

(4.6.2)

where P = dtgt(t,'^) is the connection associated with the initial frame (t,q') (cf. (4.5.4)). One can think of the right-hand side of the equation (4.6.2) as being the general coordinate expression for an inertial force in non-relativistic mechanics. The corresponding dynamic connection 7^ on the affine jet bundle J ' Q —► Q reads

7i = dkr 75 = ^

dqmdq3dqk{ql + ^ - 7 * 1 * .

>'

(4.6.3)

It is affine. By virtue of Proposition 4.4.3, this dynamic connection defines a linear connection K on the tangent bundle TQ —> Q, whose curvature necessarily vanishes. Thus, we come to the following criterion of a dynamic equation to be a free motion equation. PROPOSITION 4.6.2. If £ is a free motion equation on a configuration space Q, it is quadratic, and the corresponding symmetric linear connection (4.4.10) on the tangent bundle TQ —» Q is a curvature-free connection. □

180

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

This criterion is not a sufficient condition because it may happen that the com­ ponents of a curvature-free symmetric linear connection on TQ —► Q vanish with respect to the coordinates on Q which are not compatible with the fibration Q —> R. The similar criterion involves the curvature of a dynamic connection (4.6.3) of a free motion equation. PROPOSITION 4.6.3. If f is a free motion equation, then the curvature R (4.3.3) of the corresponding dynamic connection 7^ is equal to 0, and so are the tensor field R (4.3.4) and the scalar field R (4.3.5). D Proposition 4.6.3 also fails to be a sufficient condition. If the curvature R (4.3.3) of a dynamic connection 7^ vanishes, it may happen that components of 7^ are equal to 0 with respect to non-holonomic bundle coordinates on the affine jet bundle

JlQ-^Q. Nevertheless, we can formulate the necessary and sufficient condition of the ex­ istence of a free motion equation on a configuration space Q. PROPOSITION 4.6.4. A free motion equation on a fibre bundle Q —> R exists if and only if the typical fibre M of Q admits a curvature-free symmetric linear connection. D

Proof. Let a free motion equation take the form (4.6.1) with respect to some atlas of local constant trivializations of a fibre bundle Q —> R. By virtue of Proposition 4.3.2, there exists an affine dynamic connection 7 on the affine jet bundle JlQ —» Q whose components relative to this atlas are equal to 0. Given a trivialization chart of this atlas, the connection 7 defines the curvature-free symmetric linear connection (4.4.13) on M. The converse statement follows at once from Proposition 4.4.7. QED

The free motion equation (4.6.2) is simplified if the coordinate transition func­ tions q1 —* q1 are affine in the coordinates
qxtt = dtP - PdjP + 2q1td]r.

(4.6.4)

E x a m p l e 4.6.1. Let us consider a free motion on a plane R 2 . The corresponding configuration bundle is R 3 -» R, coordinated by (t,r). The dynamic equation of this motion is f = 0.

(4.6.5)

4.6. FREE MOTION

EQUATIONS

181

Let us choose the rotatory reference frame with the adapted coordinates r = AT,

A _ /coswt-sinwA \ sin art cos art )

(4.6.6)

Relative to these coordinates, the connection T corresponding to the initial reference frame reads r = dtr = dtA ■ A-XT. Then the free motion equation (4.6.5) with respect to the rotatory reference frame (4.6.6) takes the familiar form rM =

■M")*

(4.6.7)

The first term in the right-hand side of the equation (4.6.7) is the centrifugal force (—PojjT'), while the second one is the Coriolis force (2q\djYx). • The following lemma shows that the free motion equation (4.6.4) is affine in the coordinates q' and q\. LEMMA 4.6.5. Let (£,
q* = o£(t)g» + b'(t), then we have

r H O ^ a ^ - f r O + d^. Conversely, let

r = c)(t)q> + s'(t) such that the homogeneous linear system of differential equations

aJ
(4.6.8)

182

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

has a solution on R. Let a)(t) be the mxm matrix whose columns are m fundamental solutions qi{t),--,qm(t) of the equations (4.6.8) (see, e.g., [22]). This matrix is invertible for all t 6 R. Then the connection T is associated with the coordinates ((,5") such that

9i=oi(*)r+6'(o, where functions bl(t) fulfill the equations

dtV = ( 0 £ W " + *'■ QED One can easily find the geodesic reference frames for the free motion equation (4.6.9)

lit = o.

They are F* = V* = const. By virtue of Lemma 4.6.5, these reference frames define the adapted coordinates 5* = Vjq> - v't - a\

k'j = const.,

vx = const.,

a' = const. (4.6.10)

The equation (4.6.9) obviously keeps its free motion form under the transformations (4.6.10) between the geodesic reference frames. It is readily observed that these transformations are precisely the elements of the Galilei group.

4.7

Relative acceleration

In comparison with the notion of a relative velocity, that of a relative acceleration is more intricate. To consider a relative acceleration with respect to a reference frame T, one should prolong the connection T on the configuration bundle Q —* R to a holonomic connection £ r on the jet bundle JlQ -> R. Note that the jet prolongation JV (4.1.24) of r onto JlQ —» R is not holonomic. We can construct the desired prolongation by means of a dynamic connection 7 on the affine jet bundle JlQ —> Q. LEMMA 4.7.1. Let us consider the composite bundle (4.1.10). Given a frame T on Q —> R and a dynamic connections 7 on J ' Q -» Q, there exists a dynamic connection 7 on J ' Q —► Q with the components

7i = 7i,

7; = tkr - Ykrk.

(4.7.1)

4.7. RELATIVE

ACCELERATION

183

D

Proof. Combining the connection r on Q -> R and the connection 7 on JlQ -> Q gives the composite connection (1.6.8) on JlQ -> R which reads

B = dt ® (ft + Pft + ( 7 jp + 7*)ft). Let j r be the jet prolongation (4.1.24) of the connection T on JlQ -* R. Then the difference JT-B

= dt® {dtV -

7'r*

- 7J)9f

is a VQJ'Q-valued soldering form on the jet bundle J1Q -» R, which is also a soldering form on the affine jet bundle JlQ -+ Q. The desired connection (4.7.1) is

7 = 7 + j r - B = dt ® (ft + (d(r - 7*r*)ft) + A?* ® (ft + 7 j$). QED

Now, we construct a certain soldering form on the affine jet bundle JlQ —^ Q and add it to this connection. Let us apply the canonical projection T'Q —> V'Q and then the imbedding T : V'Q -> T'Q to the covariant derivative (4.5.7) of the reference frame T with respect to the dynamic connection 7. We obtain the VQJ1Q- valued 1-form

a = [-P(ftP - 7* o r)dt + (ftp - 7* o r)dq'} ® ft on Q whose pull-back onto JlQ is the desired soldering form. The sum def~ .

7r = 7 + °, called the frame connection, reads

7r{, = *r* - 7Jr* - p(ftp - 7 j o r), 7ri = 7l + dkr - 7* o r. This connection yields the desired holonomic connection

& = ckv + (ftp + yt - 7 ' o r)(q* - r") on the jet bundle JXQ -» R.

(4.7.2)

184

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

Let £ be a dynamic equation and 7 = 7^ the connection (4.3.10) associated with £• Then one can think of the vertical vector field

ar=g-fr = (f-f r )ff

(4.7.3)

on the affine jet bundle JlQ —» Q as being a relative acceleration with respect to the reference frame T in comparison with the absolute acceleration £. For instance, let us consider a reference frame which is geodesic for the dynamic equation £, i.e., the relation (4.5.8) holds. Then the relative acceleration of a motion c with respect to the reference frame T is

tt - fr) ° r = 0. Let £ now be an arbitrary dynamic equation, written with respect to coordinates (t, ql) adapted to the reference frame T, i.e., P = 0. In these coordinates, the relative acceleration with respect to the reference frame T is

4 = e W .
(4.7.4)

Given another bundle coordinates {t,q") on Q —* R, this dynamic equation takes the form (4.5.5), while the relative acceleration (4.7.4) with respect to the reference frame T reads

4 = S,g''4. Then we can write a dynamic equation (4.2.2) in the form which is covariant under coordinate transformations: ■P-rrgl = ^91 ~ £r = a r ,

(4.7.5)

where D 7 r is the vertical covariant differential (4.3.9) with respect to the frame connection 7r (4.7.2) on the affine jet bundle J ' Q —► Q. In particular, if £ is a free motion equation which takes the form (4.6.1) with respect to a reference frame V, then D^q\ = 0 relative to arbitrary bundle coordinates on the configuration bundle Q —> R.

4.7. RELATIVE

185

ACCELERATION

The left-hand side of the dynamic equation (4.7.5) can also be expressed in the relative velocities such that this dynamic equation takes the form (4.7.6)

^ 9 r - yr'kQr = ar

which is the covariant form of the equation (4.5.6). The concept of a relative acceleration is understood better when we deal with the quadratic dynamic equation £, and the corresponding dynamic connection 7 is affine. LEMMA 4.7.2. If a dynamic connection 7 is affine, i.e., 7A — 7AO + 7Ak9t! so is a frame connection 7 r for any frame I\

□

Proof. The proof follows from direct computation. We have

= dtr + (dJri-1'k]rk)(
1rl

7ri = a t r + 7 i J ( ^ - P ) or 7r}k

7j*>

TTofc

&r-7-fcF,

l

-rr oo = dtr-PdJr

7rio =

dkT*-T^P,

(4.7.7)

k

+ Y}krr . QED

In particular, we obtain 7r}/t = Y]k,

7rok = 7rlM) = 7rw = 0

relative to the coordinates adapted to a reference frame T. A glance at the expression (4.7.7) shows that, if a dynamic connection 7 is symmetric, so is a frame connection jp. COROLLARY 4.7.3. If a dynamic equation £ is quadratic, the relative acceleration ar (4.7.3) is always affine, and it admits the decomposition a'r

= -(r A vir + 2^vir),

(4.7.8)

186

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

where 7 = 7c is the dynamic connection (4.3.10), and 9r = 9t - T \

9? = 1,

r° = i,

is the relative velocity with respect to the reference frame T.

□

Note that the splitting (4.7.8) gives a generalized Coriolis theorem. In particu­ lar, the well-known analogy between inertia! and electromagnetic forces is restated. Corollary 4.7.3 shows that this analogy can be extended to arbitrary quadratic dy­ namic equation.

4.8

Lagrangian systems

A velocity phase space JlQ of time-dependent mechanics does not admit any canon­ ical structure mentioned in the previous Chapter. Since J1Q is an odd-dimensional manifold, it has no symplectic structure, whereas presymplectic and Poisson struc­ tures are defined only for Lagrangian systems, and depend on a Lagrangian. By a Lagrangian system is meant a mechanical system whose motions are solutions of Lagrange equations for some Lagrangian on a velocity phase space J ' Q . Obviously, a mechanical system is not necessarily Lagrangian, whereas Lagrange equations are not necessarily dynamic equations. Moreover, it may happen that Lagrange equa­ tions fail to be differential equations in a strict sense (see Definition 1.3.3). Note that, in the framework of Lagrangian formalism, one also meets Cartan and Hamilton-De Donder equations, besides the Lagrange one. These equations are equivalent to each other and to Lagrange equations in the case of a regular Lagrangian. A Lagrangian of a mechanical system is defined as a horizontal density L = £dt,

(4.8.1)

l

C : J Q -> R, on the velocity phase space JlQ, where £ is called a Lagrangian function or simply a Lagrangian if there is no danger of confusion. With respect to the universal unit system, a Lagrangian (4.8.1) is physically dimensionless. Here, we do not study the calculus of variations in depth, but apply in a straight­ forward manner the first variational formula [57]. Let us consider a projectable vector field u = uldt + u'di,

u« = 0,1,

(4.8.2)

4.8. LAGRANGIAN

SYSTEMS

187

on a configuration bundle Q —> M and calculate the Lie derivative of a Lagrangian (4.8.1) along the jet prolongation u = tfdt + u{d, +

dtu%

dt = dt + q\di + q\tdl (1.3.7) of u. We obtain l^L = (u\dC)dt = (u'dt + u% + dtu'dDCdt.

(4.8.3)

The first variational formula provides the following canonical decomposition of the Lie derivative (4.8.3) in accordance with the variational problem: u\dC = (u' - u'qpS, +

dt{u\HL),

(4.8.4)

where HL = v'idL) + L = ntdq{ - {iriq\ - C)dt

(4.8.5)

is the Poincare-Cartan form (see (4.1.14)), and £L

: J2Q -»

VQ,

£L = S.dq' = (d, - dtdPCdtf

(4.8.6)

is the Euler-La.gra.nge operator for a Lagrangian L. The latter can be seen as a 2-form Si = (d< - dtdl)Cdq% A dt. Its coefficients Si are called variational derivatives. We will use the notation iTi = d\C,

7rJt = d\d\C

The kernel Ker£ L C J2Q of the Euler-Lagrange operator (4.8.6) defines the system of second order differential equations on Q (ft - dtdj)C = 0,

(4.8.7)

caUed the Lagrange equations. Its solutions are (local) sections c of the fibre bundle Q —► R whose second order jet prolongations c five in (4.8.7). They obey the equations dtC o c — -y{^i o c) = 0. dt

(4.8.8)

CHAPTER

188

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

Remark 4.8.1. The kernel of an Euler-Lagrange operator Ker£j, fails to be a subbundle of the fibre bundle J2Q —► R in general. Therefore, it may happen that, as was mentioned above, the Lagrange equations (4.8.7) are not differential equations in a strict sense. In particular, by virtue of Proposition 1.3.4, it is a differential equation when the morphism (4.8.6) is of constant rank, e.g., if a Lagrangian L is regular. Recall that a Lagrangian L is called regular (non-degenerate), if det 7r^ ^ 0 everywhere on the velocity phase space J1Q. If a Lagrangian L is non-degenerate, the Lagrange equations can be solved algebraically for second order derivatives, and are equivalent to a dynamic equation. • Example 4.8.2. Let Q = R2 —► K be a configuration space, coordinated by {t,q). The corresponding velocity phase space JlQ is equipped with the adapted coordinates (t,q,qt). The Lagrangian L=

-q2q2dt

on JlQ leads to the Euler-Lagrange operator £L = \qq2t ~ dt(q2qt)]dq whose kernel is not a submanifold at the point q = 0. • Every Lagrangian L on the jet manifold JlQ yields the Legendre map L:JlQ^

V'Q,

(4.8.9)

P , O L = 7T,,

where (t,g',Pi) are coordinates on the vertical cotangent bundle V'Q. Indeed, due to the vertical splitting (4.1.8), the vertical tangent morphism VL to L yields the linear morphism

VL: JlQx

VQ-+R

and, consequently, the morphism (4.8.9). In time-dependent mechanics on a configuration space Q —» R, the vertical cotangent bundle V'Q plays the role of a momentum phase space (see Chapter 5).

4.8. LAGRANGIAN

189

SYSTEMS

Remark 4.8.3. The Legendre map (4.8.9) is a local diffeomorphism, i.e., it is of maximal constant rank if and only if a Lagrangian L is regular. A Lagrangian L is called hypenegular if the Legendre map L is a diffeomorphism. • In Section 5.1, we will show that the vertical cotangent bundle V'Q is provided with the canonical 3-form ft d = dPi A dq' A dt,

(4.8.10)

derived from the polysymplectic form (2.9.9). Let us consider the pull-back ilL = L'U = d-Ki A dqi A dt

(4.8.11)

on JlQ of the canonical form (4.8.10) by means of the Legendre map L (4.8.9). The form fi^ provides Lagrangian formalism with the construction similar to Hamiltonian mechanics, but which depends on the choice of a Lagrangian L. In the framework of this construction, the Lagrangian counterpart of a Hamiltonian form is the Poincare-Cartan form (4.8.5). If a Lagrangian L is regular, the form Q,L (4.8.11) defines a Poisson structure on the jet manifold JiQ as follows. By means of f2L, every vertical vector field

d = tf% + &% on the jet bundle JlQ —> R corresponds to the 2-form tfjftz, = {[<5%, + ^{djin

- BkKjffl

~ $%dqi}

A dt.

This is one-to-one correspondence if a Lagrangian L is regular. Indeed, given an arbitrary 2-form

= [4>idqi + fadql) A dt on JlQ, the algebraic equations &nji+#>(dini-#Kfi

dtirj) =
=
have a unique solution

&= $

=

-{*-ifL, ( T -i)*»[^ + ( j r 1 ) * " ^ * * - ft»r*)].

CHAPTER

190

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

In particular, every function / on JlQ determines a vertical vector field

#f = fr-yfye, - (n-ly%f + (Tr-^&jid^ - di^d)

(4.8.12)

on J1Q —> K in accordance with the relation ■d,\UL = -df Adt. Then the Poisson bracket

{/,$}x.*=* f J*/jni. {f,9}i

l 3

= (*- Y

f,9eO°(JlQ), +

(4.8.13)

(difdJg-dtigd]f)

(ft.»r i k -Sl k ir n )(ir- 1 )«(ir- 1 )^/ajff, can be defined on the space D°(JlQ) of functions on the jet manifold JlQ. In particular, the vertical vector field d} (4.8.12) is the Hamiltonian vector field for the function / with respect to the Poisson structure (4.8.13). The Poisson structure (4.8.13) defines the corresponding symplectic foliation on the jet manifold J1Q, which coincides with the fibration JlQ —>R. The symplectic form on the leaf J]Q of this foliation is f2t = dirx A dq'

[57, 182]. Example 4.8.4. If a Lagrangian L is hyperregular, the Poisson structure (4.8.13) is isomorphic to the canonical Poisson structure on the momentum phase space V'Q (see Section 5.1). Indeed, the Poisson bracket (4.8.13) reads {Wj}

= {q',qi} = 0,

{7r,,g,} = ff.

• Remark 4.8.5. If a Lagrangian is degenerate, one may try to introduce a Poisson structure on JlQ corresponding to the presymplectic form dHL (see (4.8.23) below) in accordance with the procedure suggested in [49] (see Remark 2.5.1). •

D E F I N I T I O N 4.8.1. A connection

ZL = dt + f%+edl

4.8. LAGRANGIAN

SYSTEMS

191

on the jet bundle J ' Q —» H is said to be a Lagrangian connection for the Lagrangian L if it obeys the equation

q j n L = d//L

(4.8.14)

which takes the coordinate form

(/' - # " * = o,

(4.8.15a)

a,£ - a(7r, - fd^i - e*it + {p - 4)d^ = o.

(4.8.15b)

□ Example 4.8.6. A glance at the equations (4.8.15a) - (4.8.15b) shows that, for a regular Lagrangian L, a Lagrangian connection is necessarily holonomic and unique.

In order to clarify the meaning of the equation (4.8.14), let us consider the Lagrangian

Cd^C(t,q',qi) +

(qlt)-qi)nl(t,q\ql)

on the repeated jet manifold J1JiQ. The corresponding Euler-Lagrange operator, called the Euler-Lagrange-Cartan operator, reads £z : JlJ'Q

-*

T'J'Q,

£T = [(diC + diTTjiq!^ -qi)-

dtn,)dq' + 7r 0 (^ () - q})dq\} A dt,

(4.8.16)

dt = dt + q\t)di + q\td\. Then the equation (4.8.14) is equivalent to the condition UiJ'Q)

C Ker£r

which is the first order differential equation on the jet manifold J ' Q , called the Cartan equations. They read

*v(4t) ~ d) = °diC - dtiTi +

{qj{t)

- qi)diir} = 0.

(4.8.17a)

(4.8.17b)

Remark 4.8.7. In Section 5.2, we will show that the Euler-Lagrange-Cartan operator (4.8.16) is the Lagrangian counterpart of a Hamilton operator. •

CHAPTER

192

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

Solutions of the Cartan equations (4.8.17a) - (4.8.17b) are (local) integral sec­ tions £:(}—» JlQ of Lagrangian connections &, for a Lagrangian L, i.e., Jlc = £L°C. Furthermore, the equation (4.8.14) for these integral sections is equivalent to the relation ?{d\dHL)

(4.8.18)

= 0

which is assumed to hold for any vertical vector field d on the jet bundle JlQ —» R. It is easily seen that the restriction Ker£ L = J2Q n K e r £ r

(4.8.19)

of the Euler-Lagrange-Cartan operator £ j (4.8.16) to the second order jet manifold J2Q C JlJlQ recovers the Euler-Lagrange operator £L (4.8.6) for the Lagrangian L. Therefore, the Lagrange equations (4.8.7) are equivalent to the Cartan equations (4.8.17a) - (4.8.17b) on the holonomic sections c = Jlc of the jet bundle JlQ -> R. It follows that solutions of the Lagrange equations (4.8.7) are integral sections of holonomic Lagrangian connections for the Lagrangian L, which take their values into the kernel of the Euler-Lagrange operator (4.8.19). Different holonomic Lagrangian connections lead to different dynamic equations associated with the same system of Lagrange equations. In the case of a regular Lagrangian L, the Cartan equations (4.8.17a) - (4.8.17b) are equivalent to the Lagrange equations (4.8.7), and lead to the unique dynamic equation q3tt = (n-1Y3[-diC

+ dtnl + qtdkirl}

(4.8.20)

on the configuration space Q. By very definition, the Poincare-Cartan form Hi (4.8.5) defines the fibred morphism HL-.J'Q

{pt,p)oHL

(4.8.21)

-^T'Q,

= (TT,,

C-ntqi),

from the affine jet bundle JXQ —> Q to the cotangent bundle T'Q —» Q of Q, which plays the role of the homogeneous Legendre bundle (2.9.3), equipped with the coordinates (t,q',p,pi). The morphism (4.8.21) is termed the Legendre morphism

4.8. LAGRANGIAN

SYSTEMS

193

associated with Hi. There is the following relation between the Legendre map L (4.8.9) and the Legendre morphism HL (4.8.21): (4.8.22)

L = C o HL ,

where C is the canonical projection T'Q —> V'Q (1.1.7). Furthermore, it is readily observed that the Poincare—Cartan form Hi is the pull-back of the canonical Liouville form E = pdt + pidqx on the cotangent bundle T'Q by the associated Legendre morphism (4.8.21). Ac­ cordingly, the presymplectic form dHL = d7r, A dq* - d^tf

(4.8.23)

-C)Adt

on the jet manifold J1Q is the pull-back of the canonical symplectic form dz. = dp Adt + dpi A dql on the cotangent bundle T'Q of the configuration space Q. Remark 4.8.8. The presymplectic form dHL (4.8.23), together with the 1-form dt, define a copresymplectic structure on the configuration space JlQ [37, 86]. Let us take the exterior product (dHL)m

Adt = det(7Ty)( ] T dql A dq*)™ A dt.

(4.8.24)

I

If it is nowhere vanishing, the pair (dHi,dt) is called a cosymplectic structure on the (2m + l)-dimensional manifold J1Q. A glance at the expression (4.8.24) shows that the pair (dHi, dt) is cosymplectic if and only if the Lagrangian L is regular. A cosymplectic structure (dHi,dt) on JlQ defines the isomorphism TJlQ

9 » H v\dHL + {v\dt)dt e

T'JlQ.

In particular, let T be a subbundle of the tangent bundle TJlQ define the orthocomplement T 1 = {v € TJlQ of T in TJlQ

: v\dHL + (v\dt)dt € Ann (T)}

with respect to (dHt, dt).

—» J1Q. One can

194

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

Note that the classification of Lagrangians on the product R x M by the proper­ ties of the intersection KerdHi n Kei dt has been suggested [86]. This classification remains also true for Lagrangians on an arbitrary fibre bundle Q —» R, though not all its constructions (e.g., the tangent-valued forms S and ~S (see [86])) are maintained under time-dependent transformations. • Given a Lagrangian L, let the image ZL of the configuration space JlQ by the Legendre morphism Hi (4.8.21) be an imbedded subbundle k ■ ZL *-* T'Q of the cotangent bundle T'Q. It is provided with the pull-back De Dondei form =.L = i'LE.

By analogy with the Cartan equations (4.8.18), the corresponding Hamilton-De Donder equations for sections r of the fibre bundle Zi —» R are written as the condition T'{u\dEL) = 0

(4.8.25)

where u is an arbitrary vertical vector field on ZL —> R. To obtain these equations in an explicit form, one should substitute the solutions q,t{.t,qi,pj),

£{t,q>,pj,p)

of the equations Pi =

*,{t,qj,q{),

p = C{t,q>,4) -

ni(t,q>,qi)q\

(4.8.26)

in the Cartan equations (4.8.18). If a Lagrangian L is regular, the equations (4.8.26) have a unique solution. Then the Hamilton-De Donder equations take the coordinate form

dtr = -dlr, (4.8.27)

dtT{ = dir,

r =

C(t,q\g't{t,g],pJ))-plq't(t,qi,p1),

and are equivalent to the Cartan equations. If a Lagrangian L is degenerate, the equations (4.8.26) may admit different solutions, or no solutions at all. More can be said in the following case.

4.9. NEWTONIAN

SYSTEMS

195

PROPOSITION 4.8.2. Let the Legendre morphism HL : JlQ —► ZL be a submersion. Then a section c of the jet bundle JlQ —> K is a solution of the Cartan equations (4.8.18) if and only if HL ° c is a solution of the Hamilton-De Donder equations (4.8.25) [63]. a Remark 4.8.9. In the case of a regular Lagrangian L, the Hamilton-De Donder equations (4.8.27) can be seen as the equations of motion in variant (B) of the Dirac system on the symplectic manifold T'Q in the case of: (i) a zero Hamiltonian, (ii) the primary constraint space N = ZL (4.8.26), and (iii) with the additional condition that a solution v of the equation (3.6.1) has the component v° = 1. The latter condition guarantees that the motion parameter is t. • In conclusion, let us touch briefly on the case of conservative Lagrangians. Let us suppose that, given a trivialization ip:Q = R x M in the coordinates (£,
C).

Thus, a conservative Lagrangian system reduces to the presymplectic Hamiltonian system whose Hamiltonian is the energy function (^gj — C) (see Examples 3.3.2 and 3.4.2). In particular, if a Lagrangian is degenerate, one can apply the procedure from Section 3.6 to investigate it [138]. We refer the reader to [31, 37, 111] for extension of this procedure to time-dependent degenerate Lagrangian systems on the product R x M. We will investigate degenerate Lagrangian systems in the framework of Hamiltonian formalism (see Section 5.5).

4.9

Newtonian systems

Let L be a Lagrangian on a velocity phase space J1Q and L the Legendre map (4.8.9). Due to the vertical splitting (1.1.6) of VV'Q, the vertical tangent map VL

CHAPTER

196

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

to L reads VL : VQJXQ - . V'Q x V'Q. Q

It yields the linear fibred morphism b d 5= f (Id J 1 Q ,pr 2 oVL) : VQJlQ -

V^'Q,

(4.9.1)

b : d[ i-> ffyd^, where {dg?} are bases for the fibres of the vertical tangent bundle VA JlQ —> J ' Q . The morphism (4.9.1) defines the mapping JlQ -* VAJ'Q 9 W

J'Q

VAJlQ *

and, due to the splitting (4.1.9), also the mapping m:JlQ

—>VQ®VQ,

"*ij = P,j o m = 7Ty,

where (£,(?', p tJ ) are holonomic coordinates on V'Q®V'Q. 2

Thus, n^ = mtJ are

___

components of the W ' Q - v a l u e d field m on the velocity phase space J Q. It is called the mass tensor. Let a Lagrangian L be regular. Then the mass tensor is non-degenerate, and defines a fibre metric, called mass metric, in the vertical tangent bundle VQJ^Q —» JlQ. Let us recall that, if a Lagrangian L is regular, there exists a unique Lagrangian connection £& for L which is holonomic in accordance with the equation (4.8.15a), and obeys the equation mutt* = -dtn

- djTTi^t + d,£.

(4.9.2)

This holonomic connection defines the dynamic equation (4.8.20). At the same time, the equation (4.9.2) leads to the commutative diagram VQJ'Q

-*-»

VAJ'Q

\/eL

DKL

J2Q SL = \>ODU, £> =

mxk{q'!t-(,kL),

(4.9.3)

4.9. NEWTONIAN

197

SYSTEMS

where D^L is the covariant differential (4.1.17) relative to the connection fa. Fur­ thermore, the derivation of (4.9.2) with respect to q\ results in the relation (4.9.4)

fa\dmij + 771^7* + mjknff = 0, where li

— 2 °«SL

are coefficients of the symmetric dynamic connection 7 ^ (4.3.10) corresponding to the dynamic equation fa. Thus, each regular Lagrangian L defines the dynamic equation fa, related to the Euler-Lagrange operator £L by means of the equality (4.9.3), and the non-degenerate mass tensor m^, related to the dynamic equation fa by means of the relation (4.9.4). This is a Newtonian system in accordance with the following definition. DEFINITION 4.9.1. Let Q —> R be a fibre bundle together with • a (non-degenerate) fibre metric m in the fibre bundle VQJ1Q

—> JYQ:

m:JlQ-+VQ®VQ, Q m = -mijdq'

V dq3,

satisfying the symmetry condition dlrriij = d'rriik,

(4.9.5)

• and a holonomic connection f (4.1.25) on the jet bundle JlQ —► K, related to the fibre metric m by the compatibility condition (4.9.4). The triple (Q, m, ?) is called a Newtonian system.

This is not the terminology of

[31]. □ Note that the compatibility condition (4.9.4) can also be introduced in an in­ trinsic way as V ? m = 0,

198

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

where by V is the covariant derivative with respect to the connection 7^ on the vertical cotangent bundle VqJlQ —* J1Q, which is dual of the connection \ (4.3.23) on the vertical tangent bundle VQ JlQ —> JlQ. Definition 4.9.1 generalizes the second Newton law of point mechanics. Indeed, the dynamic equation for a Newtonian system is equivalent to the equation (4.9.6)

m,k(qu - £*) = 0.

There are two main reasons for considering Newtonian systems. From the physical viewpoint, with a mass tensor, we can introduce the notion of an external force. Note that, in the universal unit system, the mass tensor m is dimensional. For instance, the dimension of a mass tensor of a point mass with respect to Cartesian coordinates q' is [length] -1 , while that with respect to the angle coordinates is [length]. DEFINITION 4.9.2. An external force is defined as a section of the vertical cotangent bundle VQJ1Q —> JlQ. Let us also bear in mind the isomorphism (4.1.9). □ Note that there are no canonical isomorphisms between the vertical cotangent bundle VQ JlQ and the vertical tangent bundle VQ J ' Q of JlQ. Therefore, one should distinguish forces and accelerations which are related by means of a mass metric (see also Remark 4.9.2 below). Let (Q, m, £) be a Newtonian system and / an external force. Then

o=r+(m-n

(4.9.7)

is a dynamic equation, but the triple (Q, m, £/) is not a Newtonian system in general. As follows from direct computation, if and only if an external force possesses the property

af/i + fl}/i = o,

(4.9.8)

then £/ (4.9.7) fulfills the relation (4.9.4), and (Q, m, £/) is also a Newtonian system. Example 4.9.1. For instance, the Lorentz force }i = eF^q?,

9? = 1,

(4.9.9)

where F\» = dxA^ - d^Ax

(4.9.10)

4.9. NEWTONIAN

199

SYSTEMS

is the electromagnetic strength, obeys the condition (4.9.8). Note that the Lorentz force (4.9.9), just as other forces, can be expressed in the relative velocities qr with respect to an arbitrary reference frame T:

*-£($*-*+4 where q are the coordinates adapted to the reference frame T, and JP is the electro­ magnetic strength, written with respect to these coordinates. • Remark 4.9.2. The contribution of an external force / to a dynamic equation

qit-Sl = (m-l)ikfk of a Newtonian system obviously depends on a mass tensor. It should be empha­ sized that, besides external forces, we have a universai force which is a holonomic connection C = ^A
Q? = 1,

associated with a symmetric linear connection K (4.4.3) on the tangent bundle TQ —> Q. From the physical viewpoint, this is a non-relativistic gravitational force, including an inertial force, whose contribution to a dynamic equation is independent of a mass tensor. • From the mathematical viewpoint, the equation (4.9.6) is the kernel of an EulerLagrange-type operator (see (4.9.25) below). By an appropriate choice of a mass tensor, one may hope to bring it into Lagrange equations. We have seen that a non-degenerate Lagrangian system is necessarily a Newtonian one, while a converse statement is not generally true. Example 4.9.3. Let us consider a non-degenerate quadratic Lagrangian

£ = ^ ( ^ ) * r f + w ) ? ; +

(4.9.11)

where the mass tensor m tJ is a Riemannian metric in the vertical tangent bundle VQ —» Q (see the isomorphism (4.1.8)). Then the Lagrangian L (4.9.11) can be written as

£ = -^Sa^gfgf,


(4.9.12)

200

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

where g is the metric Poo = -2,

9i3 = - " > ■ )

9oi — ~k<>

(4.9.13)

on the tangent bundle TQ. The corresponding Lagrange equations take the form

q\t =

-{m-l),kUMtll

9? = 1,

(4.9.14)

where {\nv} = —Adxg^u + d^g^x - d^gx,,) are the Christoffel symbols of the metric (4.9.13). Let us assume that this metric is non-degenerate. By virtue of Corollary 4.4.4, the dynamic equation (4.9.14) gives rise to the geodesic equation (4.4.9) on the tangent bundle TQ, which reads 9fl = 0,

9° = 1,

«* = { A V H V -
£ = ^ y (O(?;-r)te-r') + 0'(9"),

(4.9.15)

where T is a Lagrangian frame connection o n Q - > R which takes its values into the kernel of the Legendre map L (see Section 5.6). This connection defines an atlas of local constant trivializations of the fibre bundle Q —» R and the corresponding coordinates (t,5*) on Q such that the transition functions q1 —► q" are independent oft, and P = 0 with respect to (£,
£■=^m+wof

))•

(4.9.16)

Let us assume that ' is a nowhere vanishing function on Q. Then the Lagrange equations (4.9.14) take the form

Qu =

{x\}t?t,

3? = l,

where {>'„} are the Christoffel symbols of the metric (4.9.13), whose components with respect to the coordinates ((,?") read goo = - 2 0 ' ,

9oi = 0,

9x] = - ? R y .

(4.9.17)

4.9. NEWTONIAN

201

SYSTEMS

Then the spatial part of the corresponding geodesic equation 5° = 0,

9=1, (4.9.18)

Q ={x ■/}? q

on the tangent bundle TQ is precisely the spatial part of the geodesic equation with respect to the Levi-Civita connection for the metric (4.9.17) on TQ. • This example shows that a mass tensor may be treated sometimes as a field variable. A Newtonian system (Q, rh, £) is said to be standard, if rh is the pull-back on VQJ1Q of a fibre metric in the vertical tangent bundle VQ —► Q in accordance with the isomorphisms (4.1.8) and (4.1.9), i.e., the mass tensor rh is independent of the velocity coordinates q\. It is readily observed that any fibre metric rh in VQ —» Q can be seen as a mass metric of a standard Newtonian system, given by the Lagrangian

c = \nrnWM - n(4 - n,

(4.9.19)

where T is a reference frame. If rh is a Riemannian metric, one can think of the Lagrangian function (4.9.19) as being a kinetic energy with respect to the reference frame T. E x a m p l e 4.9.4. Let us consider a system of n distinguishable particles with masses ( m i , . . . , mn) in a 3-dimensional Euclidean space R 3 . Their positions ( r i , . . . , r„) span the configuration space R 3n . The total kinetic energy is

Z

A=l

that corresponds to the mass tensor TnABtj = S/igSiftrtAi

A,B = l , . . . n ,

i,j = 1,2,3,

on the configuration space R 3n . To separate the translation degrees of freedom, one performs a linear coordinate transformation ( r i , . . . , r n ) (-► ( 7 » i , . . . , p „ _ i , R ) ,

202

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

where R is the centre of mass, while the n - 1 vectors (pu.. ., p„-i) are massweighted Jacobi vectors (see their definition below) [119, 120). The Jacobi vectors PA are chosen so that the kinetic energy about the centre of mass has the form in-l

(4.9.20) Z

A=l

that corresponds to the Euclidean mass tensor TKABi] =

^AB^ij,

A,B = l , . . . n - l ,

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

on the translation-reduced configuration space R ~ . The usual procedure for defin­ ing Jacobi vectors involves organizing the particles into a hierarchy of clusters, in which every cluster consists of one or more particles, and where each Jacobi vector joins the centres of mass of two clusters, thereby creating a larger cluster. A Jacobi vector, weighted by the square root of the reduced mass of the two clusters it joins, is the above-mentioned mass-weighted Jacobi vector. For example, in the four-body problem, one can use the following clustering of the particles: Pi =

y/fH(T2-n),

P2 = \/^2(r4 - r3),

-

.— / m 3 r 3 + m 4 r 4

P 3 = V^3 I \ m 3 + m_ 4L Hi

mi

m2'

mir 1 + m 2 r 2 \

I, J_ mi + m / 2 J_ J_ u.3

1

1

m\ + m.2

m3 + m4

p.2 m3 m4' Different clusterings lead to different collections of Jacobi vectors, which are related by linear transformations. Since these transformations maintain the Euclidean form (4.9.20) of the kinetic energy, they are elements of the group 0(n — 1), called the "democracy group". • Now let us turn to the conditions for a Newtonian system to be a Lagrangian one. This is the well-known inverse problem formulated for time-dependent mechamcs. We will investigate it as the particular inverse problem for dynamic systems on fibre bundles [57]. The equation (4.9.6) is the kernel of the second order differential Euler-Lagrange type operator £:J2Q^

V'Q,

£ = mik(Zk - qktt)dqx.

(4.9.21)

4.9. NEWTONIAN

203

SYSTEMS

One can write such an operator as a 2-form £ = eiei A dt,

(4.9.22)

where 6l are contact forms (1.3.6). Obviously, the class of Euler-Lagrange type operators includes the Euler-Lagrange operators (4.8.6). Thus, the inverse problem reduces to the conditions for the Euler-Lagrange type operators to be the EulerLagrange ones. Given a fibre bundle Q —» R, we have the so called variationaJ sequence 0^R^O°(Q)^D°'1(J1Q)

^Qll(J2Q)

6

-^0%l(J3Q)

^••■,

(4.9.23)

where O0,l(J1Q) is the space of horizontal densities Cdt on JlQ, while D1,l(J2Q) is the space of 2-forms £id8' A dt on J2Q, and so on [46, 57, 179]. In particular, the elements of D0,1(JlQ) are the Lagrangians Cdt, while those of Dl,1(J2Q) are the Euler-Lagrange-type operators written as 2-forms (4.9.22). The key point lies in the fact that the variationaJ sequence (4.9.23) is a complex. It implies that S o dH = 0,

(4.9.24)

<52 o 8 = 0,

(4.9.25)

where dH(f)

= dtfdt,

feO°{JlQ),

is the horizontal exterior

differential,

6(Cdt) = (di - dtdDLP A dt is the Euler-Lagrange

(4.9.26)

map (or the variationai operator), and

^ ( ^ A dt) = [(2d; - djdj + dldf^P j

A 6' +

{d)8i + dlSj - 2dtdf£i)6\ A 9 + {df£{ - df^)^

A 6*} A dt = 0

is the Helmholtz-Sonin map. A glance at the variationai operator (4.9.26) shows that the image Im<5 of the variationai operator (4.9.26) consists of the Euler-Lagrange operators, and Im<5 C Ker<52 in accordance with the equality (4.9.25). It follows that 62(£) = 0

(4.9.27)

204

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

is the necessary condition for an Euler-Lagrange type operator £ to be an EulerLagrange one £ = 6(L)

(4.9.28)

for some Lagrangian L. The condition (4.9.27) takes the coordinate form

d3£, - di£} + fkiBtSj - Sfa) = 0,

(4.9.29a)

d)£i + d\£j - 2dtdf£x = 0,

(4.9.29b)

df£i - df£i = 0.

(4.9.29c)

The obstruction preventing the condition (4.9.27) from implying the equality (4.9.28) is topological [46, 179). If Q = R m + 1 -> R (in particular, locally), the variational sequence (4.9.23) is exact, i.e., Im6 = Ker6 2 . and the equality (4.9.27) is also a sufficient condition for an Euler-Lagrange type operator £ to be an Euler-Lagrange one. Remark 4.9.5. A glance at the equality (4.9.24) shows that the Lagrangian L in the equality (4.9.28) is defined modulo the variationally trivial Lagrangians

L = ckfdt

(4.9.30)

where / is a function on Q. • Applying the condition (4.9.27) to the operator (4.9.21), we restate the wellknown Helmholtz conditions (see [34, 78, 81, 143, 156] and references therein). It is readily observed, that the condition (4.9.29c) is satisfied since the mass tensor is symmetric. The condition (4.9.29b) holds due to the equality (4.9.4) and the property (4.9.5). Thus, it suffices to verify the condition (4.9.29a) for a Newtonian system to be a Lagrangian one. Example 4.9.6. Let £ be a free motion equation which takes the form (4.6.9) with respect to a reference frame (i.g 1 ), and let m be a mass tensor which depends only on the velocity coordinates q\. Such a mass tensor may exist in accordance with affine coordinate transformations (4.6.10) which maintain the equation (4.6.9). Then £ and m make up a Newtonian system. This system is a Lagrangian one if

4.9. NEWTONIAN

205

SYSTEMS

m is constant with respect to the above-mentioned reference frame (t,if). Relative to arbitrary coordinates on a configuration space Q, the corresponding Lagrangian takes the form (4.9.19), where T is the connection associated with the reference frame {t,qi). • Example 4.9.7. Let us consider the 1-dimensional motion of a point mass mo subject to friction. It is described by the equation 77io«jH =

fc>0,

-kqt,

(4.9.31)

2

on the configuration space R —» K, coordinated by [t, q). This mechanical system is characterized by the mass function m = mo and the holonomic connection

f = 8 t + q td q

k

m

qtfr

(4.9.32)

but it is neither a Newtonian nor a Lagrangian system. The conditions (4.9.29a) and (4.9.29c) are satisfied for an arbitrary mass function m(t,q,qt), whereas the conditions (4.9.4) and (4.9.29b) take the form —kqtd^m — km + dtm + qtdqm = 0. The mass function m = const, equation (4.9.33) has a solution ' Jfc m = mo exp — t [mo

fails to satisfy this relation.

.

(4.9.33) Nevertheless, the

(4.9.34)

The mechanical system characterized by the mass function (4.9.34) and the holo­ nomic connection (4.9.32) is both a Newtonian and a Lagrangian system with the Havas Lagrangian k

, 1 [ 12 (4.9.35) C = - m o exp — t q, 2 [m0 J [151]. The corresponding Lagrange equations are equivalent to the equation of motion (4.9.31). • Example 4.9.8. [143]. The dynamic equation Ztt - Vt = 0,

Vtt - y = 0

on the configuration space R 3 —► K is not equivalent to any system of Lagrange equations. •

206

CHAPTER

4.10

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

Holonomic constraints

Let (Q, m, £) be a Newtonian system, where m is a Riemannian metric in the vertical tangent bundle VQJ1Q —» JlQ. This Section is devoted to the restriction of (<5, m, £) to a closed imbedded fibred submanifold iN : N «-» Q of the configuration bundle Q —» R, which is treated as a holonomic constraint. We refer the reader to [24] and references therein for the geometric description of holonomic constraints in conservative mechanics. With a holonomic constraint N C Q, we have the following imbedding diagrams y 2 /v - >

VN

I N

I

<-. VQ

I «-+ Q . \ / R

J2<2

I

I

I

N

^-> Q \ / R

Furthermore, there exists an open tubular neighbourhood U of the submanifold N, which is a fibred manifold over TV [109]. Therefore, one can provide the configuration space Q with the atlas of bundle coordinates (t, aT, ql) such that
ql |JIJV= o,

9M \J*N= 0.

We will continue to use the notation qx for the whole coordinate collection (t,
—> J*N.

Its components are

nrs = mTS \jiN . Since d^,

= d^rrir,

\jiN,

the Riemannian metric n satisfies the symmetry condition (4.9.5). Using the Riemannian metric m in VQJ1Q —> J ' Q , we can define the orthocomplement V -» JlN of the subbundle VNJlN C VQJ1Q \JIN and the corresponding splitting VQJ'Q

\JIN=

VNJ'N

© V

(4.10.1)

4.10. HOLONOMIC

CONSTRAINTS

207

together with the canonical projections Prj : VQJlQ \fiN-> t

P^i(d r)=dtr,

VNJ'N,

ptx(flf) = n"msidl

(4.10.2)

and pr 2 : VQJ'Q

| J I A f - V,

pr 2 (9j) = d\ - nT'm3ld'r = tft.

pr 2 (5*) = 0,

Since the restriction J2Q \j\N—> JYN is an affine bundle modelled over the vector bundle VQJ1Q \JIN, the splitting (4.10.1) defines the corresponding decomposition J2 Q

U A T = J2N

0

V.

J*N

Then the dynamic equation £ : JlQ -» J2Q of our Newtonian system splits in the following way: (4.10.3)

f | j ' w = (IN + r, where &v ■ JlN

~* J2N,

CN =

(t;T+nr'msie)\jiN,

(4.10.4)

and

r = Z% : JlN

— V.

(4.10.5)

It follows that the dynamic equation f on the configuration space Q induces the dynamic equation £N (4.10.3) on the holonomic constraint N. In an equivalent way, the induced dynamic equation £^ is characterized by the relation KTSCN = ("V.£' + "hi?)

Ij'AT ■

(4.10.6)

PROPOSITION 4.10.1. Let (<2,m,£) be a Newtonian system on a configuration space Q, and N C Q a holonomic constraint. Then (Af, n,£jv) is a Newtonian system. □ Proof. The proof of the compatibility condition (4.9.4) for £N and n follows from direct computation. QED

208

CHAPTER 4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

Let 7 ? be the dynamic connection (4.3.10) associated with a dynamic equation £ of the Newtonian system in question. It can be represented by the tangent valued form (1.4.4) which reads 7{

: JlQ -» T'J'Q

® VQJlQ,

7« = (dq\ - 7 i V ) ® 9, + (d< - 7XV) ® &• Let us consider the map 7« : J*N

«-. J ' Q

IJIJV ^ T ' J ' Q

® VQJ1*? | 7 , W

—>

rv'jv ® vw1^, where the dual morphism (1.1.3) ( . / % ) • : T V ' Q | j . w - > TV'TV and the morphism pr, (4.10.2) are utilized. We obtain

%=

{dort-%dt-r3do°)®dT,

lo = (7o + ™r'"i5.7o) \j*N,

7p = (7p + nTSmstjlp)

(4.10.7) \jiN

It follows that the map 7^ is a connection on the affine jet bundle J N —» N. PROPOSITION 4.10.2. The dynamic connection 7^ (4.10.7), induced on the jet bundle JlN —► N of the holonomic constraint N by the dynamic connection 7^, defines a dynamic equation on N which is precisely £N (4.10.4), induced on N by the dynamic equation {, □ Proof. The proof is straightforward.

QED

It is readily observed that the dynamic connection 75 differs in a torsion from the symmetric dynamic connection -yN, associated with the dynamic equation £#. We have the relation

r, = \Ti+yNi

% = -\TZ + INI 77 = -d^n^m^e

~

nrhdlmhtC-

(4.10.8)

Let now a Newtonian system (Q, m,£) be a Lagrangian system for a regular Lagrangian L on the velocity phase space J*Q. The pull-back Ls = i*NL of L by

4.10. HOLONOMIC

CONSTRAINTS

209

the imbedding JliN is a Lagrangian on the holonomic constraint N. We come to the following assertion. PROPOSITION 4.10.3. The dynamic equation £LN, associated with the induced Lagrangian L^ on N, coincides with the dynamic equation £/,# (4.10.4) induced on N by the dynamic equation £L, associated with the Lagrangian L. □ Proof. We obtain the relation ( m r . a + mrt&) \jxN= -dtdlrLN

- a°td,drLN + d'rLN = n r 3 £l„

(see (4.9.2)), where nTS = mrs \jiN=

dffiLrf.

This is precisely the relation (4.10.6).

QED

It is readily observed that the Lagrange equations for a Lagrangian L^ on the holonomic constraint N are locally equivalent to those of the local Lagrangian L + Xiy' on Q, where X{ are the Lagrange multipliers. Consequently, the dynamic equation £LM = £LH describes a motion on an ideal holonomic constraint N. This fact also remains true for an arbitrary Newtonian system. Indeed, the splitting (4.10.3) shows that one can think of the dynamic equation £# as being the dynamic equation £ at the points of N plus the additional acceleration —r (4.10.5), treated as a constraint reaction acceleration. Then, bearing in mind the isomorphism (4.1.8), it is readily observed that this acceleration is orthogonal to the holonomic constraint N with respect to the mass metric m, i.e., m(u, s) = m^v!1^

— m/ lr ii''n rs m SJ £' 7 = 0

for any local vertical tangent vector field uhdh on N —» K. Thus, we generalize the D'Alembert principle (the principle of virtual displace­ ments) for an arbitrary Newtonian system. In accordance with this principle, a constraint reaction acceleration of a Newtonian system on an ideal holonomic con­ straint is orthogonal to the constraint manifold with respect to the mass metric of this system. One can say more when (Q,m,£) is a standard Newtonian system, i.e., m is a Riemannian metric in the vertical tangent bundle VQ —> Q. This metric induces

CHAPTER

210

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

the Riemannian metric n on the vertical tangent bundle VN —► N. Its components are TV, = m r s \N .

We have the splitting TQ\N=TN®V N

with the canonical projections

prt(ar) = ar,

Wi(9i) =

pr 2 (d r ) = 0,

nr'msidr,

pr 2 (9i) = d\ - rfm^dr

= du

and the corresponding splitting of the first order jet manifold JXQ\N=

JlN®V. N

In particular, any reference frame T on a configuration space Q defines a reference frame r \ on the holonomic constraint TV in accordance with the decomposition

r

\N=

nrsr%

rN + av, = ( m r s r S + TTViP) I AT .

(4.10.9)

Moreover, the expression (4.10.8) shows that, in the case of a standard Newtonian system, the dynamic connection 7 4 (4.10.7) is precisely the dynamic connection y^N, associated with the dynamic equation fa on the holonomic constraint N. Example 4.10.1. Let (Q,m,£) be a Newtonian system for a free motion equation £. Then, fa fails to be a free motion equation on a holonomic constraint N in general. In particular, if L is a Lagrangian (4.9.19), its restriction to a holonomic. constraint N reads LN = 2nT-(°l ~ rN)(o! ~ Tsw) + where I \ is the induced frame (4.10.9). •

^n{ov,ov),

4.11. NON-HOLONOMIC

4.11

CONSTRAINTS

211

Non-holonomic constraints

Here we are concerned with the geometric theory of non-holonomic constraints in time-dependent mechanics. We refer the reader to [25, 30, 112, 113, 124, 183, 184] and references therein for the geometric description of non-holonomic constraints in conservative mechanics. The key problem is that the method of the Lagrange multipliers is not appropriate to non-holonomic constraints in general (see [7, 25, 112, 114]). Let the jet manifold JlQ be a velocity phase space of time-dependent mechanics on a configuration bundle Q —» R. The most general non-holonomic constraints considered in the literature are given by codistributions S or, accordingly, by distri­ butions Ann(S) on the jet manifold JlQ [55, 132]. In conservative mechanics on a configuration space M, this is the case of a codistribution on TM [183]. Submanifolds of the jet manifold JlQ [88, 106] and distributions on a configuration space Q [114] can also be seen as non-holonomic constraints. In connection with non-holonomic constraints in time-dependent mechanics, one usually studies the following problem. Given a configuration space Q, let £ be a dynamic equation on Q, and S a codistribution on JlQ whose annihilator Ann (S) is treated as a non-holonomic constraint. Similarly to the case of holonomic con­ straints, the goal is a decomposition

£ = I + r,

(4.11.1)

where £ is a dynamic equation obeying the condition |cAnn(S). Then one can think of the dynamic equation £ as describing a mechanical system subject to the non-holonomic constraint S, while (— r) is said to be the constraint reaction acceleration. Let us assume that the codistribution S has dimension n and is locally spanned by the 1-forms s" = sa0dt + s°dq' + s°dq\ on the jet manifold JlQ. Then a dynamic equation £ is compatible with the nonholonomic constraint S if

Sa(£)=Z]sa = Sao + S°ql + S°e = 0.

212

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

This equation is algebraically solvable for n components of £ if and only if the nxm matrix s ° ( g \ g | ) has everywhere maximal rank n < m. Therefore, we will restrict our consideration to the non-holonomic constraints, called admissible, such that dimS = dim?(S), where v* is the vertical endomorphism (4.1.14) Example 4.11.1. Note that holonomic constraints can also be seen as the nonholonomic ones which, however, are not admissible. • If a non-holonomic constraint is admissible, there exists a local m x n matrix sJ,(gA,
4*J =
(4.11.2)

Then, the local decomposition (4.11.1) of a dynamic equation £ can be written in the form (4.11.3)

g = g + JJafjQ. The global decomposition (4.11.1) exists by virtue of the following lemma. LEMMA 4.11.1. [55]. The intersection F=

J2QnAnn(S)

is an affine bundle over J ' Q , modelled over the vector bundle F = VQJlQn

Ann (S).

n Proof. The intersection F consists of the vertical vectors vxd\ 6 VQJ1Q the conditions

which fulfill

*?(«*,«*)«* = 0. Since the non-holonomic constraint S is admissible, every fibre of F is of dimension m-n, i.e., F is a vector bundle, while F is an affine bundle. QED Since the intersection J2Q n Ann (S) —> J1Q is an affine bundle, it has always a global section f by virtue of Theorem 1.1.2.

4.11. NON-HOLONOMIC

213

CONSTRAINTS

To construct the global decomposition (4.11.1), one should perform a splitting of the vertical tangent bundle V Q Jl Q = F © V

(4.11.4)

in order to obtain the corresponding splitting of the second order jet manifold J2Q = F © V

(4.11.5)

which is modelled over (4.11.4). Here, V —» J ' Q should be interpreted as the bundle of possible constraint reaction accelerations. If an admissible non-holonomic constraint S is of dimension n = m, a dynamic equation £ is decomposed in a unique fashion. If n < m, the decomposition (4.11.1) is indeterminate. Different variants of this decomposition lead to different con­ straint reaction accelerations which, from the physical viewpoint, characterize dif­ ferent types of non-holonomic constraints. Let us turn now to some important particular cases of non-holonomic constraints. Example 4.11.2. Let N be a closed imbedded submanifold of the velocity phase space JlQ, defined locally by the equations

/a(g\
a = 1 , . . . ,n < m.

(4.11.6)

One can treat AT as a non-holonomic constraint at points N C J ' Q , given by the codistribution S = Ann(TN) on JlQ \s- This codistribution is locally spanned by the 1-forms

sa = dfa = dtfadt + d]}adq> + aj/°d^. The non-holonomic constraint ,/V is admissible if and only if the matrix (9|/°) is of maximal rank n. It follows that N is a fibred submanifold of the affine jet bundle JiQ —» Q. Moreover, it is possible to separate locally some velocities <£, a = 1 , . . . , n, expressed as functions of the remaining coordinates (qx, qf , • • •,
{Q ,Qt

(4.11.7)

i • • •. It ) = 0-

Then we have sg =

-dtga,

«? = ~dwa,

S? = 6f - dig",

(4.11.8)

214

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

where dlbga = 0 for all indices a, b = 1 , . . . , n. Now, applying the above procedure to the codistribution (4.11.8), one can obtain a dynamic equation on a submanifold N, compatible with the constraint Ann (TN). For instance, one can use the particular solution s\ = 6la of the equations (4.11.2), where the matrix s? is given by the expression (4.11.8) [106]. Then the (local) dynamic equation £ (4.11.3), compatible with the constraint (4.11.7), reads

£ ,/a = e,

£° = {• _ s*(£) = dkgae + dtga + qfrg*.

(4.11.9)

• Example 4.11.3. A non-holonomic constraint N is said to be linear if it is an affine subbundle of the affine jet bundle JlQ —> Q, which is given by the local equations

r=/oV)+/f(
(4.11.10)

where the matrix / t a is of maximal rank. A linear non-holonomic constraint is always admissible. Since N is an affine subbundle of JlQ —> Q, it has a global section V which is a connection on the configuration bundle Q —> R, called the constraint reference frame. With this connection F, the constraints (4.11.10) take the form

f°(qxM - n = o.

(4.11.11)

We may say that the linear constraint is stationary with respect to the constraint reference frame T. Then, one can think of
Example 4.11.4. Let a configuration space Q admit a composite fibration Q —» E -> R, where 7TQE : Q — E

is a fibre bundle, and let (t, aT, ga) be coordinates on Q, compatible with this fibra­ tion. Given a connection B = dt ® (dt + Bada) + daT ® (dT + B?da)

(4.11.12)

4.11. NON-HOLONOMIC

CONSTRAINTS

215

on the fibre bundle Q —» E, we obtain the corresponding horizontal splitting (1.4.1) of the tangent bundle TQ. Restricted to the jet manifold JXQ C TQ, this splitting reads J1Q =

B(n'QEJ1Yi)®VEQ,

dt + c\dT + q?da = [(dt + Bada) + art{dT + BaTda)] +

[tf - Ba - o\B?\da, where 7TQE J l E is the pull-back of the affine jet bundle J ' E —> E onto Q. It is readily observed that N =

B(n'QSJlZ)

is an affine subbundle of the affine jet bundle JlQ equations

—► Q, defined locally by the

qf - oTtB°(qX) ~ Ba(qx) = 0 (cf. (4.11.7)). Then this subbundle yields a linear non-holonomic constraint, given by the codistribution S = Ann (TN) [162, 163]. This codistribution is locally spanned by the 1-forms sa = -{dtBa a

(dbB

+ o\dtB«)dt r

+a

a b tdbB T)dq

- (daBa + ortdsB*)do" + dqf -

-

(4.11.13)

B«do\.

With the connection (4.11.12), we also have the splitting (1.6.10a) of the vertical tangent bundle VQ of Q —> R and the corresponding splitting of the vertical tangent bundle VQJ1Q (see the canonical isomorphism (4.1.21)). The latter splitting reads V Q Jl Q = F 0 V , a[dl + qfdl = art(% + B^a)

+ (q? - B r V [ ) ^ .

(4.11.14)

It is readily observed that F |/v consists of vertical vectors which are the annihilators of the codistribution (4.11.13). The splitting (4.11.14) yields the corresponding splitting (4.11.5) of the second order jet manifold J2Q. Then, given a dynamic equation f on JlQ, we obtain the decomposition (4.11.1) which reads

f = c,

e=r - s°(o

216

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

(cf. (4.11.9)). • Now we consider Newtonian systems because they provide the vertical tangent bundle VQJ1Q with a non-degenerate fibre metric m. Let us assume that m is a Riemannian metric. With this metric, we immediately obtain the splitting (4.11.4), where V is the orthocomplement of F. Then the decomposition (4.11.3) takes the form

f = | i + matm"5fot(0.

(4.11.15)

where m,^ is the inverse matrix of mab = s's^mv [55] By definition, the decomposition (4.11.15) satisfies the generalized D'AIembert principle. The constraint reaction acceleration -r' =

-mabmlJs°sb(Z)

(4.11.16)

is orthogonal to every element of VQjiQnArm (S) with respect to the mass metric m. Since elements of VQ J'QflAnn (S) can be treated as the virtual accelerations relative to the non-holonomic constraint S, the constraint reaction acceleration (4.11.16) characterizes S as an ideai non-holonomic constraint. The Gauss principle is also fulfilled as follows. Given a dynamic equation £ and the above-mentioned fibre metric m, let us define a positive function G(w) on J^Q as G(w) = rh (£(ir?(uO) - U/,£(JT?(IU)) - w) ,

G(q\ q\, qltt) = mi}(q\
IMI =

^GH

is a norm of w 6 J2Q. PROPOSITION 4.11.2. [55]. Among all dynamic equations compatible with a non-holonomic constraint, the dynamic equation £ defined by the decomposition (4.11.15) is that of least norm. D

4.11. NON-HOLONOMIC

CONSTRAINTS

217

Proof. Let £ be another dynamic equation which takes its values into F. Then

Then we obtain

K l l - « a - f + € - C . € - € + € - 0 = llCll + S(C-C.C-0. i.e., |Cn > llltl-

QED

Now we will show that, in the case of non-degenerate Lagrangian system and linear non-holonomic constraints, the decomposition (4.11.15) satisfies the traditional D'Alembert principle. Let S be an admissible non-holonomic constraint on the velocity phase space JXQ, and L a regular Lagrangian on JlQ with a Riemannian mass metric ml} = nl}. Since this is a particular Newtonian system, we obtain the dynamic equation & = ft - f w ' ^ ^ g i + jjq* + «&]

(4.H.17)

which is compatible with the constraint S, treated as an ideal non-holonomic constraint. This is the system of Lagrange equations in the presence of a non-Lagrangian external force

F, = -rH^Afe) (see the equation (4.12.9) below). It is a constraint reaction force. Let us consider the energy-momenturn conservation law in the presence of this force. It reads LpL - frft =

-d(Tr,

where LfL is the Lie derivative of the Lagrangian L along a reference frame T and Tr is the energy function relative to this reference frame (see (4.12.15), (4.12.16), and (4.12.19) below). In particular, if a non-holonomic constraint is linear and T is a constraint reference frame (see Example 4.11.3), the constraint reaction force does not contribute to the energy conservation law. It follows that, in this case, the standard D'Alembert principle holds, while the equation (4.11.17) describes a motion in the presence of an ideal non-holonomic constraint in the sense of this D'Alembert principle.

CHAPTER

218

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

The constrained motion equation (4.11.17) on a configuration space Q is neither a system of Lagrange equations nor a dynamic equation of a Newtonian system. In Section 5.11, we will show that this can be seen as Hamilton equations in the framework of the Hamiltonian formalism extended to the configuration space VQ.

4.12

Lagrangian conservation laws

Given a Lagrangian system ( J ' Q , L), its integrals of motion can be found if a La­ grangian L possesses symmetries. In time-dependent mechanics, these integrals of motion should be covariant under time-dependent transformations. For instance, the canonical energy function (4.12.1)

EL = mqxt - £

fails to be such an integral of motion. There are different approaches in order to obtain the conservation laws in Lagrangian dynamics (see [1, 29, 51, 52, 57, 160] for the geometric methods). As in field theory, we will use the first variational formula of the calculus of variations [57, 160]. Let L be a Lagrangian (4.8.1) on the velocity phase space JlQ and u = uldt + u'dx,

ul = 0,1,

(4.12.2)

a projectable vector field (4.8.2) on the configuration bundle Q —► R. By virtue of Theorem 1.2.1, this vector field may be treated as the generator of a 1-parameter group of local automorphisms, called gauge transformations, of the fibre bundle Q —> R. In particular, if ul = 0, the vertical vector field (4.12.2) is the generator of vertical automorphisms of the fibre bundle Q —> R projected over the identity transformation of the base R. If ul = 1, the vector field u (4.12.2) is projected over the standard vector field dt on the base R, which is the generator of the group of translations of R. Let us apply the first variational formula (4.8.4) to a Lagrangian L and to a vector field u (4.12.2). On the shell (4.8.7), the identity (4.8.4) is brought into the weak identity u\dC »

-dtl,

(u'dt + u% + dtu'dl)C w -(f,(7r,(u'g; - u') - u'C),

(4.12.3)

4.12. LAGRANGIAN

CONSERVATION

LAWS

219

where, by analogy with field theory, T = -u\HL

= ■Ki{utq\ - u{) - ulC

(4.12.4)

is said to be the current along the vector field u. The symbol " ~ " stands throughout for weak equalities fulfilled on-shell. If the Lie derivative L^L of a Lagrangian L along a vector field u vanishes, i.e., L is invariant under the 1-parameter group generated by u, we have the weak conservation law 0 « -d,[7T,(u'qJ - u*) - «'£].

(4.12.5)

It is brought into the differential conservation law 0 ~ —T[( W > ° c)(u'3[c' - u ' o c ) - u'C o c] at on solutions c of the Lagrange equations (4.8.8). A glance at this expression shows that, in time-dependent mechanics, the conserved current (4.12.4) plays the role of an integral of motion. Remark 4.12.1. Let L be a Lagrangian on the velocity phase space J ' Q . Every integral of motion (4.12.4) defines a non-holonomic constraint dT = 0 on JlQ such that any Lagrangian connection fx, for L is a dynamic equation compatible with this constraint. • Remark 4.12.2. The conservation law (4.12.5) is broken if a Lagrangian depends on variable parameters which do not live in the dynamic shell (4.8.7) (see Section 5.9). In order to obtain a conservation law in the presence of such parameters, let us consider a bundle product (4.12.6)

Qtot = Q x Y

of a configuration bundle Q —> R with coordinates [t, q') and a fibre bundle Y —> R with coordinates (t, yA), whose sections are the variable parameters which take the background values

yB =
„f = d,4>B{x).

A Lagrangian L is defined on the total velocity phase space ./'Qtot-

220

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

Let u be a projectable vector field (4.12.2) on Qtot which projects also onto Y because the transformations of parameters do not depend on the dynamic variables. This vector field takes the coordinate form (4.12.7)

u = uldt + uA(t, yB)dA + u'{t,yB,
Substitution of the vector field (4.12.7) into the first variational formula (4.8.4) leads to the first variationai formula in the presence of parameters

[u'd, + uAdA + u'd, + dtuAdA + dtu'dflC = (uA - ytul)dAC +

(4.12.8)

vAdt(v.A - j / t V ) + (W - qlu^Si - dt[«i(uWt ~ «*) - «*£]• Then the weak identity [u% + uAdA + u'd, + dtUAdlA + dtUl%]C » {uA - VS)9A^ A

A l

■nAdt(u - y u ) - MM^Ql

+

u

~ "') - ' £ ]

is fulfilled on the dynamic shell (4.8.7). If a Lagrangian L is invariant under gauge transformations of the product (4.12.6) whose generator is the vector field u (4.12.7), we obtain the weaic conservation Jaw in the presence of parameters (uA - y^u{)dAC + nAdt(uA

- yAul) « dt{K,{ulq't - u') - u*£].

• Remark 4.12.3. The first variational formula (4.8.4) can also be utilized when a Lagrangian possesses symmetries, but a motion equation is seen as Lagrange equations plus additional non-Lagrangian external forces and reads [di-dtdf)C

+ Fi{t,qi,
(4.12.9)

Let us substitute £ = —F, from this equality in the first variational formula (4.8.4) and assume that the Lie derivative of the Lagrangian L along a vector field u van­ ishes. Then, we have the conservation law (it1 - q\)F, g - d t M u ' q j - u') - u'£].

•

(4.12.10)

4.12. LAGRANGIAN

CONSERVATION

LAWS

221

It is easy to see that the weak identity (4.12.3) is linear in the vector field u. Therefore, one can consider superposition of the weak identities (4.12.3) associated with different vector fields. For instance, if u and v! are projectable vector fields (4.12.2), projected onto the standard vector field dt on R, the difference of the corresponding weak identities (4.12.3) results in the weak identity (4.12.3) associated with the vertical vector field u — v!. Conversely, every vector field u (4.12.2), projected onto dt, can be written as the sum

u = Y + ti

(4.12.11)

of some reference frame

r = dt + rdt

(4.12.12)

and a vertical vector field $ on Q. It follows that the weak identity (4.12.3) associated with an arbitrary vector field u (4.12.2) can be represented as the superposition of those associated with a reference frame V (4.12.12) and some vertical vector field d. If u = d is a vertical field, the weak identity (4.12.3) reads

(^di +

dt^dDC^dtin^).

If the Lie derivative of L along -d vanishes, we obtain from (4.12.5) the weak con­ servation law 0w

Mviti')

(4.12.13)

and the integral of motion 1 = -TTjg.

(4.12.14)

By analogy with field theory, (4.12.13) is called the Noether conservation law for the Noether current (4.12.14). Example 4.12.4. Let assume that, given a trivialization Q = R x M in coordinates (t, q'), a Lagrangian L is independent of a coordinate g1. Then the Lie derivative of L along the vertical vector field d = d\ equals zero, and we have the conserved Noether current (4.12.14) which reduces to the momentum X = —TTI.. With respect to arbitrary coordinates (t,^'), this conserved Noether current takes the form 1 =

-

^

222

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

In particular, the free motion Lagrangian admits m conserved Noether currents. • In the case of a reference frame T (4.12.12), where u' = 1, the weak identity (4.12.3) reads

{dt + rdi + d(raj)£ « -Mniqt - V) - c),

(4.12.15)

X r = 7r,(gt' - P ) - C

(4.12.16)

where

is said to be the energy function relative to the reference frame T [51, 57, 161). With respect to the coordinates adapted to the reference frame T, the weak identity (4.12.15) takes the form of the familiar energy conservation law (4.12.17)

dtC ss -dt(niq't - £),

and Xr coincides with the canonical energy function Ei (4.12.1). It follows that the canonical energy function Ei is not a unique existent energy function. Each reference frame defines an energy function. Example 4.12.5. Let us consider a free motion on a configuration space Q. It is described by the Lagrangian £ = 2 m 'j9t9t-

m = const.,

(4.12.18)

written with respect to a reference frame (£,
x r = m(q\ -r)-C

= ImjWM-n(qi

- P).

Now we generalize this example for a motion described by the equation (4.12.9), where £ is the free motion Lagrangian (4.12.18) and F is an external force. The Lie derivative of the Lagrangian (4.12.18) along the reference frame T vanishes, and we have the weak equality (4.12.10) which reads q\.Ft «

dtlr,

(4.12.19)

4.12. LAGRANGIAN

CONSERVATION

LAWS

223

where q'r is the relative velocity. This is the well-known physical law whose left-hand side is the power of an external force. • Example 4.12.6. Let us consider a 1-dimensional motion of a point mass mo subject to friction on the configuration space R 2 —► R, coordinated by (t,q) (see Example 4.9.7). It is described by the dynamic equation (4.9.31) which is the system of Lagrange equations for the Lagrangian L (4.9.35). It is readily observed that the Lie derivative (4.8.3) of this Lagrangian along the vector field

T=

d -~qd 2 tmo q

(4.12.20)

vanishes. Hence, we have the conserved energy function (4.12.16) with respect to the reference frame T (4.12.20). This energy function reads _ 1 [ k , , k . 1 ., mk? o 2 T = ^wi 0 exp — ' Qt(Qt + —Q) = ^rnqr - — j g , 2 [m0 J mo 2 8mo where m is the mass function (4.9.34). • Since any vector field u (4.12.2) can be represented as the sum (4.12.11) of a reference frame T (4.12.12) and a vertical vector field $, each current (4.12.4) along a vector field u (4.12.2) is the sum of a Noether current (4.12.14) along the vertical vector field t? and the energy function (4.12.16) relative to the reference frame T [51, 57, 161]. Conversely, energy functions relative to different reference frames V and T' differ from each other in the Noether current along the vertical vector field

r-r. In conclusion, we touch on gauge transformations with a generator u (4.12.2), which preserve the Euler-Lagrange operator SL, but not necessarily a Lagrangian L. They are called generalized invariant transformations [57, 105, 174]. We use the formula Lj2 u £ t — £L_L [54, 57], where J2u = u'dt + u% + dtu'dj + dtdttfd? is the second order jet prolongation of a vector field (4.12.2). The following two assertions are immediate corollaries of this formula.

224

CHAPTER

4. LAGRANGIAN

TIME-DEPENDENT

MECHANICS

COROLLARY 4.12.1. If a Lagrangian L is invariant under a 1-parameter gauge group whose generator is a vector field u (4.12.2), so is the associated Euler-Lagrange operator EL. D □ COROLLARY 4.12.2. If an Euler-Lagrange operator £/,, associated with a Lagran­ gian L, is invariant under a 1-parameter gauge group whose generator is a vector field u (4.12.2), the Lie derivative h^L is a variationally trivial Lagrangian (4.9.30). D □ Let us consider conservation laws in the case of generalized invariant transfor­ mations. Let L be a Lagrangian and EL the associated Euler-Lagrange operator. Let u be a vector field (4.12.2) which is the generator of a local 1-parameter group of gauge transformations such that (4.12.21)

L/2u£z, = 0. Then we have the equality LuL = d,/, where / is a function on Q. In this case, the weak identity (4.12.3) reads

dtf^dti^iiS-qb+u'C), and we obtain the weak equality OwdeMu' -
(4.12.22)

Example 4.12.7. Let L be the free motion Lagrangian (4.12.18). The correspond­ ing Euler-Lagrange operator EL = -rriijqltdq' is invariant under the Galilei transformations with the generator ui = vit + ai,

v' = const.,

a1 = const.,

(4.12.23)

(see (4.6.10)). At the same time, the Lie derivative of the free motion Lagrangian (4.12.18) along the vector field (4.12.23) does not vanish, and we have LjjL = mijVl(ft = d t ( m l ; v V + c),

c = const.,

4.12. LAGRANGIAN

CONSERVATION

LAWS

225

[148]. Then the weak equality (4.12.22) shows that (q\t — q%) is a constant of motion.

• Remark 4.12.8. The invariance condition (4.12.21) is generalized as L^u^L ~ 0

if one deals with symmetry transformations of the differential equation Si = 0 [93, 94]. For instance, the Galilei transformations (4.6.10) are the symmetry trans­ formations of the free motion equation. ••

Chapter 5 Hamiltonian time-dependent mechanics This Chapter is devoted to the Hamiltonian formulation of time-dependent mechan­ ics with respect to an arbitrary reference frame. Let us recall that a configuration bundle Q —> R of time-dependent mechanics is isomorphic to the product R x M, where M is a typical fibre of Q, but this isomorphism is not canonical in general. Different trivializations Q = R x M correspond to different reference frames. For this reason, many constructions of Hamiltonian conservative mechanics on a symplectic manifold Z and its extension to the product I x Z can not be applied to mechanical systems which are subject to time-dependent transformations, including canonical transformations and reference-frame transformations. Let us summarize the main peculiarities of Hamiltonian time-dependent mechan­ ics. • A momentum phase space of time-dependent mechanics is provided with the canonical degenerate Poisson structure. However, Hamiltonian time-dependent mechanics does not reduce to a Poisson Hamiltonian system, since a Hamilto­ nian on a momentum phase space of time-dependent mechanics is not a scalar function under time-dependent transformations (see Remark 5.1.1 below). • As a consequence, the evolution equation of time-dependent mechanics is not expressed in terms of a Poisson bracket, and integrals of motion cannot be defined as functions in involution with a Hamiltonian. For the same reason, the familiar procedure of describing constraint Hamiltonian systems in Section 227

228

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

3.6 cannot be applied to time-dependent mechanics. • Hamiltonian and Lagrangian formulations of time-dependent mechanics are equivalent in the case of hyperregular Lagrangians. A degenerate Lagrangian requires a set of associated Hamiltonians in order to exhaust all solutions of the Lagrange equations. We follow the notation of the previous Chapter, where Q —» K is a fibre bundle (4.0.1), coordinated by (t, ql), whose typical fibre M is an m-dimensional manifold, while its base R is equipped with the Cartesian coordinate t possessing the transition functions t' = t+ const.

5.1

Canonical Poisson structure

As was mentioned, the momentum phase space of time-dependent mechanics is the vertical cotangent bundle (5.1.1)

7rn : V'Q - Q

of a configuration bundle Q —> R. This is the particular Legendre bundle II (2.9.7) over a fibre bundle Q —> X when X = R. The vertical cotangent bundle (5.1.1) is equipped with the coordinates (£, X when X = R. The cotangent bundle T'Q admits the canonical Liouville form 3 = pdt + p,dq'

(5.1.2)

and the canonical symplectic form dE = dp A dt 4- dpi A dq'.

(5.1.3)

The corresponding Poisson bracket on the vector space D°(T'Q) T'Q reads

{/, 9} = Vfdtg - Vgdtf + d'fdl9 -

d'gdj.

of functions on

(5.1.4)

5.1. CANONICAL

POISSON

STRUCTURE

229

Let us consider the subspace of D°(T'Q) which comprises the pull-backs £*/ on T'Q of functions / on the vertical cotangent bundle V'Q by the canonical projection T'Q —> V ' Q (1.1.7). It is easily seen that this subspace is closed under the Poisson bracket (5.1.4). By virtue of Proposition 2.3.1, there exists the canonical Poisson structure

difdig-digdJ

{!,9}v =

(5.1.5)

on momentum phase space V'Q induced by (5.1.4), i.e.,

C{f,9}v = {Cf,Cg}The corresponding Poisson bivector field

w{df,dg) =

{f,g}v

on V'Q —> R is vertical with respect to the fibration V'Q —► M, and reads w'J = 0,

wi} - 0,

W*j = 1.

(5.1.6)

A glance at this expression shows that the holonomic coordinates of V'Q are canon­ ical for the Poisson structure (5.1.5). Since the rank of the bivector field w (5.1.6) is constant, the Poisson structure (5.1.5) is regular. This Poisson structure is obviously degenerate. Given the Poisson bracket (5.1.5), the Hamiltonian vector field ■Of for a function / on the momentum phase space V'Q is defined by the relation

{f,9h = 4f\dg,

VP 6 D°(V'Q).

It is the vertical vector field

0/ = #fdj

- d,fd'

(5.1.7)

on the fibre bundle V'Q —► R. Hence, the characteristic distribution of the Poisson structure (5.1.5) is precisely the vertical tangent bundle VV'Q C TV'Q of the fibre bundle V'Q — R. In accordance with Theorem 2.3.3, the Poisson structure (5.1.5) defines the symplectic foliation on the momentum phase space V'Q, which coincides with the fibra­ tion V'Q —* R. The symplectic forms on the fibres of V'Q -> R are the pull-backs Qt = dpi ^ dq'

230

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

of the canonical symplectic form on the typical fibre T'M of the fibre bundle V'Q —♦ R with respect to trivialization morphisms [27, 74, 161]. Given such a trivialization V'Q & R x

T'M,

(5.1.8)

the Poisson structure (5.1.5) is isomorphic to the direct product of the zero Poisson structure on R and the canonical symplectic structure on T'M (see Example 2.3.7). The Poisson structure (5.1.5) can be introduced in a different way [57, 161]. The polysymplectic form (2.9.9) on the Legendre bundle V'Q -* Q (5.1.1) reads A = dp, Adq{

Adt®dt,

and reduces to the canonical exterior 3-form n = dp, A dq' A dt.

(5.1.9)

This is the exterior differential of the canonical 2-form 0 = Pidq' A dt

(5.1.10)

on the momentum phase space V'Q. The canonical forms (5.1.9) and (5.1.10) are maintained under any holonomic coordinate transformation of V'Q. Given the canonical form (5.1.9), every function / on the momentum phase space V'Q determines the corresponding Hamiltonian vector field $ / (5.1.7) by the relation tf,Jft =

-dfAdt,

(5.1.11)

while the Poisson bracket (5.1.5) is defined by the condition {f,g}vdt

=

4g\4f\n.

R e m a r k 5.1.1. It should be emphasized that, though the momentum phase space V'Q of time-dependent mechanics is provided with the canonical Poisson structure, non-conservative mechanical systems are not reduced to the Poisson Hamiltonian systems of Section 3.2. Given a trivialization (5.1.8), i.e., a reference frame, one can write the Hamilton equations (5.2.18a) - (5.2.18b) (see below) as the equations of the Hamiltonian vector field d-n (5.1.7) for a Hamiltonian Ti with respect to the Poisson structure (5.1.5). Moreover, by very definition of the canonical form ft (5.1.9), the reference frame transformations are canonical transformations for the

5.2. HAMILTONIAN

CONNECTIONS

AND HAMILTONIAN

FORMS

231

Poisson structure (5.1.5) (see Section 5.3). However, a Hamiltonian H (see (5.2.10) below) is not a scalar function under reference frame transformations in general. Therefore these transformations are not the equivalence transformation of a Poisson Hamiltonian system. •

5.2

Hamiltonian connections and Hamiltonian forms

The dynamics of time-dependent mechanics on a phase phase space V*Q is described by first order dynamic equation. DEFINITION 5.2.1. Let 7

= dt + j% + < * #

(5.2.1)

be a connection on the fibre bundle V'Q —> M. In accordance with Definition 4.2.1, it defines the first order dynamic equation on the momentum phase space V'Q —► K written as

q\ = V,

Pu = 7i

with respect to the adapted coordinates {t,q\Pi,q't,Pti) on the jet manifold J1 V'Q. a Example 5.2.1. Let us consider the product (J = R x M - t I , coordinated by (t, ql). We have the canonical isomorphism

V'Q = R x TM, with the coordinates p< = Qj, and the corresponding isomorphism JW'Q^RXTT'M,

with the coordinates q\ = i- Every vector field u = u'dt + Uid'

(5.2.2)

232

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

on the cotangent bundle T'M defines the first order dynamic equation

7 = dt + u% + u,9* on V'Q (5.2.2). Its projection qx = u\

p, = ux

onto T'M is an autonomous first order dynamic equation in conservative mechanics (see Definition 3.1.1). • Let J2Q be the second order jet manifold of a configuration space Q —» K provided with the adapted coordinates (*,?', «*.««)• Recall that it coincides with the sesquiholonomic jet manifold T*Q. Following the general scheme of polysymplectic formalism in field theory (see Section 3.8), we say that a connection (5.2.1) on the Legendre bundle V'Q —» R is locally Hamiltonian if the exterior form 7jfi is closed, i.e.,

L,n = d(7jn) = o.

(5.2.3)

It is readily observed that a connection 7 (5.2.1) on V'Q —» R is locally Hamiltonian if and only if it obeys the conditions

a y - tvy = 0,

(5.2.4a)

drf] - djli = 0,

(5.2.4b)

dj-y' + a y = 0.

(5.2.4c)

Example 5.2.2. Every connection r : Q -» J'Q C TQ, (5.2.5)

r = 9, + Fdit

on the configuration bundle Q —» R gives rise to the locally Hamiltonian connection

r = KT = a( + r i a i -p,a J r , a j

(5.2.6)

(1.6.13) on the momentum phase space V'Q —► R such that

T\n = dHr, Hr = pjdtf -

pjFdt.

(5.2.7)

5.2. HAMILTONIAN

CONNECTIONS

AND HAMILTONIAN

233

FORMS

• Locally Hamiltonian connections constitute an affine space modelled over the linear space of vertical vector fields fl on the Legendre bundle V'Q —> R, which fulfill the same condition

d(tfjn) = 0

(5.2.8)

as (5.2.3). These vector fields are locally the Hamiltonian vector fields in accordance with the following lemma. LEMMA 5.2.2. Every closed form -y\fl on the fibre bundle V'Q -» R is exact.

□

Proof. Let us consider the decomposition

7=r+tf,

(5.2.9)

where T is a connection on Q —> R and f is its lift (5.2.6) onto the fibre bundle V'Q —> R, while -d is a vertical vector field on V'Q satisfying the relation (5.2.8). It is easily seen that

■d\n = aAdt, where a is a 1-form on V'Q. Using the properties of the De Rham cohomology of a manifold product, one can show that every closed 2-form a A dt on a manifold V'Q, diffeomorphic to the product R x T'M, is exact, and so is 7JO [57]. Moreover, in accordance with the relative Poincare lemma, we can write locally

ti\n = df Adt, where / is a local function on V'Q.

QBD

DEFINITION 5.2.3. An exterior 1-form H on the momentum phase space V'Q is said to be a locally Hamiltonian form if 7jn

= dH

for a connection 7 on the fibre bundle V'Q —> R. □

234

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

By virtue of Proposition 5.2.2, there is one-to-one correspondence between the locally Hamiltonian connections and locally Hamiltonian forms, considered through­ out modulo closed forms. In particular, the exterior form Hr (5.2.7) is a locally Hamiltonian form. DEFINITION 5.2.4. An exterior 1-form H on the momentum phase space V'Q is called a Hamiltonian form if it is the pull-back H = h*S - pidq* -

Hit

(5.2.10)

of the Liouville form S (5.1.2) on the homogeneous Legendre bundle T'Q by a section h of the fibre bundle

C : TQ -» V'Q.

(5.2.11)

a Remark 5.2.3. Note that, with respect to the universal unit system, a Hamiltonian form is physically dimensionless. • Remark 5.2.4. Given a trivialization Q = R x M, the Hamiltonian form (5.2.10) is the well-known integral invariant of Poincare-Cartan [6]. Therefore, the coefficient H of the horizontal part of the Hamiltonian form (5.2.10) is said to be a Hamiltonian. A glance at the expression (5.2.10) shows that Hamiltonians fail to be functions, i.e., H $. £)°(V'Q), but make up an affine space modelled over the linear space of functions on V'Q. * For instance, every connection T on a configuration bundle Q —» R is an affine section

p o r = -p.r of the fibre bundle (5.2.11) (see (1.3.12)), and defines the Hamiltonian form Hr (5.2.7) on the momentum phase space V'Q, where its Hamiltonian is H =

PiF.

It follows that any Hamiltonian form on the momentum phase space V'Q admits the splitting H = HT-

Hrdt = Pldql - {p,F + Hr)dt,

(5.2.12)

5.2. HAMILTONIAN

CONNECTIONS

AND HAMILTONIAN

235

FORMS

where T is a connection on Q -» R, and Hr is a real function on V'Q, called the Hamiltonian function. Since a connection T on a configuration space Q —» K is treated as a refer­ ence frame, i.e., as a kinematic object, one may say that, in accordance with the decomposition (5.2.12), every Hamiltonian form H is split into kinematic and dy­ namic parts, and so is its Hamiltonian H. In Section 5.7, we will show that the Hamiltonian function HT in the splitting (5.2.12) is the Hamiltonian energy func­ tion relative to the reference frame I\ Of course, the splitting (5.2.12) is not unique. Nevertheless, every Hamiltonian form H admits the canonical splitting (5.2.12) as follows. We mean by a Hamiltonian map any fibred morphism

*:VQfJlQ, 5t°$=

&(p),

Pe

V'Q,

(5.2.13)

over Q from the momentum phase space V'Q to the velocity phase space J1Q. Its composition with the canonical morphism A (4.1.4) yields the fibred morphism 9: V'Q

^TQ,

$ = dt + Vd{.

(5.2.14)

In particular, every connection T on a configuration bundle Q —> R defines the Hamiltonian map f = T o Trn : V'Q - Q -» JxQt

T = dt + r%. Conversely, every Hamiltonian map $ (5.2.13) yields the associated connection

r* = $ o 6 on the configuration bundle Q —» R, where 0 is the global zero section of the vertical cotangent bundle V'Q —» Q. For instance, we have

r~ = r PROPOSITION 5.2.5. Every Hamiltonian map (5.2.14) defines the associated Ha­ miltonian form H* = - $ J 0 = (A o $)*H = pidq1 -

Pi&dt

236

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

on V'Q, where 0 is the canonical 2-form (5.1.10). □ PROPOSITION 5.2.6. Every Hamiltonian form H on the momentum phase space V'Q defines the associated Hamiltonian map H : V'Q -> JlQ, q]oH = dxU.

□ Proof. Let h : V'Q --> T'Q be a section of the fibre bundle (5.2.11), which corresponds to the Hamiltonian form H, i.e., H = h'E. The vertical tangent map Vh of the morphism h defines the linear fibred morphism Vh : VV'Q

-> V'Q x T'Q Q

over V'Q. Therefore, it can be represented by the section Vh = dp, ® {dq* -

d'Hdt)

of the fibre bundle VV'Q

® T'Q -> V'Q.

After natural contractions, this section is brought into the section

Vh = {dq* - d'Hdt) ® ft of the pull-back

VQx(T'Q®VQ)->VQ, Q

and takes its values into the image of the jet manifold J1Q by its canonical imbedding (1.3.5) into T*Q®VQ. QED COROLLARY 5.2.7. Every Hamiltonian form H on the momentum phase space V'Q determines the associated connection TH = H o 0

5.2. HAMILTONIAN

CONNECTIONS

AND HAMILTONIAN

FORMS

237

on the configuration bundle Q —► R. It is called a Hamiltonian frame connectioi

□ In particular, we have

r Wr = r, where Hr is the Hamiltonian form (5.2.7) associated with the connection T on the fibre bundle Q -» R. COROLLARY 5.2.8. Every Hamiltonian form H (5.2.10) on the momentum phase space V'Q admits the canonical splitting H = HrH — Hdt. a Let us turn now to the relationship between the locally Hamiltonian forms and the Hamiltonian ones. PROPOSITION 5.2.9. Every locally Hamiltonian form on the momentum phase space V'Q is locally a Hamiltonian form. □ Proof. Given two locally Hamiltonian forms H*, and Hy on the momentum phase space V'Q, their difference (7 = fly — Hy,

Ar = ( 7 - V ) j n , is a 1-form on V'Q such that the 2-form a A dt is closed and, consequently, exact by virtue of Lemma 5.2.2. Hence, in accordance with the relative Poincare lemma, we have
= fdt + dg,

where / and g are local functions on V'Q. Then, we deduce from the splitting (5.2.9) that, in a neighbourhood of every point p € V'Q, a locally Hamiltonian form H~! can be written as H^ = Hr + fdt,

238

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

and coincides with the pull-back of the Liouville form E on the cotangent bundle T*Q by the local section (t,q',Pi) -» (t,q\Pi,P

= -P,F + f)

of the fibre bundle (5.2.11).

QED

A converse of Proposition 5.2.9 is the following. PROPOSITION 5.2.10. Given a Hamiltonian form H on the momentum phase space V'Q, there exists a unique connection JH on the Legendre bundle V'Q —» R, called the Hamiltonian connection, such that (5.2.15)

y„\n = dH. a

Proof. As in the polysymplectic case, let us introduce the first order Hamilton operator on the phase space V'Q, which is associated with the Hamiltonian form H. This Hamilton operator reads £H : J'V'Q

^

AT'V'Q,

£H d=dH - AJfi = [(
(5.2.16)

where A is the canonical monomorphism (4.1.4) and J 1 V'Q is the jet manifold of the fibre bundle V'Q —► R, coordinated by {t,q\p„q],Pti) It is readily observed that the kernel of the Hamilton operator EH (5.2.16) is an image of the global section 7w = dt + d'Hdt -

dindi

(5.2.17)

of the affine jet bundle JlV"Q —» V'Q, which is precisely the desired Hamiltonian connection (see Remark 5.2.6 below for a different proof). QED Let us recall that Hamiltonian forms H on the momentum phase space V'Q make up an affine space modelled over the linear space of horizontal densities fdt

5.2. HAMILTONIAN

CONNECTIONS

AND HAMILTONIAN

FORMS

239

on the fibre bundle V'Q -> R. Then, as follows from the relation (5.2.15), Hamil­ tonian connections 7# constitute an affine space modelled over the linear space of Hamiltonian vector fields (5.1.11):

H-H' = df,

7 « - 7tf' = - $ / ■

The kernel of the covariant differential D1H, associated with a Hamiltonian con­ nection (5.2.17), is a closed imbedded subbundle of the jet bundle J ' V Q -» R, and defines the system of first order differential equations on the momentum phase space V ' Q . These are called the Hamilton equations

g| = d{n,

(5.2.18a)

Pti =

(5.2.18b)

-diH

for the Hamiltonian form H. Note that, in comparison with the Lagrange equations, the Hamilton ones are always well defined differential equations. The integral sec­ tions r : R 3 () - • V'Q of the Hamiltonian connection (5.2.17) (or equivalently, the integral curves of the horizontal vector field (5.2.17)) are solutions of the Hamilton equations (5.2.18a) - (5.2.18b). Moreover, a glance at the equation (5.2.18a) shows that the relation J1{TTU or)

= H

(5.2.19)

or

is fulfilled for any solution r of the Hamilton equations (5.2.18a) - (5.2.18b). Remark 5.2.5. The Hamilton equations (5.2.18a) - (5.2.18b) are the particular case of the first order dynamic equations

qt = Y,

(5.2.20)

Pti = 7.

on the momentum phase space V'Q —» R, where 7 is a connection (5.2.1) on the fibre bundle V'Q — R. The first order reduction of the equation of a motion of a point mass m subject to friction in Example 4.9.7 exemplifies first order dynamic equations which are not Hamilton ones. These equations read 1 Qt = — P , m0

k Pt = m0

The connection 1 7 = dt + —pdq m

P

k pdp m

(5.2.21)

240

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

does not obey the condition (5.2.4c) for a locally Hamiltonian connection. At the same time, the equations (5.2.21) are equivalent to the Hamilton equations for the Hamiltonian associated with the Lagrangian (4.9.35). • Note that the Hamilton equations (5.2.18a) - (5.2.18b) can be introduced without appealing to the Hamilton operator. On sections r of the fibre bundle V'Q —» R, these equations (5.2.22)

r< = -d,H o r

r' = &H o r,

are equivalent to the relation r'{u\dH)

(5.2.23)

= 0

which is assumed to hold for any vertical vector field u on V'Q —► R. Remark 5.2.6. Every Hamiltonian form H on a. momentum phase space V'Q defines a presymplectic structure. A Hamiltonian form H = h'E, by definition, is a pull-back of the canonical Liouville form 5 (5.1.2) by means of a section h of the fibre bundle T'Q —> V'Q. Accordingly, its exterior differential dH = h'dE = {dpi + diHdt) A {dq? - d'Tidt)

(5.2.24)

is the pull-back of the canonical symplectic form (5.1.3), and is a presymplectic form. The presymplectic form (5.2.24) has constant rank 2m since the form {dH)m = {dPi A dg') m - m{dpx A dq')m-1

AdH/\dt

(5.2.25)

is nowhere vanishing. It is also easily seen that {dH)m

Adty^O.

It follows that the pair (dH, dt) defines a cosymplectic structure on V'Q (see Remark 4.8.8). Since the presymplectic form dH is of constant rank 2m, its kernel Ker dH is a 1-dimensional distribution which is spanned by the vector field ^H (5.2.17). Hence, there is a unique vector field u = 7// on V'Q such that u\dH = 0,

u\dt = 1

(see [37]). This is a different proof of Proposition 5.2.10. •

5.2. HAMILTONIAN

CONNECTIONS

AND HAMILTONIAN

FORMS

241

PROPOSITION 5.2.11. The Hamiltonian form (5.2.6) is a contact form on the mo­ mentum phase space V'Q if the function jH\H

=

(5.2.26)

pidiH-H=[H]

nowhere vanishes [116]. □ Proof. Since a Hamiltonian connection 7# (5.2.17) is a nowhere vanishing vector field, the condition H A {dH)m ± 0 for a Hamiltonian form H to be a contact form is equivalent to the condition lH\{H

A (dH)m) = {lH\H){dH)m

= [H\(dHr

± 0.

The result follows because the form (dH)m (5.2.25) is nowhere vanishing.

QED

Note that one may try to add some exact form, e.g., the form cdt, c = const., to a Hamiltonian form H in order to make the function \H] nowhere vanishing. For instance, the Hamiltonian form i / r (5.2.7) fails to be a contact one since [Hr] = 0, but the equivalent Hamiltonian form Hr - dt, where [Hr — dt] = 1, is a contact form. If a Hamiltonian form H is a contact one, the corresponding Reeb vector field (2.2.5) reads

EH = [WTV

(5.2.27)

By virtue of Proposition 2.2.6, one can then introduce the Jacobi bracket defined by the vector field (5.2.27) and by the bivector field wH on V'Q derived from the relations (2.2.8) which read wH(,.)}dH=-(4>-(EH\)H), whenever 0 is a 1-form on V'Q. We find wH{i + Pia'EH\(p - p{(j>'EH\a,

242

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

where 4>, a axe arbitrary 1-forms on V'Q. The corresponding Jacobi bracket on the momentum phase space V'Q is {J,9}H

= wH(4>,a) + EH\(fdg-gdf)

=

V,9}v + [HT'fobtfJtf - lfhH\dg), where {/, g}v is the Poisson bracket (5.1.5) and [g]=Pidig-g.

[/]=?.£>'/-/,

Remark 5.2.7. Given the Poisson bracket (5.1.5) on the momentum phase space V'Q, one can introduce the generalized Poisson bracket {., .}w (2.8.3) on the exterior algebra Q'(V'Q), and the bracket {., .}d (2.8.5) on the quotient

0'(VQ)/dD'(VQ). In particular, the generalized Poisson bracket (2.8.3) of Hamiltonian forms H and H' reads {H, H'}w = Pi(d"H' -

d'H)dt.

•

5.3

Canonical transformations

Canonical transformations in time-dependent mechanics are not compatible with the fibration V'Q —► Q of the momentum phase space in general. DEFINITION 5.3.1. By a canonical automorphism is meant a vertical automorphism p of the fibre bundle V'Q —» R, which preserves the canonical Poisson structure (5.1.5) on the momentum phase space V'Q, i.e.,

{f°P,g°p}v

=

{f,g}v°P-

D

It is easily seen that an automorphism p of V ' Q —» R is canonical if and only if it preserves the canonical form SI (5.1.9) on V'Q, i.e.,

n = p'n.

5.3. CANONICAL

TRANSFORMATIONS

243

DEFINITION 5.3.2. The bundle coordinates of the fibre bundle V'Q -» R are called canonical if they are canonical for the Poisson structure (5.1.5). □ It is readily observed that canonical coordinate transformations satisfy the rela­ tion dp, dpk

dpk dp,

dpUW dqi dqk

dqk dqi

dpk dqi

dq> dpk

= 0, = 0, ' =

3*

'

By very definition of the canonical form il, the holonomic coordinates of the vertical cotangent bundle V'Q —► Q are the canonical coordinates. Accordingly, holonomic automorphisms i—+ (
(
(5.3.1)

of V'Q —► Q, induced by the vertical automorphisms of the configuration bundle Q —► R are also canonical. PROPOSITION 5.3.3. Locally Hamiltonian connections are transformed into each other by canonical automorphisms, and so are locally Hamiltonian forms. □ Proof. If 7 is a locally Hamiltonian connection for H, we have

7>(7)jn = (p-'rwn) = d[fjrlYH\ and Tp(y) is also a locally Hamiltonian connection.

QED

PROPOSITION 5.3.4. Let 7 be a complete locally Hamiltonian connection on V'Q —> R, i.e., the vector field (5.2.1) is complete. There exist canonical coordinates on V'Q such that 7 = dt. D □ Proof. A glance at the relation (5.2.3) shows that each locally Hamiltonian con­ nection 7 is the generator of a local 1-parameter group of canonical automorphisms of the fibre bundle V'Q -» R. Let V0'Q be the fibre of V'Q -> R at 0 G R. Then

244

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

canonical coordinates of the symplectic manifold V£Q = T'M dragged along inte­ gral curves of the complete vector field 7 determine the desired canonical coordinates QED on V'Q. In other words, a complete locally Hamiltonian connection 7 on the momentum phase space V'Q in accordance with Proposition 4.1.2 defines a trivialization (5.3.2)

0 : V'Q - R x V0'Q

of the fibre bundle V'Q —> R such that the corresponding coordinates of V'Q, compatible with this trivialization, are canonical. However, it should be emphasized that, although the fibre V0*Q is diffeomorphic to T'M, the trivialization (5.3.2) fails to be a trivialization of the type V'Q = R x T'M since the trivialization morphism rp is not a bundle morphism of the fibre bundle V'Q —» Q. In particular, let H be a Hamiltonian form (5.2.12) such that the correspond­ ing Hamiltonian connection -yH (5.2.17) is complete. By virtue of Proposition 5.3.4, there exists a trivialization of the phase space V'Q with respect to the global canon­ ical coordinates {qA,PA) (where qA are not coordinates on Q in general) such that ft = dpA A dqA A dt,

1H

= dt,

dH = dpA A dqA,

and H reduces to the Hamiltonian form H = pAdqA with the Hamiltonian H = 0. Then the corresponding Hamilton equations take the form of the equilibrium equations qt = 0,

PtA = 0

(5.3.3)

such that qA(t,qt,pl) and PA(t,ql,Pi) are constants of motion. Accordingly, any Hamiltonian Ti can be locally brought into zero, and the cor­ responding Hamilton equations are reduced to the equilibrium ones (5.3.3) by local canonical coordinate transformations. E x a m p l e 5.3.1. Let us consider the 1-dimensional motion with constant acceler­ ation a with respect to the coordinates (t,q). Its Hamiltonian on the momentum phase space R 3 —» R reads P2

5.3. CANONICAL

TRANSFORMATIONS

245

The associated Hamiltonian connection is 1H

= dt+ pdq + adp.

This Hamiltonian connection is complete. The canonical coordinate transformations q =q-pt

+

at2 —,

p' =p — at

(5.3.4)

bring it into fn = dt. Then, the functions q'(t, q%p) and p'(t,p) (5.3.4) are constants of motion. • Example 5.3.2. Let us consider the 1-dimensional oscillator with respect to the same coordinates as in the previous Example. Its Hamiltonian on the momentum phase space R 3 —* K reads K=\{p2

+ q2)-

The associated Hamiltonian connection is lH = dt+ pdq -

qdp,

which is complete. The canonical coordinate transformations q' = qcost — psint,

p' = p cos t + q sin t

bring it into JH = dt. Then, the functions q^(i, q,p) and p'(t,q,p) stants of motion. •

(5.3.5) (5.3.5) are con­

It should be emphasized that, in general, canonical automorphisms do not trans­ form Hamiltonian forms into Hamiltonian forms, but only locally. Let H be a Hamiltonian form (5.2.10) on a momentum phase space V'Q equipped with the coordinates (t,qx,Pi). Given a canonical automorphism p, we have d(p*H - H) = 0, where p'H = pidp1 -Ho

pdt.

(5.3.6)

Therefore, we can write locally H -p'H

= dS,

(5.3.7)

246

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

where S(t, q',Pi) is a local function on a momentum phase space V'Q. It should be emphasized that, if a configuration space Q is a contractible mani­ fold, the equality (5.3.7) is fulfilled globally, and the function S is defined everywhere on V'Q. Substituting (5.3.6) in (5.3.7), we obtain the relations diS = Pil

Pjdtp', x

dS =

-pjd (P,

H-H'

= p,dtp' - dtS.

Taken on the graph A„ = {(p,p(p)) 6 V'Q x V'Q, p e

V'Q}

of the canonical automorphism, the function S plays the role of a generating function of canonical transformations. Example 5.3.3. Given a canonical automorphism p, let (5.3.8)

det(dV)^0. Then, if the graph A„ can be coordinated by (t,q',qn

qtop),

=

we obtain the familiar relations dS Pi=

dq~'' ,_ dS p '~ do"' H-H'

=

dtS(t,q\q").

(5.3.9)

• Example 5.3.4. For instance, the generating function of a holonomic automor­ phism (5.3.1), where det(<9i<j'J) ^ 0, is S(t,q'3,Pi)

= -ql(t,

q'j)Pl.

• Let H be a Hamiltonian form whose associated Hamiltonian connection is com­ plete. By virtue of Proposition 5.3.4, there exists a canonical automorphism p which bring its Hamiltonian into zero. If the condition (5.3.8) holds, then (5.3.9) is the Hamilton-Jacobi equation.

5.4. THE EVOLUTION

5.4

247

EQUATION

The evolution equation

Given a Hamiltonian form H (5.2.12) and the corresponding Hamiltonian connection 7 H (5.2.17) on the momentum phase space V'Q, let us consider the Lie derivative of a function / e D°(V'Q) along the horizontal vector field 7/f, which reads

!*»»/ = 7«J4f = (dt + VHdi -

W#)J.

(5.4.1)

This equality is the evolution equation in time-dependent mechanics. Substituting a solution r of the Hamilton equations (5.2.22) in (5.4.1), we obtain the time evolution of / along this solution: L

-m/ o r = JtU°r)-

It should be emphasized that, in comparison with the evolution equation (3.2.5) in symplectic mechanics, the right-hand side of the evolution equation (5.4.1) is not reduced to the Poisson bracket since a Hamiltonian H o n a momentum phase space of time-dependent mechanics is not a function, i.e., 7i £ D°(V'Q). Of course, one can define locally the Poisson bracket {H,f}v of a Hamiltonian H with a function / on V'Q. However, being equal to zero with respect to some coordinates, this Poisson bracket does not necessarily vanish with respect to other coordinates. Given the splitting (5.2.12) of a Hamiltonian form H, the evolution equation (5.4.1) is written as

L7„/ = dtf+(r% -

diVp^f+iHrJh-

(5.4.2)

In particular, the following two consequences of this form of the evolution equation should be pointed out. • Since the evolution equation (5.4.2) is not reduced to the Poisson bracket, the integrals of motion in time-dependent mechanics cannot be defined as functions on a momentum phase space, which are in involution with a Hamil­ tonian. Therefore, to obtain conservation laws in Hamiltonian time-dependent mechanics, we follow the methods of field theory (see Section 5.7). • The second (kinematic) term in the right-hand side of this equality plays an essential role under quantization. It makes the quantization procedure depen­ dent on a reference frame. In Appendix B, we will show that quantizations under different reference frames fail to be equivalent in general.

248

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

Note that the kinematic term in the evolution equation (5.4.2) can be eliminated at least locally by means of canonical transformations. Let a connection T in the splitting of a Hamiltonian form (5.2.12) be complete. With respect to the coordinate system (t, q') adapted to the reference frame T, the configuration bundle Q is triv­ ialized, and the corresponding holonomic coordinates (t, q',pi) on the momentum phase space V'Q are canonical. With respect to these coordinates, the evolution equation (5.4.2) takes the familiar form

L^Hf = dtf + {H,f}v.

5.5

Degenerate systems

This Section is devoted to the relationship between Lagrangian and Hamiltonian formalisms of time-dependent mechanics. This relationship is characterized by the diagram

V'Q ■£*

z\

JIQ

_ j.

X

JQ

^-V'Q

which fails to commute in general. Its jet extension is jiy'Q J'L

J

^JlJlQ

| l l

JJQ

I

PL

l

i^-J V'Q

where the jet prolongations of Hamiltonian and Legendre maps read

JlH:JlV'Q^JlJlQ, (
= (q\ d'H, q\, dt&H), = (g\7r„ q\ty d,7r,).

As we will show below, in the case of a hyperregular Lagrangian when the Leg­ endre map L (4.8.9) is a diffeomorphism, Lagrangian and Hamiltonian formalisms are equivalent. Conversely, let H be a Hamiltonian form and JH the corresponding

5.5. DEGENERATE

249

SYSTEMS

Hamiltonian connection (5.2.17) on the momentum phase space V'Q —» K. Let us consider the composition of morphisms

JlHolH:V'Q^J2QdJlJlQ,

( < M 0 . < 4 W t f °7/f =

(5.5.1)

(d'Kd'KjHlMd'H)).

If the Hamiltonian map H is a diffeomorphism, then JlH 0 7 ^ 0 H~l is a holonomic connection on the jet bundle JlQ —► M, which defines a dynamic equation on the velocity configuration space J1Q. This dynamic equation is equivalent to the system of Lagrange equations for the Lagrangian function £ =

H-U[H]

which is the pull-back on JXQ of the function [H\ (5.2.26) by the morphism / / " ' . Let us consider more general conditions for solutions of Hamilton equations on a momentum phase space to be solutions of Lagrange equations and second order dynamic equations on a velocity phase space, and vice versa. Following the general scheme of polysymplectic Hamiltonian formalism [57, 158, 159], we say that a Hamiltonian form H on the momentum phase space V'Q is associated with a Lagrangian L on the velocity phase space JlQ if H obeys the conditions L°HoL

= L,

(5.5.2a)

H = H& + LoH

(5.5.2b)

[161]. A glance at the condition (5.5.2a) shows that the composition L o H is the projection operator p,(p) = d , £ ( t , ^ , c ^ ( p ) ) , |

peQ,

(5.5.3)

from V'Q onto a submanifold N = L{JXQ) C V'Q,

(5.5.4)

called the Lagrangian constraint space. Given a Hamiltonian form H associated wit! the Lagrangian L, this constraint space is defined by the relation (5.5.3). Accord­ ingly, the composition HoL is the projection operator from JiQ onto a submanifold H(N) C JlQ. The relation (5.5.2b) takes the form

CoH

= [H]

(5.5.5)

250

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

everywhere on V'Q. Remark 5.5.1. Unless otherwise stated, we regard N as a subset of the momentum phase space V'Q, without a manifold structure. All objects are defined on the whole phase space V'Q, and their restriction to N means only that their values at the points of N C V'Q are considered. • PROPOSITION 5.5.1. A Hamiltonian form H associated with a Lagrangian L obeys the relation H \s= H'HL

(5.5.6)

\N,

where Hi is the Poincare-Cartan form (4.8.5). □ Proof. The proof follows from direct computation.

QED

Acting on both sides of the equality (5.5.5) by the exterior differential, we obtain the relations dtH(p) =

-(dt£)(t,q>,d>H(p)),

peN,

(5.5.7)

dlH{p) =

-{diC){t^,&H{p)),

peN,

(5.5.8)

(P, - (a^)(t,^,^H))a'd°w = o.

(5.5.9)

A glance at the relation (5.5.9) shows that: • the condition (5.5.2a) is a corollary of the condition (5.5.2b) if the Hamiltonian map H is regular, i.e., d e t ( a ' ^ W ) yt 0 at all points of the Lagrangian constraint space N; • the Hamiltonian map H is always degenerate outside the Lagrangian con­ straint space N. Example 5.5.2. Let L = 0 be the zero Lagrangian. In this case, the Lagrangian constraint space is N = 0(Q), where 0 is the canonical zero section of the Legendre bundle V'Q —» Q. The condition (5.5.2a) is fulfilled trivially for every Hamiltonian map, while the condition (5.5.2b) takes the coordinate form H = Pid{H.

5.5.

DEGENERATE

SYSTEMS

251

Any Hamiltonian form Hr (5.2.7) obeys this condition, and is associated with the Lagrangian L = 0. • If a Lagrangian L is hyperregular, there exists a unique associated Hamiltonian form H = #£_, +

(L-l)'L

(5.5.10)

such that Pi=nl(q\d^H(qX,Pk)),

H = L~\

g{ saw,*,(**,rf}).

(5.5.11)

As an immediate consequence of (5.5.11), we have JlH

=

(J1L)-\

PROPOSITION 5.5.2. Let L be a hyperregular Lagrangian, and H the associated Hamiltonian form. The following relations hold: (5.5.12)

HL = L'H,

eI = £B =

1

(j ly£H,

(5.5.13)

l

{J H)'£T,

(5.5.14)

where EH is the Hamilton operator (5.2.16) for H, and £j is the Euler-LagrangeCartan operator (4.8.16) for L. O Proof. The proof follows from direct computation.

QED

A glance at the relations (5.5.6) and (5.5.12) shows that the Poincare-Cartan form is the Lagrangian counterpart of a Hamiltonian form, whereas it follows from (5.5.13) and (5.5.14) that the Lagrangian counterpart of the Hamilton operator is the Euler-Lagrange-Cartan operator £ j (4.8.16). In particular, if 7// is a Hamiltonian connection for the associated Hamiltonian form H (5.5.10), then, by virtue of the equality (5.5.14), the composition JlH o^H (5.5.1) takes its values into the kernel of the Euler-Lagrange-Cartan operator £^ or. more exactly, in the kernel of the Euler-Lagrange operator £L- Then the composition J1/fo7„oL:;'(j3(t,l/i,g;)M (t, q\ q\ = VH o L, q\t) = d'H o L, q\t = lH\d(d'H) (t, q\ q\, q\t) =
2

o L) €e J Q

o L) =

252

CHAPTER

5. HAMILTONIAN

is a holonomic Lagrangian connection for L. connection for L, then

yH = JlL°£Lo

TIME-DEPENDENT

MECHANICS

Conversely, if £L is a Lagrangian

H

is a Hamiltonian connection for H. This proves the following assertion. PROPOSITION 5.5.3. Let L be a hyperregular Lagrangian and H the associated Hamiltonian form (5.5.10). (i) Let a section r of the fibre bundle V'Q -» 1 be a solution of the Hamilton equations (5.2.18a) - (5.2.18b) for the Hamiltonian form H. Then the section c = 7rn o r of the fibre bundle Q —» R is a solution of the Lagrange equations (4.8.7) for the Lagrangian L, while its first order jet prolongation c satisfies the Cartan equations (4.8.17a) - (4.8.17b). (ii) Conversely, if a section c of the jet bundle J1Q —► R is a solution of the Cartan equations for the Lagrangian L, the section r = Loc of the fibre bundle V'Q —> R satisfies the Hamilton equations (5.2.18a) - (5.2.18b) for the Hamiltonian form H. O It follows that, given a hyperregular Lagrangian, there is one-to-one correspon­ dence between the solutions of the Lagrange equations (and, consequently, of the Cartan equations) and the solutions of the Hamilton equations for the associated Hamiltonian form. In the case of regular Lagrangian L, the Lagrangian constraint space N is an open subbundle of the Legendre bundle V'Q —» Q. U N jt V'Q, an associated Hamiltonian form fails to be defined everywhere on V'Q in general. At the same time, an open constraint subbundle N can be provided with the appropriate pullback structure with respect to the imbedding N •—» V'Q so that we may restrict our consideration to Hamiltonian forms on N. If a regular Lagrangian is additionally semiregular (see Definition 5.5.4 below), the associated Legendre morphism is a diffeomorphism of JlQ onto the Lagrangian constraint space N and, on N, we can restate all results true for hyperregular Lagrangians.

5.5.

DEGENERATE

253

SYSTEMS

Example 5.5.3. Let Q be the fibre bundle R 2 - » R with coordinates (t,q). Its jet manifold JlQ 3 K3 and its Legendre bundle V'Q = R 3 are coordinated by (t,q,qt) and (tf,g,p), respectively. Put L=

exp(qt)dt.

(5.5.15)

This Lagrangian is regular, but not hyperregular. The corresponding Legendre map reads p o L = expift. It follows that the Lagrangian constraint space N is given by the coordinate relation p > 0. This is an open subbundle of the Legendre bundle, and L is a diffeomorphism of JXQ onto N. Hence, there is a unique Hamiltonian form H on N which is associated with the Lagrangian (5.5.15). Its Hamiltonian function reads H = p{\np-

1).

This Hamiltonian form, however, fails to be smoothly extended to V'Q.

•

Hereafter, we will restrict our consideration of degenerate systems described by semiregular Lagrangians. Most physically interesting Lagrangians, including the quadratic ones, are of this type. On the other hand, there is a comprehensive relationship between Lagrangian and Hamiltonian formalisms in this case [57, 158, 159, 190]. DEFINITION 5.5.4. A Lagrangian L is said to be semiregular, if the pre-image L'1(p) of any point p € N is a connected submanifold of the velocity phase space

JlQ. □ PROPOSITION 5.5.5. All Hamiltonian forms H associated with a semiregular La­ grangian L coincide with each other at the points of the Lagrangian constraint space N, i.e., H \N— H' |/V . Moreover, the Poincare—Cartan form Hi for the Lagrangian L is the pull-back HL = L'H,

faqi - C)dt = H(t,qi,n)dt,

(5.5.16)

CHAPTER

254

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

of any associated Hamiltonian form H by the Legendre map L. □ Proof. Let u be a vertical vector field on the jet bundle J ' Q -» Q. If u takes its values into the kernel Ker TL of the tangent morphism to L, it is easy to see that L U H £ = 0. Hence, the Poincare-Cartan form HL for a semiregular Lagrangian L is constant on the connected pre-image £~'(p) of each point p € N. Then results follow from QED (5.5.6). Note that the Hamilton operators for Hamiltonian forms in Proposition 5.5.5 do not necessarily coincide at points of the Lagrangian constraint N because of the derivatives of these forms. Let H be a Hamiltonian form associated with a semiregular Lagrangian L. Act­ ing by the exterior differential on the relation (5.5.16), we obtain the equality {q\ - d"H o L)d-nx A dt - (diC + di(H o L))dqi Adt = 0

(5.5.17)

or, equivalently, the equalities 7 r t > ( ^ - 3 > W o L ) = 0,

Bi*M ~ &H o £) - (diC + (diH) o L) = 0. Using the equality (5.5.17), one can extend the relation

£z =

(jlly£H

(5.5.18)

(5.5.13), but not necessarily the relation (5.5.14) to semiregular Lagrangians. The relation (5.5.18) enables us to generalize item (i) of Proposition 5.5.3 for Hamiltonian forms associated with semiregular Lagrangians. PROPOSITION 5.5.6. Let a section r of the fibre bundle V'Q —> R be a solution of the Hamilton equations for a Hamiltonian form H associated with a semiregular Lagrangian L If r lives in the Lagrangian constraint space ./V, the section c=

nnor

of the fibre bundle Q —> R satisfies the Lagrange equations for L, while its first order jet prolongation c=ffor=c

5.5.

DEGENERATE

255

SYSTEMS

obeys the corresponding Cartan equations (4.8.17a) - (4.8.17b). □ Proof. Put c = H o r. Since r(R) C N, then f= J ' L o Jlc.

r = L o c,

If r is a solution of the Hamilton equations, the exterior form £H vanishes at points of r(R). Hence, the pull-back form

£T = (j1ly£H vanishes at points of c(R). It follows that the section c of the fibre bundle J ' Q —> R obeys the Cartan equations. By virtue of the equation (5.2.18a), we have c = c, and the section c is a solution of the Lagrange equations, which lives in the submanifold H(N). QED In the case of semiregular Lagrangians, item (ii) of Proposition 5.5.3 can be modified as follows. PROPOSITION 5.5.7. Given a semiregular Lagrangian L, let a section c of the jet bundle JlQ —* R be a solution of the Cartan equations (4.8.17a) - (4.8.17b). Let H be a Hamiltonian form associated with L so that the corresponding Hamiltonian map satisfies the condition Jl(fo°4

HoLo-5=

(5.5.19)

Then the section r = L o c, r,

=TTi(t,Cj,4),

r* = c\

of the fibre bundle V'Q —» R is a solution of the Hamilton equations (5.2.18a) (5.2.18b) for H, which lives in the Lagrangian constraint space TV. D Proof. The Hamilton equations (5.2.18a) hold by virtue of the condition (5.5.19). Using the relations (5.5.17) and (5.5.19), the Hamilton equation (5.2.18b) is brought into the Cartan equation (4.8.17b): dtTCi o c =

-{diH)oloc

= -(3-&H°l°c)dtn3

(dtct — cDdiKj o c + d{C o c.

oc + diCoc =

256

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS QED

Proposition 5.5.6 shows that, if H is a Hamiltonian form associated with a semiregular Lagrangian L, every solution of the corresponding Hamilton equations, which lives in the Lagrangian constraint space N, yields a solution of the Cartan equations and of the Lagrange equations for L. Thus, the counterpart of degenerate Lagrangian systems are the constraint Hamiltonian systems. We will restrict our consideration to the case of a Lagrangian constraint space N. This plays the role of a primary constraint space. In order that local solutions of the Hamilton equations for a Hamiltonian form H exists on N, the Hamiltonian connection 7# must be tangent to TV. This condition is given by the equations (5.5.20a)

Pi = %C(t,<j>,9iH), (9, + d>Hd, - djH&)\d{pt

- d\C{t, q>,d>H)) = 0,

(5.5.20b)

where (5.5.20a) is the equation (5.5.3) of a Lagrangian constraint space, while (5.5.20b) requires that the horizontal vector field JH is tangent to N at the point (t, g1, Pi). In contrast with the case of autonomous constraint systems, the left-hand side of the equation (5.5.20b) is not reduced to the Poisson bracket. Therefore, we cannot follow the familiar procedure of constructing the final constraint space in Section 3.5. There is another possibility. Given a solution of the Lagrange equations, one can try to find an associated Hamiltonian form H such that this solution is a solution of the Hamilton equations for H. It may happen that different solutions of the Lagrange equations require different Hamiltonian forms in general. Thus, degenerate Lagrangian systems are described as multi-Hamiltonian systems. Note that, in the case of semiregular Lagrangians, the condition (5.5.19) is the obstruction for a solution c of the Cartan equations to be a solution of the Hamilton equations and the Lagrange equations. At the same time, one can try to find a family of associated Hamiltonian forms such that each solution of the Lagrange equations is a solution of the Hamilton equations for some Hamiltonian form from this family. DEFINITION 5.5.8. We will say that a family of Hamiltonian forms H, associated with a Lagrangian L, is complete if, for each solution c of the Lagrange equations, there exists a solution r of the Hamilton equations for a Hamiltonian form H from

5.5. DEGENERATE

SYSTEMS

257

this family so that c = 7Tn°r,

r = Lo c

(5.5.21)

(see the relation (5.2.19)). □ Let L be a semiregular Lagrangian. Then, by virtue of Proposition 5.5.7, such a complete family of associated Hamiltonian forms exists if and only if, for every solution c of the Lagrange equations for L, there is a Hamiltonian form H from this family such that H o L o c = c.

(5.5.22)

Example 5.5.4. Let L = 0. This Lagrangian is semiregular. Its Lagrange equations reduce to the identity 0 = 0. Every section c of the configuration bundle Q —► R is a solution of this equation. Given a section c, let T be a connection on the fibre bundle Q —> R such that c is an integral section of T. The Hamiltonian form Hr (5.2.7) is associated with L = 0, and the Hamiltonian map H^ obeys the relation (5.5.22). The corresponding Hamilton equations read

?! = r1,

Pu = -PjdiT3.

They have a solution r = L o c, r' = c \

n = o,

which lives in the Lagrangian constraint space p, = 0. • The example below shows that a complete family of associated Hamiltonian forms may exist when a Lagrangian is not necessarily semiregular. Example 5.5.5. Let Q be the fibre bundle R 2 —♦ R 1 in Example 5.5.3 with coor­ dinates (t,q). Put

£ = i(*)3The associated Legendre map reads

poL

=
(5.5.23)

258

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

The corresponding Lagrangian constraint space N is given by the coordinate relation p > 0. It fails to be a submanifold of the momentum phase space V'Q. There exist two associated Hamiltonian forms H+ = pdq -

-p3'2dt,

H-=pdq+-p3/2dt, on N, which correspond to the two different solutions <7t = VP,

Qt =

-y/p

of the equation (5.5.23). The Hamiltonian forms H+ and # _ make up a complete family. • Remark 5.5.6. Let 7# be a Hamiltonian connection for a Hamiltonian form H associated with a semiregular Lagrangian L. By virtue of the relation (5.5.18), the composition JlH o -yH (5.5.1) takes its values into the kernel of the Euler-Lagrange operator £{,. Since HoL\S(Q)

=

UH(Q),

the morphism j'Ho-yol:

j'Qgft/oilH

(t, q\ qt = d'H o L, q\t) = Shi ° L, q\t = yH\d(d>H) o L) restricted to H(Q) is the holonomic section over H(Q) C JlQ of the affine jet bundle J2Q —> JlQ. Let H(Q) be a closed submanifold, e.g., when a Lagrangian L is almost regular (see Definition 5.5.9 below). Then the section JlH o^HoL can be extended to a holonomic connection on the jet bundle JlQ —» R, which defines a dynamic equation on the configuration space Q. In this case, given a solution r of the Hamilton equations for H which lives in the Lagrangian constraint space ,/V, its projection 7rn o r is a solution of both the Lagrange equations and the dynamic one. •

DEFINITION 5.5.9. A semiregular Lagrangian L is called almost regular if: (i) the Lagrangian constraint space N is a closed imbedded subbundle i^ : N ^-> V'Q of

5.5.

DEGENERATE

SYSTEMS

ZiO\y

the Legendre bundle V'Q —» Q and (ii) the Legendre map L : JXQ —» N is a fibre bundle. D

PROPOSITION 5.5.10. [57, 190]. Let L be an almost regular Lagrangian. On an open neighbourhood in V*Q of each point q € N, there exists a complete family of local Hamiltonian forms associated with L. O Let L be an almost regular Lagrangian L. Given an associated Hamiltonian form H, the Hamiltonian map H\N:N^

JlQ

restricted to N is a section of the fibre bundle L : JlQ —» N, and its image H(N) is a closed imbedded submanifold of the configuration space JlQ. Therefore, it follows from Remark 5.5.6 that each solution of the Lagrange equations for L, which is a solution of Hamilton equations, is a solution of a dynamic equation. Since solutions of the Lagrange equations for a degenerate Lagrangian may cor­ respond to solutions of different Hamilton equations, we can conclude that, roughly speaking, the Hamilton equations involve some additional conditions in comparison with the Lagrange equations. Therefore, let us separate a part of the Hamilton equations which are defined on the Lagrangian constraint space Q in the case of an almost regular Lagrangian. Let IT

"N

def .» Tf

—lN™

be the restriction of a Hamiltonian form H associated with an almost regular La­ grangian L to the Lagrangian constraint space N. By virtue of Proposition 5.5.5, this restriction, called the constrained Hamiltonian form, is uniquely defined. More­ over, the Poincare—Cartan form is the pull-back He = L'Hs of the constrained Hamiltonian form HN. On sections r of the fibre bundle Q —» R, we can write the constrained Hamilton equations r'{uN\dHN)

= 0,

(5.5.24)

where us is an arbitrary vertical vector field on the fibre bundle N —> R [56, 57, 161]. For the sake of simplicity, we can identify a vertical vector field uN on TV —> R with

260

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

its image Tz^(u^) and can bring the constrained Hamilton equations (5.5.24) into the form r'{uN\dH)

(5.5.25)

= 0,

where r is a section of the fibre bundle TV —> R, and u^ is an arbitrary vertical vector field on N —» R. It should be emphasized that these equations fail to be equivalent to the Hamilton equations (5.2.23) restricted to the Lagrangian constraint space N. If an almost regular Lagrangian admits associated Hamiltonian forms (see Propo­ sition 5.5.10), the following three assertions together with Proposition 5.5.7 give the relationship between Cartan, Hamilton-De Donder, Hamilton, and constrained Hamilton equations [57, 161]. PROPOSITION 5.5.11. For any Hamiltonian form H associated with an almost regular Lagrangian L, every solution r of the Hamilton equations which lives in the Lagrangian constraint space N is a solution of the constrained Hamilton equations (5.5.25). □ Proof. For any vertical vector field Ufj on the fibre bundle N —* K, the vector field Tiw(ww) is obviously a local vertical vector field on the fibre bundle V'Q —» R. Then we have r'(uN\dHN)

=

r'(uN\i%dH) = T'[TiN{uN)\dH)

= 0

if r is a solution of the Hamilton equations (5.2.23) for the Hamiltonian form H. QED

PROPOSITION 5.5.12. A section c of the jet bundle JlQ -> R is a solution of the Cartan equations (4.8.18) if and only if Loc is a solution of the constrained Hamilton equations (5.5.25). □ Proof. Let uN be a vertical vector field on the fibre bundle N —► R. Since L is a submersion, there exists a vertical vector field v on the jet bundle JlQ —> R such that TL ov = us-

5.5. DEGENERATE

SYSTEMS

261

For instance, v is the horizontal lift of u by means of a connection on the affine jet bundle JlQ —► Q. Let a section c : M «—► JlQ be a solution of the Cartan equations (4.8.18). Then we have (Loc)*(uN\dHN)

= c*{v\dHL) = 0.

(5.5.26)

It follows that the section L oc : R ^* N is a. solution of the equations (5.5.25). The converse is obtained by running (5.5.26) in reverse, bearing in mind that the restriction of any vector field v on JlQ to c(R) is projectable by L. QED

PROPOSITION 5.5.13. The constrained Hamilton equations (5.5.24) are equivalent to the Hamilton-De Donder equations (4.8.25). □ Proof. Let H be a Hamiltonian form associated with an almost regular Lagrangian L, and let h be the corresponding section of the fibre bundle

C : T*Q -» VQ (see Definition 5.2.4). This section yields the morphism hN = h o iN : N -» T*Q which does not depend on the choice of H. This is a local section of the fibre bundle T*Q -» V*Q over N C VQ, i.e, (5.5.27)

C,ohN = lAN. Moreover, we have HN = i*NH = i*N(h*E) = h*NE,

whenever H is a Hamiltonian form associated with the Lagrangian L. In accordance with the relation (5.5.16), (5.5.28)

HL = hN o L,

where HL is the Legendre morphism (4.8.21) associated with the Poincare-Cartan form Hi. Substituting (4.8.22) in (5.5.28), we obtain HL = hNoC,o

HL.

262

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

It follows that (5.5.29)

hN°C \zL W i i ( Z L ) ,

where iL(ZL) = HL(JlQ) is the image of the Legendre morphism HL. A glance at the relations (5.5.27) and (5.5.29) shows that there is the bundle isomorphism

ZL •

N

over Q. Since H^ HN

h*NE and EL

{iLl o hNyzL,

~L

ilH, we have

{c°hyHN.

Hence, the Hamilton-De Donder equations (4.8.25) are equivalent to the constrained Hamilton equations (5.5.24). QED

5.6

Quadratic degenerate systems

As an important illustration of Propositions 5.5.6 and 5.5.7, let us describe the complete families of Hamiltoman forms associated with almost regular quadratic Lagrangians. The Hamiltonians of these Hamiltonian forms are quadratic in mo­ menta. Remark 5.6.1. Since Hamiltonians in time-dependent mechanics are not scalar functions on a momentum phase space, one cannot apply to them the well-known analysis of the normal forms [18] (e.g., quadratic Hamiltonians [6] in symplectic mechanics). • Given a configuration bundle Q —► R, let us consider a quadratic Lagrangian C

-aijq\4, + biql + c,

(5.6.1)

where a, b and c are local functions on Q. This property is global due to the transformation law of the velocity coordinates q\. The associated Legendre map reads Pi o L = a^ql + bi.

(5.6.2)

5.6. QUADRATIC

DEGENERATE

263

SYSTEMS

LEMMA 5.6.1. The Lagrangian (5.6.1) is semiregular.

□

Proof. For any point p of the Lagrangian constraint space N (5.5.4), the system of linear algebraic equations (5.6.2) for q\ has solutions which make up an afHne space modelled over the linear space of solutions of the homogeneous linear algebraic equations 0 = Oy^, where q> are the coordinates on the vertical tangent bundle VQ. This affine space is obviously connected. QED Let us assume that the Lagrangian L (5.6.1) is almost regular, i.e., the matrix a„ is of constant rank. The Legendre map (5.6.2) is an affine morphism over Q. It defines the corre­ sponding linear morphism L:VQ

^VQ,

p{oL = ai:jq>, whose image N is a linear subbundle of the Legendre bundle V*Q ^ Q. Accordingly, the Lagrangian constraint space N, given by the equations (5.6.2), is an affine subbundle, modelled over N, of the momentum phase space VQ —» Q. Hence, the constraint fibre bundle N —> Q has a global section. For the sake of simplicity, let us assume that this is the canonical zero section 0(Q) of the vertical cotangent bundle VQ -> Q. Then N = N. The kernel KerL = L- 1 (0(Q)) of the Legendre map is an affine subbundle of the affine jet bundle JXQ —> Q, which is modelled over the vector bundle KeiL =

L~\0(Q))cVQ.

Then there exists a connection T : Q -» Ker L,

(5.6.3)

aijP + h = 0,

(5.6.4)

CHAPTER 5. HAMILTONIAN

264

TIME-DEPENDENT

MECHANICS

on the configuration bundle Q - » R , which takes its values into KerL. It is called a Lagrangian frame connection. With this connection, the quadratic Lagrangian (5.6.1) can be brought into the form c

= 2 a y9r?f + c'>

* = 4 - r'For instance, if the quadratic Lagrangian (5.6.1) is regular, there is a unique solution (5.6.3) of the algebraic equations (5.6.4). PROPOSITION 5.6.2. There exists a linear map (5.6.5)

VQ, l

t]

q oa = a pi, over Q such that L o a o iN = i^. D

Proof. The map (5.6.5) is a solution of the algebraic equations dija^akb

(5.6.6)

= (lib-

The matrix ay is symmetric. After diagonalization, let this matrix have nonvanishing components aAA, A & I. Then a solution of the equations (5.6.6) takes the form 1

°AA = ^X'

OAA'

= 0,

A, A1 6 / ,

A?

A',

while the remaining components are arbitrary. In particular, there is a solution 1

°AA = ^ u -

OAA' = 0,

OCB = 0,

B*I,

(5.6.7)

which satisfies the relation a = a o L o a,
ik

(5.6.8) b

a akb^ .

5.6. QUADRATIC

DEGENERATE

SYSTEMS

265

QED From now on, by a is meant the solution (5.6.7). If the Lagrangian (5.6.1) is regular, the map (5.6.5) is uniquely determined by the equation (5.6.6). The Lagrangian frame connection (5.6.3) and the map (5.6.5) play a prominent role in the construction below. PROPOSITION 5.6.3. The matrix a defines the splitting J1Q = K e r Z e I m ( a o Z ) , Q

(5.6.9)

4 = [qj -
Ker

a®N, Q

Pi

\pi - aijcr^pk] + [aija3kpk].

a COROLLARY 5.6.5. Every vertical vector field u = u'di + Uid' on the momentum phase space V*Q —> M admits the decomposition U — (u — Upf) + UN,

m = [Ui - aija3kuk] + [atja^Uk], where uN = uldi + aijt7jkukd'

(5.6.10)

CHAPTER 5. HAMILTONIAN

266

TIME-DEPENDENT

is a vertical vector field on the Lagrangian constraint space N —> R.

MECHANICS

□

Given the linear map a (5.6.5) and the Lagrangian frame connection T (5.6.3), let us consider the affine Hamiltonian map (5.6.11)

$ = T + a : V'Q -> J'Q, $' = F(g)+<7

y

Pi,

and the Hamiltonian form H = H* + Ldt, H = ntf

- [r(Pi

- -h)

+ -O»KPS

-

c]dt.

(5.6.12)

It is readily observed that H = $ and, by virtue of (5.6.8), the Hamiltonian form H is associated with the quadratic Lagrangian (5.6.1). We aim to show that the Hamiltonian forms (5.6.12), parameterized by the Lagrangian frame connections T (5.6.3), constitute a complete family. Given the Hamiltonian form (5.6.12), let us consider the Hamilton equations (5.2.18a) for sections r of the fibre bundle V'Q -» R. They read c = (f + a) o r,

c = 7Tn o r,

(5.6.13)

or V(ri =

aijrjt

where y t r ' = < V - ( T o c)1 is the covariant derivative relative to the connection T. With the splitting (5.6.9), we have the surjections S = prl:

JlQ-*KetL,

S : q\ — ql - oik{akjqi

+ bk),

and T = pr 2 : JlQ -» Im(cr o Z), F:qi->aik(akjqi

+ bk).

5.6. QUADRATIC

DEGENERATE

SYSTEMS

267

With respect to these surjections, the Hamilton equations (5.6.13) are split into the following two parts: S o c = T o c, Vtr

{

ik

= a (akjdtri

(5.6.14) + bk),

and T o c = a o r, ik

a {akjdtT

J

(5.6.15) ik

+ h) = a rk.

The Hamilton equations (5.6.14) are independent of the canonical momenta rk and play the role of gauge conditions. Moreover, for every section c of the fibre bundle Q —► R (in particular, for every solution of the Lagrange equations for the Lagrangian L), there exists a Lagrangian frame connection T (5.6.3) such that the equation (5.6.14) holds. Indeed, let F* be a connection on the fibre bundle Q —> R whose integral section is c. Put

T^SoT1,

r = r' i -<7 ilc (a fcJ r' i + 6fc). In this case, the Hamiltonian map (5.6.11) satisfies the relation (5.5.22) for c, i.e., $ o L o c = c. Hence, the Hamiltonian forms (5.6.12) constitute a complete family. The Hamil­ tonian forms from this family differ from each other only in the Lagrangian frame connections T (5.6.3) which lead to the different gauge conditions (5.6.14). It is readily observed that a Lagrangian frame connection V is also a Hamiltonian frame connection for the corresponding Hamiltonian form (5.6.12). PROPOSITION 5.6.6. For every Hamiltonian form H (5.6.12), the Hamilton equa­ tions (5.2.18b) and (5.6.15) on sections of the constraint fibre bundle N —> R are equivalent to the constrained Hamilton equations (5.5.25). □ Proof. In accordance with the decomposition (5.6.10) of a vertical vector field u on the momentum phase space V*Q —» R, the constrained Hamilton equations (5.5.25) take the form r*(aija:>kdi\dH)=0,

(5.6.16a)

r*{dt\dH) = 0.

(5.6.16b)

268

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

The equations (5.6.16b) are obviously the Hamilton equations (5.2.18b) for H. Bear­ ing in mind the relations (5.6.4) and (5.6.8), one can easily bring the equations (5.6.16a) into the form (5.6.15). QED It follows that the equations (5.6.14) are the additional conditions which make the Hamilton equations differ from the constrained Hamilton equations (5.5.25) and the Lagrange equations in the case of a quadratic Lagrangian. COROLLARY 5.6.7. By virtue of Proposition 5.5.12, a section c of the jet bundle JlQ -» 1 is a solution of the Cartan equations (4.8.17a) - (4.8.17b) for the almost regular Lagrangian (5.6.1) if and only if Loc is a solution of the constrained Hamilton equations (5.2.18b) and (5.6.15). □ It follows that the equations (5.6.14) are responsible for the obstruction (5.5.19) for solutions c of the Cartan equations to be solutions of the equations and of the Lagrange equations for the Lagrangian (5.6.1). It seen that the equations (5.6.15) do not contribute to the obstruction (5.5.19). If r = L o c, these equations hold for solutions c of the Cartan (4.8.17a).

condition Hamilton is readily condition equations

PROPOSITION 5.6.8. Let c be a solution of the Cartan equations for the almost regular quadratic Lagrangian L (5.6.1). Let Co be a section of the fibre bundle VQ —► R, which takes its values into Ker L and projects onto c = 7TQ O C. Then the sum c + Co over Q is also a solution of the Cartan equations. □ Proof. The proof is an immediate consequence of the relation r = L o c = L o ( c + Co). QED

Since Hamiltonian forms associated with an almost regular quadratic Lagrangian (5.6.1) constitute a complete family, that is, each solution of the Lagrange equations is also a solution of some Hamilton equations, we can conclude that each solution of the Lagrange equations is a solution of some dynamic equation.

5.7. HAMILTONIAN

5.7

CONSERVATION

269

LAWS

Hamiltonian conservation laws

As was mentioned above, the notion of integrals of motion as functions in involution with a Hamiltonian can not be extended to time-dependent mechanics because its Hamiltonian is not a scalar function under time-dependent transformations. In Section 4.12, we have studied the conservation laws in Lagrangian timedependent mechanics. Therefore, in order to discover the conservation laws in Hamiltonian time-dependent mechanics, let us consider the following construction. Given a Hamiltonian form H (5.2.10) on a momentum phase space V*Q, let us consider the Lagrangian LHd=(Piqi-H)dt

(5.7.1)

(cf. (4.4.6)) on the jet manifold JlV*Q of the momentum phase space V*Q —* M. Note that the Lagrangian (5.7.1) plays a prominent role in the functional integral approach to mechanics (see Remark 5.11.4 below). Here, our utilization of this Lagrangian is based on the following two assertions. LEMMA 5.7.1. The Poincare-Cartan form for the Lagrangian LH (5.7.1) coin­ cides with the pull-back onto JXV*Q of the Hamiltonian form H by the projection J1V*Q ~* V*Q, while the Euler-Lagrange operator for LH coincides with the pullback onto J2V*Q of the Hamilton operator for H by projection J2V*Q —> JlV*Q.

a Proof. The proof is straightforward.

QED

It follows from this Lemma that the Lagrange equations for the Lagrangian LH (5.7.1) are equivalent to the Hamilton equations for the Hamiltonian form H. LEMMA 5.7.2. Let u be a vector field (4.12.2) on a configuration space Q, and u = uldt + u'di — diUjpjd'

(5.7.2)

its lift onto the momentum phase space V*Q in accordance with the formula (1.6.13). Then, there is the equality L;H =

Lji~LH,

where Jlu is the jet prolongation (1.3.7) of the vector field u onto JlVQ.

□

CHAPTER

270

5. HAMILTONIAN

TIME-DEPENDENT

Proof. The proof follows from direct computation.

MECHANICS QED

Thus, we can use the general procedure of constructing Lagrangian conservation laws in Section 4.12 for the Lagrangian LH in order to obtain the Hamiltonian conservation laws. Let us apply the first variational formula (4.8.4) to the Lagrangian (5.7.1). Then the weak identity (4.12.3) reads -u%H

- u'dtH + Pidttf ~ -dt%,

X = -pitf

(5.7.3) (5.7.4)

+ u'H.

In the case of a vertical vector field u, when u ( = 0, the weak identity (5.7.3) leads to the weak equality L J I ^ H ~ dt{piU%).

If \JJI~LH

= 0, we have the weak conservation law

0 pa dt(piu') of the Noether current % =

(5.7.5)

-piU*.

For a reference frame Y (5.2.5) on the configuration bundle Q —> K, the weak identity (5.7.3) takes the form -dtH

- rdiH+PidtV

«

-dtHT,

where H? = Ji — P i P is the Hamiltonian function in the splitting (5.2.12). In the case of semiregular Lagrangians, there is the following relationship between Lagrangian and Hamiltonian conserved currents. PROPOSITION 5.7.3. Let H be a Hamiltonian form on the momentum phase space V*Q, which is associated with a semiregular Lagrangian L on the velocity phase space J 1 ^ . Let r be a solution of the Hamilton equations (5.2.18a) - (5.2.18b) for H, which lives in the Lagrangian constraint space N, and c the associated solution of the Lagrange equations for the Lagrangian L so that the conditions (5.5.21) are

5.8. TIME-DEPENDENT

SYSTEMS

WITH

SYMMETRIES

271

satisfied. Let u be a vector field (4.12.2) on the configuration space Q —> R, Then we have

T{r) =

1{Hor),

X(Zoc)=X(c), where X is the current (4.12.4) on the configuration space JXQ, and X is the current (5.7.4) on the phase space V*Q. □ Proof. The proof is straightforward.

QED

In accordance with Proposition 5.7.3, the Hamiltonian counterpart of the Lagrangian energy function Xr (4.12.16) relative to a reference frame T is the Hamiltonian function HT in the splitting (5.2.12). Therefore, one can think of the Hamiltonian function Hr as being the Hamiltonian energy function relative to the reference frame T. In particular, if P = 0, we obtain the well-known energy conservation law dtH « dtH relative to the coordinates adapted to the reference frame V. This is the Hamiltonian variant of the Lagrangian energy conservation law (4.12.17).

5.8

Time-dependent systems with symmetries

In mechanics, just as in field theory, systems with symmetries can be described in terms of bundles with structure groups. Let 7Tp : P —► R be a principal bundle with a structure Lie group G. Let Q = (P x

M)/G

(5.8.1)

be a P-associated fibre bundle. By gauge transformations of a principal bundle P are meant its vertical auto­ morphisms $ which are equivariant with respect to the canonical action (1.5.4) of the structure group G on P on the right, i.e., $ o rg — rg o $ ,

VgeG.

(5.8.2)

Due to the property (5.8.2), gauge transformations $ of a principal bundle yield the gauge transformations of a P-associated fibre bundle $ Q : ( P x M)/G -» ( $ ( P ) x

M)/G.

272

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

Thus, one can treat a fibre bundle Q (5.8.1) as a configuration space of a mechanical system with the gauge symmetry group G. Every gauge transformation of the configuration bundle Q - » l gives rise canonically to the holonomic vertical automorphisms (5.3.1) of the momentum phase space VQ —> M. They are canonical automorphisms for the canonical Poisson structure (5.1.5) on VQ. Moreover, this group action is Hamiltonian as can be seen in the following way (see Definition 3.7.1). Any 1-parameter group of gauge transformations of a principal bundle P defines a vector field (its generator) on P which is invariant under the canonical action of the group G on P. There is one-to-one correspondence between these right-invariant vector fields on P, called principal vector fields, and the sections of the gauge algebra bundle VQP (1.5.6), whose typical fibre is the right Lie algebra jj r . Accordingly, any 1-parameter group of gauge transformations of a configuration space Q, associated with P , defines a principal vertical vector field C = C(g)d, on the configuration space Q. This vector field gives rise to the vector field

Z^cat-atfpjtf on the momentum phase space (see (5.7.2)). It is readily observed that this is the Hamiltonian vector field for the Noether current J = Pi€(q)

(5.8.3)

(5.7.5) which is defined globally on VQ (see Example 3.7.4). It follows that, given a finite dimensional Lie group of gauge transformations, one may apply the familiar theorems of reduction, e.g., Theorem 3.7.9 to time-dependent mechanics, but not Theorem 3.7.10 and other results which involve the condition of an invariance of a Hamiltonian under a group action. Hamiltonians in time-dependent mechanics fail to be invariant under time-dependent transformations, including gauge transforma­ tions. Remark 5.8.1. It should be emphasized that gauge transformations make up a group whose suitable Sobolev completion is a Banach Lie group. The possible ex­ tension of the above-mentioned theorems to this infinite-dimensional Lie group in the spirit of [36] seems promising. At the same time, we will show below that a Ha­ miltonian form may possess a finite-dimensional Lie group of gauge transformations. •

5.8. TIME-DEPENDENT

SYSTEMS

WITH

SYMMETRIES

273

Let us restrict our consideration to the case of a vector bundle Q —> R. A principal vector field on this vector bundle reads £=

ar(t)emiiq'ai,

(5.8.4)

where e m 'j are generators of the action of the group G on the typical fibre M, and am(t) are local functions on R. The vector field (5.8.4) gives rise to the vector field

| = am(t)Emy9, -

am(t)emijPld>

on the momentum phase space VQ. It is readily observed that this vector field is the Hamiltonian vector field for the Noether current p;£' (5.8.3) relative to the canonical Poisson structure on VQ. Moreover, these Noether currents constitute the Lie algebra with respect to the Poisson bracket

{pie,pd'j}v=Pi[^']i.

(5.8.5)

In particular, let pf£* and Pi£" be conserved Noether currents for a Hamiltonian form H on the momentum phase space VQ. Since

K,e] = [f,r],

Jl[U'\ = [J1lJ1Z]

and Lji^Lff

Ljiji/f = 0,

= 0,

we obtain L

JM«'1LH

=

°'

i.e., the Noether current (5.8.5) is also conserved. Thus, conserved Noether currents in time-dependent mechanics constitute a Lie algebra (cf. Example 3.7.4). This algebra obviously is finite-dimensional, and corresponds to some finite-dimensional Lie group of gauge symmetries of a Hamiltonian form H. As was shown, an action of this group on a momentum phase space VQ is Hamiltonian, with an equivariant momentum mapping. This result remains true for Noether currents (4.12.14) in Lagrangian mechanics. Given a Lagrangian L, the Noether currents (4.12.14) [conserved Noether currents (4.12.14)] along vector fields (5.8.4) constitute a Lie algebra with respect to the Poisson bracket (4.8.13).

274

CHAPTER

5.9

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

Systems with time-dependent parameters

Let us consider a configuration space which is a composite fibre bundle (5.9.1)

Q^E->R,

coordinated by (£, am, q') where (t, am) are coordinates of the fibre bundle E —> R. We will treat sections h of the fibre bundle £ —► R as time-dependent parameters, and say that the configuration space (5.9.1) describes a mechanical system with time-dependent parameters. Let us call E -» R the parameter bundle. Note that the fibre bundle Q —» E is not necessarily trivial. Let us recall that, by virtue of Proposition 1.6.2, every section h of the parameter bundle E —> R defines the restriction (5.9.2)

Qh = h'Q

of the fibre bundle Q —> E to h(R) C E, which is a subbundle ih : QH *-» Q of the fibre bundle Q —► R. One can think of the fibre bundle Qh —> R as being a configuration space of a mechanical system with the background parameter function h(t). The velocity momentum space of a mechanical system with parameters is the jet manifold JXQ of the composite fibre bundle (5.9.1) which is equipped with the adapted coordinates {t,o

,q ,at

,qt).

Let the fibre bundle Q —> E be provided with a connection A = dt ® (dt + A\d{) + dam (dm + <,<%).

(5.9.3)

Then the corresponding vertical covariant differential D:JXQ^

D = (qi-A\-

VZQ,

A^o?)*,

(5.9.4)

(1.6.11) is defined on the configuration space Q. Given a section h of the parameter bundle E —> R, its restriction to Pihi^Qh) C JlQ is precisely the familiar covariant differential on Qh corresponding to the restriction

Ah=dt + {Aimdthm + (Aoh)\)di

(5.9.5)

5.9. SYSTEMS

WITH TIME-DEPENDENT

275

PARAMETERS

of the connection A to h(R) C E. Therefore, one may use the vertical covariant differential D in order to construct Lagrangians for a mechanical system with parameters on the configuration space Q (5.9.1). We will suppose that a Lagrangian L depends on derivatives of parameters a™ only through the vertical covariant differential D (5.9.4), i.e., L = C{t, a m ) q\q\ ~ A* -

(5.9.6)

A^o?)*.

Such a Lagrangian is obviously degenerate because of the constraint condition nm + A^i

= 0.

As a consequence, the total system of the Lagrange equations (di - dt%)C = 0,

(5.9.7)

(dm - dt&JC = 0

(5.9.8)

admits a solution only if the very particular relation (dm + Aimdi)C + TTidtAin = 0

(5.9.9)

holds. However, we believe that parameter functions are background, i.e., inde­ pendent of a motion. Therefore, only the Lagrange equations (5.9.7) should be considered. In particular, let us apply the first variational formula (4.12.8) in order to obtain conservation laws of a mechanical system with time-dependent parameters. Let u = %% + um{t, ah)dm + u{(t, ak, qj)di,

u* = 0,1,

(5.9.10)

be a vector field which is projectable with respect to both the fibration Q - * B and Q —» E. Then the Lie derivative L^L of a Lagrangian L along the vector field u (5.9.10) reads [u% + umdm + v.% + dtiTdl (u

m

l

- a^u )dmC

+ dtu'dl}L =

+ 7r m d,(u m - a j V ) +

(«' - % - *[»ri(»*flj - u*) - u'£],

dt = dt +
+ ^mdt{um

- ffjV) - *[«&%

~ « ' ) ~ ^C]

CHAPTER

276

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

is fulfilled. If the Lie derivative L^L vanishes, we obtain the conservation law 0 w (um - aT^)dmC

+ nmdt(um - a?ul) - ^ [ ^ ( u ' g j - u*) - u«£]

(5.9.11)

for a system with time-dependent parameters. Now, let us describe such a system in the framework of Hamiltonian formalism. Its momentum phase space is the vertical cotangent bundle V*Q. Given a connection (5.9.3), we have the splitting (1.6.10b) which reads V*Q =

A{VZQ)®{QxV*Y,),

Pidq1 + pmdam

= Pi(dq' - A^da"1) + (pm +

A^da™.

Then VQ can be provided with the coordinates Pm=Pm+Alrnpi,

Pi=Pi,

compatible with this splitting. However, these coordinates fail to be canonical in general. Given a section h of the parameter bundle S - * K , the submanifold {a = h(t),

Pm

= Q}

of the momentum phase space V*Q is isomorphic to the Legendre bundle V*Qh of the subbundle (5.9.2) of the fibre bundle Q —> R, which is the configuration space of a mechanical system with the parameter function h(t). Let us consider Hamiltonian forms on the momentum phase space VQ which are associated with the Lagrangian L (5.9.6). Given a connection

r = dt + rmdm

(5.9.12)

on the parameter bundle E —* K, the desired Hamiltonian form H = {pxdq{ + Pmdam) - faA" + pmTm + H(t, am,

q^p^dt

(5.9.13)

can be found, where

B = dt + rmdm + B'd{,

B1 = Ai + y^I™,

is the composite connection (1.6.8) on the fibre bundle Q —> K, which is defined by the connection A (5.9.3) on Q —» E and the connection F (5.9.12) on E —► R. The Hamiltonian function H satisfies the conditions ni(t,qi,am,diH(t,am,qi,ni))=7Th

Pid'n-n^c^q^^^d'n)

5.9. SYSTEMS

WITH TIME-DEPENDENT

PARAMETERS

111

which axe obtained by substitution of the expression (5.9.13) in the conditions (5.5.2a) - (5.5.2b). The Hamilton equations for the Hamiltonian form (5.9.13) read

ql = A' + A^r™ + dm j

Pu = -Pj{diA + ^ r ) _m

(5.9.14a)

-

W,

run

(5.9.14b) (5.9.14c)

Ptm = -Pi(dmA{ + r n a m 4 ) -

dmn,

(5.9.14d)

whereas the Lagrangian constraint space is given by the equations pi =

TTi(t,q\(Tm,diH(t,CTm,q\pi)),

pm + AIJH = 0.

(5.9.15) (5.9.16)

The system of equations (5.9.14a) - (5.9.16) are related in the sense of Section 4.5 to the Lagrange equations (5.9.7) - (5.9.8). Because of the equations (5.9.14d) and (5.9.16), this system is overdetermined. Therefore, the Hamilton equations (5.9.14a) - (5.9.14d) admit a solution living in the Lagrangian constraint space (5.9.15) - (5.9.16) if the very particular condition, similar to the condition (5.9.9), holds. Since the Lagrange equations (5.9.8) axe not considered, we also can ignore the equation (5.9.14d). Note that the equations (5.9.14d) and (5.9.16) determine only the momenta pm and do not influence other equations. Therefore, let us consider the system of equations (5.9.14a) - (5.9.14c) and (5.9.15) - (5.9.16). Let the connection T in the equation (5.9.14c) be complete and admit the integral section h(t). This equation together with the equation (5.9.16) defines a submanifold V'Qh of the momentum phase space V*Q, which is the mo­ mentum phase space of a mechanical system with the parameter function h(t). The remaining equations (5.9.14a) - (5.9.14b) and (5.9.15) are the equations of this system on the momentum phase space V*Qh, which correspond to the Lagrange equations (5.9.7) in the presence of the background parameter function h(t). Conversely, whenever h(t) is a parameter function, there exists a connection T on the parameter bundle E —» R such that h(t) is an integral section T. Then the system of equations (5.9.14a) - (5.9.14b) and (5.9.15) - (5.9.16) describes a mechanical system with the background parameter function h(t). Moreover, we can locally restrict our consideration to the equations (5.9.14a) - (5.9.14b) and (5.9.15). The following examples illustrate the above construction.

278

CHAPTER 5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

E x a m p l e 5.9.1. Let us consider the 1-dimensional motion of a probe particle in the presence of a force field whose centre moves. The configuration space of this system is the composite fibre bundle M3 -> M2 -> E, coordinated by (t, a, q) where a is a coordinate of the field centre with respect to some inertial frame and q is a coordinate of the probe particle with respect to the field centre. There is the natural inclusion QxTY,B

{t,a,q,i,a)

^ {t,o,i,a,y

= -a)

eTQ

which defines the connection A = dt ® dt + da (d„ - dq) on the fibre bundle Q —> E. The corresponding vertical covariant differential (5.9.4) reads

D = (qt + at)d„. This is precisely the velocity of the probe particle with respect to the inertial frame. Then the Lagrangian of this particle takes the form L=[^(qt

+

at)2-V(q)}dt.

(5.9.17)

In particular, we can obtain the energy conservation law for this system. Let us consider the vector field u = dt. The Lie derivative of the Lagrangian (5.9.17) along this vector field vanishes. Using the formula (5.9.11), we obtain 0 « —TTq(7u — dt[irqqt — £], or 0 ~ dqCat ~ dt[irq{qt + ot) - £], where X = 7r,(ft+ < r t ) - £ is the energy function of the probe particle with respect to the inertial reference frame. •

5.9. SYSTEMS

WITH TIME-DEPENDENT

PARAMETERS

279

Example 5.9.2. Let us consider the 1-dimensional oscillator whose frequency is a time-dependent parameter. In accordance with the adiabatic hypothesis, there exists a coordinate q such that this oscillator differs from the standard one only in kinetic energy. Let us take the configuration space Q = RxR+xR-»R+xR-+R, provided with the coordinates (t, a > 0, q), where a is the frequency parameter. The corresponding momentum phase space V*Q is coordinated by {t,cr,q,pa,pg). The oscillator is characterized by the Hamiltonian form H = pada + pqdy — Hdt, H=paT°

+ \{oyq

+

q\

(5.9.18)

The corresponding Hamiltonian frame connection TH = dt + T°tda

(5.9.19)

on the configuration bundle Q —► R is the composite connection (1.6.8) generated by some linear connection r = dt ® (dt + T(t)oda) on the parameter bundle S = K

+

xl-.R

and by the trivial connection A = dt ® dt + da ® da on the fibre bundle Q —> E. The Hamiltonian form (5.9.18) is associated with the Lagrangian L=\{o-2c?t-qi)dt which describes the oscillator with time-dependent frequency a = h{t) in accordance with the adiabatic hypothesis.

280

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

The Hamilton equations (5.9.14a) - (5.9.14c) for the Hamiltonian form (5.9.18) read (5.9.20a)

qt = ^ 2 P„

(5.9.20b)

Pt = -9, ct = T{t)a.

(5.9.20c)

They also describe the oscillator with the variable frequency a = h{t) if we put r = h~ldth. The Lagrangian constraint space equations (5.9.15) are trivial. Let us consider the canonical transformation q = aq', 1 , ,

1 i , QPqa Relative to the new coordinates (t,a, ^ . p ' ^ . p ' , ) , the connection (5.9.19) is written Po=Va

rH = dt + T°{da - q'dq,), while the Hamiltonian form (5.9.18) reads H = p'ada + p'qdq' - Hdt,

n = p^-p'qq'T

+ \(pf +

a^).

Accordingly, the Hamilton equations (5.9.20a) - (5.9.20c) take the form

Pty = \PT q't =

- -V,

-±q>T°+p'q,

<>t = 1 7 -

(5.9.21a) (5.9.21b) (5.9.21c)

Substitution of a = h(t) and the equation (5.9.21c) in the equations (5.9.21a) and (5.9.21b) leads to the Hamilton equations corresponding to the Hamiltonian form H = p'qdq' - Hdt,

H = -i<Wp;+^(p',2+*v2).

5.9. SYSTEMS

WITH TIME-DEPENDENT

PARAMETERS

281

This Hamiltonian form describes the well-known Berry oscillator where the connec­ tion (5.9.5) is the classical variant of the Berry connection Ah = dt

a

dtaq'dq/

[135, 144]. • Example 5.9.3. Let us consider an n-body as in Example 4.9.4. Let R3™-3 be the translation-reduced configuration space of the mass-weight Jacobi vectors {pA}Two configurations {pA} and {pA} are said to define the same shape of the nbody if pA = RpA for some rotation R £ 5 0 ( 3 ) . This introduces the equivalence relation between configurations, and the shape space S of the n-body is defined as the quotient S = R3n-3/50(3) [119, 120]. Then we have the composite fibre bundle 1 x R 3 n _ 3 -» R x S -» R,

(5.9.22)

where the fibre bundle K 3 "" 3 _> S

(5.9.23)

has the structure group 5 0 ( 3 ) . The composite fibre bundle (5.9.22) is provided with the bundle coordinates(t, am,ql), where q', i = 1,2,3, are some angle coordinates, e.g., the Eulerian angles, while am, m = 1 , . . . ,3n — 6, are said to be the shape coordinates on S. A section QA{om) of the fibre bundle (5.9.23), called a gauge convention, determines an orientation of the n-body in a space. Given such a section, any point {pa} of the translation-reduced configuration space R3™-3 is written as

PA = R(qTQA{°m)This relation yields the splitting ~pA = diRcfQji + dm~eAdm of the tangent bundle TR 3 n 3 , which determines a connection A on the fibre bundle (5.9.23). This is also a connection on the fibre bundle R x R 3 " - 3 -► R x 5 ,

282

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

with the components At = 0. Then the corresponding vertical covariant differential (5.9.4) reads

D = (4 - /&?)*. With this vertical covariant differential, the total angular velocity of the n-body takes the form (5.9.24)

0 = a^D* = u + Am<77\

where &i are certain kinematic coefficients and 3 is the angular velocity of the nbody as a rigid one. In particular, we obtain the phenomenon of a faffing cat if 0 = 0 so that u = -A m
5.10

Unified Lagrangian and Hamiltonian formalism

The relationship between Lagrangian and Hamiltonian formalisms in Section 5.5 is broken by canonical transformations if the transition functions q' —♦
□

JlV"Q,

ft *—> Pu,

4t <—► P i ,

(5.10.1)

5.10. UNIFIED LAGRANGIAN

AND HAMILTONIAN

FORMALISM

283

Proof. The isomorphism (5.10.1) can be proved by an inspection of transition functions of the coordinates {%,$) and (puPti)QED Due to the isomorphism (5.10.1), one can think of II as being both the momen­ tum phase space over the velocity phase space JlQ and the velocity phase space over the momentum phase space V*Q. Hence, the space II can be utilized as the unified phase space of joint Lagrangian and Hamiltonian formalism. Remark 5.10.1. In connection with this, note that, according to [10, 127, 180], the dynamics of autonomous mechanical system described by a degenerate Lagrangian L on TM is governed by a differential equation on T*M generated by dL, due to the canonical diffeomorphism between TT'M and T*TM. • The unified phase space I I is equipped with the holonomic coordinates {t,q\qlt,PtuPi),

(5.10.2)

where {q',pu) and {q\,Pi) are canonically conjugate pairs. It is endowed with the jet prolongation of the canonical 3-form (4.8.10), which with respect to the coordinates (5.10.2) reads ft = (dpti A dq' + dpi A dq\) A dt = dt{dpi A dq' A dt),

(5.10.3)

where dt = dt + q\di + ptid' is the total derivative on II. The corresponding Poisson bracket (5.1.5) takes the form {/,ff}n

df_ 0£

df_ dg_ _ dg_dl _ dg_dl

dptidqi

dpidqt

dpudq*

dpidq\

(5.10.4)

It is readily observed that the canonical form (5.10.3) and, consequently, the Poisson bracket (5.10.4) are invariant under the transformations of the unified phase space II, which are the jet prolongation of the canonical automorphisms of the phase space V*Q. Let H

Ptidq'+ Pidq\ -

Hu(t,q',q\,pu,Pi)dt

CHAPTER

284

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

be a Hamiltonian form on II. The corresponding Hamilton equations (5.2.18a) (5.2.18b) read (5.10.5a) OPti

, ,

,

dHn dpi

(5.10.5b)

dHn

(5.10.5c)

dHn

(5.10.5d)

Substitution of (5.10.5a) in (5.10.5b) and of (5.10.5c) in (5.10.5d) leads to the equa­ tions mn = &Hn dpu dpi ' &Hn _ dHn * dq\ " dq'

(5.10.6a) (5.10.6b)

which look like the Lagrange equations for the "Lagrangian" Tin- Though Hn is not a true Lagrangian, one can put

H„ = -C + dt(Pir),

(5.10.7)

whenever L is a Lagrangian on the velocity phase space J Q. Then the equations (5.10.6a) - (5.10.6b) are equivalent to the Lagrange equations for a Lagrangian L on JlQ. However, their solutions fail to be solutions of the corresponding Ha.milton equations (5.10.5a) - (5.10.5d) for the Hamiltonian (5.10.7) in general. This illustrates the fact that solutions of the Hamilton equations (5.10.5a) - (5.10.5d) are necessarily solutions of the Lagrange equations (5.10.6a) - (5.10.6b), but the converse statement is not true. To construct the joint Lagrangian-Hamiltonian formalism, let us consider the Hamiltonian form H = pudg' + Pidgj - (dtH + {ml -H)-

C)dt

(5.10.8)

on the unified phase space II, where L is a semiregular Lagrangian on the velocity phase space JlQ and H is an associated Hamiltonian form on the momentum phase

5.11.

VERTICAL

EXTENSION

OF HAMILTONIAN

285

FORMALISM

space V*Q. The Hamilton equations (5.10.5a) - (5.10.5d) for H (5.10.8) read

<w

&H dpi'

dtq]

2m

(5.10.9a)

OPi

dtPi dtPu

+

(5.10.9b)

dpi dC

on

(5.10.9c)

dq*

-dt

an an dc dqi

+

dqi

+

dtf

(5.10.9d)

Using the relations (5.5.2a) and (5.5.7), one can show that solutions of the Hamil­ ton equations (5.2.18a) - (5.2.18b) for the Hamiltonian form H, which live in the Lagrangian constraint space L^Q) C V*Q are solutions of the equations (5.10.9a) - (5.10.9d). Now let us consider the Lagrange equations (5.10.6a) - (5.10.6b) for the Hamil­ tonian form (5.10.8). They read


dpi

o,

(5.10.10a) _dU _ d£ dqi dqr

(5.10.10b)

In accordance with Proposition 5.5.7, every solution of the Lagrange equations for the Lagrangian L such that the relation (5.5.22) holds is a solution of the equations (5.10.10a) - (5.10.10b). In particular, if the Lagrangian L is hyperregular, the equations (5.10.10a) (5.10.10b) and the equations (5.10.9a) - (5.10.9d) are equivalent to the Lagrange equations for L and the Hamilton equations for the associated Hamiltonian form.

5.11

Vertical extension of Hamiltonian formalism

Let Q —► R, coordinated by {t,ql), be a configuration space of time-dependent mechanics. Let us consider the vertical tangent bundle VQ of the fibre bundle Q —> R, coordinated by (t,q',q'). It can be seen as the new configuration space, called the vertical configuration space. Here, we will be concerned with the following three applications of Hamiltonian mechanics on this configuration space: • mechanical systems with linear deviations (see Example 5.11.1 below),

CHAPTER

286

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

• Hamiltonian description of non-holonomic constraints (see Example 5.11-3 be­ low), • Hamiltonian BRST mechanics (see Appendix A). Given VQ, the corresponding velocity phase space is the first order jet manifold JlVQ of the fibre bundle VQ -» R. It is canonically isomorphic (see (1.3.8)) to the vertical tangent bundle VJlQ of the ordinary velocity phase space JXQ -» M, coordinated by

{% M. It plays the role of a momentum phase space, called the vertical momentum phase space, of mechanical systems on the configuration space VQ. The vertical momentum phase space is canonically isomorphic (see (1.1.10)) to the vertical tangent bundle VV'Q of the ordinary mo­ mentum phase space V*Q —> K, coordinated by (t,q',Pi,q',Pi).

It is easily seen from the transformation laws Pi =

.,

djlPv

dp\ .

dp[

dq> .

dtf

that (q\fii) and {q',Pi) are canonically conjugate pairs. Thus, we have observed that the vertical velocity and momentum phase spaces, corresponding to the configuration space VQ, are the vertical tangent bundles of ordinary velocity and momentum phase spaces, respectively. The momentum phase space VV'Q is endowed with the canonical 3-form Qv = [dpi A dql + dpi A dq^] A dt. For brevity, one can write flv = dy£l,

(5.11.1)

5.11.

VERTICAL

EXTENSION

OF HAMILTONIAN

FORMALISM

287

where fi is the canonical 3-form (5.1.9) on the momentum phase space V*Q, and the formal operator of the vertical derivative dy = q'di + pjd' (1.1.4) is utilized. This operator possesses the properties dv(4> ACT) = dv() ACT +
^ae&iV'Q),

dv{dcj>) = d{dv{cj>)). The canonical 3-form (5.11.1) provides VVQ ture, given by the Poisson bracket {/, g}vv = frfdjg + d'fdjg - VgdJ

-

with the canonical Poisson struc­

frgdj.

Recall the notation (1.2.4). The notions of a Hamiltonian vector field, a Hamiltonian connection, a Hamiltonian form and others on the vertical momentum phase space VV*Q = V*VQ are introduced in accordance with the general formalism of Hamiltonian time-dependent mechanics, expressed in Section 5.2. In particular, any Hamiltonian form on VV*Q reads Hy = Pidq1 + Pidq' — Hydt. Since a Hamiltonian form is defined modulo exact forms and the function p{
(5.11.2)

The corresponding Hamilton equations read V = ql = li =Pti =

d'Hv, -diHv,

(5.11.3a) (5.11.3b)

f = 4 = d'Hv,

(5.11.3c)

7i =Pti =

(5.11.3d)

-diHv,

where 7 = dt + J% + "fid* + j % + jid*

CHAPTER

288

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

is a Hamiltonian connection on the momentum phase space VV'Q —> R. We have the following relationship between Hamiltonian formalism on the mo­ mentum phase space V'Q and that on VV'Q. PROPOSITION 5.11.1. Let 7# be a Hamiltonian connection on the ordinary mo­ mentum phase space V*Q —► R for a Hamiltonian form (5.11.4)

H = pidq* - Hdt.

Then the connection VjH (1.6.12) on the vertical momentum phase space VV'Q —► R is a Hamiltonian connection for the Hamiltonian form (5.11.5)

Hv = Ptdq' — q'dpi — dyTidt, dvH = (q% + pj&yH, called the vertical extension of the Hamiltonian form H. D Proof. A direct computation shows that, given a Hamiltonian connection lH = dt + 'yidi + 'yid', f = d'H,

7«

=

-diH,

on the fibre bundle V'Q —> R, the connection V-TH = dt + j'di + 7idl + fdi + j{d\ 7 1 = dv-y',

7i = dvlu

on the fibre bundle VV'Q -» R satisfies the Hamilton equations (5.11.3a) - (5.11.3d) for the Hamiltonian form (5.11.5), i.e.,

y = titty = d'H,

(5.11.6a)

7i = -mv

= -diH,

(5.11.6b)

f = d'Hv = dvdln,

(5.11.6c)

7i = -diHv

(5.11.6d)

=

-dvdiH.

QED In particular, given a splitting H

= p,-r + HT

5.11.

VERTICAL

EXTENSION

OF HAMILTONIAN

FORMALISM

289

of a Hamiltonian Tt with respect to a connection Y on Q —> R, there is the corre­ sponding splitting

Hv = ftP - tfi-pidjF) + dvHT of the Hamiltonian Hv (5.11.5) with respect to the connection T (5.2.6) on the fibre bundle V*Q -^ R. It is easily seen that the Hamilton equations (5.11.6a) - (5.11.6b) for the Hamil­ tonian form Hv (5.11.5) are precisely the Hamilton equations (5.2.18a) - (5.2.18b) for the Hamiltonian form H. E x a m p l e 5.11.1. In order to clarify the physical meaning of the Hamilton equations (5.11.6c) - (5.11.6d), let us suppose that Q —> R is a vector bundle. Due to the splitting

VQ^Q®Q, we can think of the vertical configuration space VQ as the configuration space of the initial mechanical system and its small linear deviations

3* + eg*,

£€R,

where £ is a small parameter. Accordingly, the vertical momentum phase space

VVQ =

VQ®VQ

can be seen in a similar way as the momentum phase space of the initial Hamiltonian mechanical system and its small linear deviations qi + eq\

Pi + epi-

Let if be a Hamiltonian form on the momentum phase space V*Q of the initial Hamiltonian system, and let r{t) be a solution of the Hamilton equations (5.11.6a) - (5.11.6b). Let r(£) be a Jacobi field, i.e.,

r(t) + ef(t) is also a solution of the same Hamilton equations modulo terms of order two in e. Then it is readily observed that the Jacobi field f(t) satisfies the Hamilton equa­ tions (5.11.6c) - (5.11.6d) for the Hamiltonian form Hv (5.11.5). Substituting a solution r(t) of the Hamilton equations (5.11.6a) - (5.11.6b) in the Hamilton equa­ tions (5.11.6c) - (5.11.6d), we obtain the system of homogeneous linear differential

CHAPTER 5. HAMILTONIAN

290

TIME-DEPENDENT

MECHANICS

equations for the Jacobi field F(t), which are actually the differential equations of linear deviations. • Returning to the definition of a Hamiltonian form Hv (5.11.5), let us consider the vertical tangent bundle VT*Q of the cotangent bundle T*Q -» E. It is equipped with holonomic coordinates {t,q',Pi,P,q',Pi,p) and endowed with the canonical form Ey = pdt + pidq1 — qldpi. One may also utilize another canonical form Ev + d{q'pi) since the exact form d(<j*p,*) is globally defined. By very definition, a Hamiltonian form H (5.11.4) on the ordinary momentum phase space V*Q is the pull-back H = h'E of the Liouville form E (5.1.2) on the cotangent bundle T*Q by a section h of the fibre bundle T'Q -> V'Q. Then we have Hv =

(VhYEv,

where Vh : VVQ

-»

VT'Q

is the vertical tangent map to h. We can also generalize Proposition 5.11.1 as follows. PROPOSITION 5.11.2. Any connection 7 on the momentum phase space V'Q —» M gives rise to a Hamiltonian connection on the vertical momentum phase space VVQ - > R . D Proof. Let us consider the Hamiltonian form Hv = pijdq1 - fdt)

- ^(dpi - -gdi) = p i dg i - tfdpj - (pg* -

tf-y^dt

on the vertical momentum phase space VV*Q. The corresponding Hamiltonian connection on VVQ —> R is given by the Hamilton equations (5.11.3a) - (5.11.3d) and reads

f = 7\

7i = 7t,

V-fcay-a'aS,

7i = -Pj-8i7 J +? 3 9i7j.

(5.11.7)

5.11.

VERTICAL

EXTENSION

OF HAMILTONIAN

FORMALISM

291

In particular, if 7 is a locally Hamiltonian connection on the fibre bundle V*Q —> R, i.e., satisfies the relations (5.2.4a) - (5.2.4c), then the Hamiltonian connection (5.11.7) is precisely the connection V7 (1.6.12). QED It follows that every first order dynamic equation (5.2.20) on the momentum phase space VQ can be seen as the Hamilton equations (5.11.3a) - (5.11.3b) for a suitable Hamiltonian form on the vertical momentum phase space. Example 5.11.2. The equations of a motion (5.2.21) of a point mass m subject to friction in Example 5.2.5 are the Hamilton equations (5.11.3a) - (5.11.3b) for the Hamiltonian Wv =p{—p-\ q) m m on the vertical momentum phase space. • Similarly to the relationship between Hamiltonian formalisms on ordinary and vertical momentum phase spaces, that between Lagrangian formalisms on ordinary and vertical velocity phase spaces can also be considered. Given a Lagrangian L = Cdt (4.8.1) on an ordinary velocity phase space JlQ, let us consider the vertical tangent morphism VC : JlQ -* R x R to the Lagrangian function C, and the Lagrangian Ly = (pr 2 ° V£)dt = Cydt, Cy=dyC=

(5.11.8)

(fdj + q\d\)C,

on the vertical velocity phase space VJlQ. It is called the vertical extension of the Lagrangian L. The corresponding Lagrange equations read

(4 - dtdt)cv = 0,

(5.11.9a)

{d, - dtdt)£v = 0.

(5.11.9b)

The Lagrange equations (5.11.9a) are precisely the Lagrange equations for the initial Lagrangian L. If a configuration space Q —> M is a vector bundle, the Lagrange equations (5.11.9b), just as the Hamilton equations (5.11.6c) - (5.11.6d) do, describe small linear deviations of the initial mechanical system.

292

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

Let us turn now to the relationship between Lagrangian and Hamiltonian for­ malisms on the vertical configuration space. Each Lagrangian Lv (5.11.8) on the vertical velocity phase space VJlQ yields the Legendre map LV = VL:VJ1Q Pi = d\CV

= TTi,

^VV'Q,

VQ

Pi =

dy^i,

(5.11.10) (5.11.11)

which is the vertical tangent map to the Legendre map L. Each Hamiltonian form Hv (5.11.5) on the vertical momentum phase space VV*Q yields the Hamiltonian map HV = VH : VV'Q qi

= ffnv = am,

-> VJXQ,

(5.11.12)

VQ

q\ = dvd'H,

which is the vertical tangent map to the Hamiltonian map H. PROPOSITION 5.11.3. If a Hamiltonian form H on the momentum phase space VQ is associated with a Lagrangian L on the velocity phase space JlQ, then its vertical extension Hv (5.11.5) on the vertical momentum phase space VV'Q is associated with the vertical extension Lv (5.11.8) of the Lagrangian L on the vertical velocity phase space V JXQ. D Proof. By virtue of the relations (5.11.10) and (5.11.12), the condition (5.5.2a) for the Hamiltonian form Hv reads VL o VH o VL = VL, and follows at once from the condition (5.5.2a) for the Hamiltonian form H. The condition (5.5.2b) for the Hamiltonian form Hv, written in the coordinate form (5.5.5), follows immediately from the condition (5.5.5) for the Hamiltonian form H. QED The equalities (5.11.11) show that, if a Lagrangian L on the velocity phase space JlQ is semiregular, so is its vertical extension Lv (5.11.8) on the vertical velocity phase space VJlQ. Therefore, Proposition 5.11.3 can also be formulated for a semiregular Lagrangian. In particular, if the relation (5.5.16) is fulfilled for H and L, it is true for the vertical extensions Hv and Ly-

5.11.

VERTICAL

EXTENSION

OF HAMILTONIAN

293

FORMALISM

Example 5.11.3. Let us consider the Hamiltonian counterpart of the constrained motion equation (4.11.15) in the case of a hyperregular Lagrangian and a nonholonomic constraint. By virtue of Proposition 5.11.2, it can be represented as the Hamilton equations (5.11.3a) - (5.11.3b) for a suitable Hamiltonian form on the vertical phase space VV*Q. Let L be a hyperregular Lagrangian with a Riemannian mass metric m and S a codistribution on the velocity phase space JXQ, whose annihilator Ann (S) is an admissible non-holonomic constraint on JlQ. In the case of a hyperregular Lagrangian, Hamiltonian and Lagrangian formalisms of time-dependent mechanics are equivalent, and we have 7H = J ' i o ^ o f f . Let introduce the notation M , J = d'&>H. There are the relations Mik(mkj

mkj(Mik o I) = 6),

o H) = 8},

TTlij = Tty.

It follows that M is a fibre metric in the vertical tangent bundle VQV'Q of the fibre bundle VQ ~* Q. Given the above-mentioned codistribution S on JXQ, let us consider the pull-back codistribution H*S on V*Q spanned locally by the 1-forms /?» = H*sa = (sg + s^dtd>n)dt + (s? + s'di&HW

+ s°MijdPj

=

ai

ffidt + (3fdq' + $ dPi. This codistribution defines a non-holonomic constraint on the momentum phase space V*Q. Given a Hamiltonian connection 7// (5.2.17), let us find its splitting 1H

= 7+ 0

(5.11.13)

where 7 is a connection on V*Q —► R which satisfies the condition 7 C Ann(iTS).

(5.11.14)

The connection 7 (5.11.14) obviously defines a first order dynamic equation on the momentum phase space V*Q, which is compatible with the non-holonomic con­ straint H*S. The decomposition (5.11.13) is not unique. Let us construct it as follows.

294

CHAPTER

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

Given a Hamiltonian connection 7//, we consider the codistribution SH on V'Q, spanned locally by the 1-forms dq' - i\jdt. Its annihilator Ann (SH) is an affine subbundle of the affine jet bundle JlV'Q -> V'Q modelled over the vertical tan­ gent bundle VQV'Q. The Hamiltonian connection j H is obviously a section of this subbundle. Let us consider the intersection (5.11.15)

W = Ann(SH)nAnn(#*S).

LEMMA 5.11.4. The intersection W (5.11.15) is an affine bundle over V'Q, modelled over the vector bundle W = VQV*Q

nAnn(H'S).

a Proof. The intersection W consists of elements v of VQV'Q ditions

which fulfill the con­

v{pa' = 0. Since the non-holonomic constraint S is admissible and the matrix M%i is nondegenerate, every fibre of W is of dimension m — n, i.e., W is a vector bundle, while W is an affine bundle. QED Then, using the fibre metric M in VQV'Q, VQV'Q

= W e

we obtain the splitting

V,

where V is the orthocomplement of W, and the associated splitting Ann (SH) = W ® V. The corresponding decomposition (5.11.13) reads 7 =

lH-MabMtJpaipb(lH)d>,

where Mat, is the inverse matrix of Mab =

fripMMij.

(5.11.16)

5.11.

VERTICAL

EXTENSION

OF HAMILTONIAN

FORMALISM

295

The splitting (5.11.16) is the Hamiltonian counterpart of the splitting (4.11.15). We have the relations ~ab

=

Mab

Q

^

/3"(7tf) = s Q ( a ) ° H,

and as a consequence 7 = J^ofoff. Note that the above procedure can be extended in a straightforward manner to any standard Newtonian system seen as a Lagrangian system with the Lagrangian (4.9.19) and an external force. Following this procedure, one may also study a nonholonomic Hamiltonian system, without appealing to its Lagrangian counterpart. The connection (5.11.16) defines the system of first order dynamic equations

«{= d*n,

Pu = -W

- Mo6My^/3»(7ff)

(5.11.17)

on the momentum phase space V*Q, which are not Hamilton equations. Neverthe­ less, in accordance with Proposition 5.11.2, one can restate the constrained motion equations (5.11.17) as the Hamilton equations (5.11.6a) - (5.11.6b) for the Hamil­ tonian form Hv = fad? - tfdpi - dvHdt -

qiMabMij$a:i(3b('yH)dt

on the vertical momentum phase space VV'Q. Its last term can be written in brief as (—<9v/J?9J$7). To exclude the remaining Hamilton equations (5.11.6c) - (5.11.6d), one can choose their solutions q' = pi = 0. • Remark 5.11.4. Let Hy be a Hamiltonian form (5.11.2) on the vertical momen­ tum phase space VV'Q. Similarly to the Lagrangian (5.7.1), let us consider the Lagrangian LH = Piq\ - qlPu - 'Hv

(5.11.18) l

on the first order jet manifold J VV*Q of the fibre bundle VV*Q —> M. It is readily observed that its Lagrange equations are precisely the Hamilton equations (5.11.3a) - (5.11.3d) for the Hamiltonian form Hy- In particular, let H be a Hamiltonian form on an ordinary momentum phase space VQ and "Hv = dy'H- In this case, the Lagrangian (5.11.18) reads

LH=Pi(qi-dlH)-qi(Pti

+ diH).

CHAPTER

296

5. HAMILTONIAN

TIME-DEPENDENT

MECHANICS

It is easily seen that this Lagrangian vanishes on solutions of the Hamilton equations for the Hamiltonian form H. For this reason, it plays a prominent role for the functional integral formulation of mechanics [65, 66] •

5.12

Appendix. Time-reparametrized mechanics

We have assumed above that the base M of a configuration space of time-dependent mechanics is parameterized by the standard coordinate t with the transition func­ tions t —> t' = t+ const. Here, we consider an arbitrary reparametrization of time

t -» a = m

(5.12.1)

which is discussed in some models of quantum mechanics [75, 153, 154]. In this case, K is not only an affine space as before, but a 1-dimensional manifold. Then the stan­ dard vector field dt and the standard 1-form dt on R are not invariant. Therefore, we should follow the polysymplectic Hamiltonian formalism of Section 3.8. Nev­ ertheless, Hamiltonian formalism of time-reparametrized mechanics possesses some peculiarities due to R. 1. There exists the invariant tangent-valued 1-form eR = dt® dt on the base R of a configuration space of time-reparametrized mechanics. 2. The velocity phase space J1Q of time-reparametrized mechanics is no more an affine subbundle of the tangent bundle TQ under time reparametrizations. Nev­ ertheless, the momentum phase space U —> Q (3.8.1) is isomorphic to the vertical cotangent bundle V*Q —> Q. It follows that the momentum phase space of timereparametrized mechanics is provided with the canonical Poisson structure (5.1.5). Moreover, this Poisson structure is invariant under time reparametrization (5.12.1) which, consequently, is a canonical transformation. 3. The momentum phase space V*Q is endowed with the canonical polysym­ plectic form A = dpi A dq' Adt®

dt.

Then the notions of a Hamiltonian connection and a Hamiltonian form are the rep­ etitions of those in Section 3.8. At the same time, since the homogeneous Legendre

5.12. APPENDIX.

TIME-REPARAMETRIZED

MECHANICS

297

bundle (2.9.3) of time-reparametrized mechanics is the cotangent bundle T*Q of Q, Hamiltonian forms and Hamilton equations of time-reparametrized mechanics have the same form as those in time-dependent mechanics. The difference is only that a Hamiltonian function Tir in the splitting (5.2.12) is a density, but not a function under the transformations (5.12.1). 4. Since Lagrangians and Hamiltonians of time-reparametrized mechanics are densities under the transformations (5.12.1), one should introduce a volume ele­ ment on the base M in order to construct them in explicit form. A key problem of models with time reparametrization lies in the fact that the volume dt on the time axis M is not invariant under time reparametrization transformations. Another problem is concerned with mass tensors which also fail to be invariant under time reparametrizations.

Chapter 6 Relativistic mechanics Let us note on the following two main peculiarities of relativistic mechanics in com­ parison with non-relativistic one. • A configuration space in general or, at least, cannot use the familiar generalize it for jets of

Q of relativistic mechanics has no fibration Q —> R it has no preferable fibration over K. Therefore, one formalism of jets of sections of fibre bundles, but must submanifolds.

• Hamiltonian relativistic mechanics is described as an autonomous constraint Dirac system on the hyperboloids of relativistic momenta in the momentum phase space T*Q. Relativistic mechanics can be formulated on an arbitrary configuration space Q. If Q is a 4-dimensional manifold provided with a pseudo-Riemannian metric g, we come to relativistic mechanics in the presence of a gravitational field g. If Q = R 4 and g is the Minkowski metric, this is the case of Special Relativity.

6.1

Jets of submanifolds

We start from the general notion of jets of submanifolds [57, 104]. DEFINITION 6.1.1. Let Z be a manifold of dimension m + n. The first order jet manifold J\Z of n-dimensional submanifolds of a manifold Z comprises the equivalence classes [S]lz, z € Z, of n-dimensional imbedded submanifolds of Z which

299

300

CHAPTER 6. RELATIVISTIC

MECHANICS

pass through z € Z and which are tangent to each other at z. There is the natural fibration J\Z -> Z. D R e m a r k 6.1.1. In fact, the definition of the jet [S}1 of submanifolds involves only local properties of submanifolds around the point z € Z. It is easily extended to higher order jets of submanifolds. By definition, J°Z = Z. • The set of jets J\Z is provided with a manifold structure as follows. Let Y —> X be an (m + n)-dimensional fibre bundle over an n-dimensional base X and let $ be an imbedding of Y into Z. Then there is the natural injection of the jet manifold JlY of the fibre bundle Y —> X into J\Z such that 7 1 * : JlY -> J\Z, Jl

S

»-» [Sjsfgfa;))!

(6.1.1) S = Im($os),

where s are sections of Y —» X. PROPOSITION 6.1.2. The injection (6.1.1) defines a chart on J^Z. Such charts cover the set J^Z, and transition functions between these charts are differentiable. They provide the set J\Z with the structure of a finite-dimensional manifold. □ Proof. The proof is based on the fact that, given a submanifold S C Z which belongs to the jet [S\l, there exists a neighbourhood Uz of the point z and the tubular neighbourhood Us of Uz D S so that the fibration

ua->u,ns takes place [109]. This means that every jet [S]l belongs to a chart of the abovementioned type. We will describe these charts and the corresponding transition functions in explicit form. QED It is convenient to use the following coordinate atlases of the jet manifold J\Z of n-dimensional submanifolds of Z. Let Z be equipped with a coordinate atlas

Mz*)},

A= 1,... ,n + m.

(6.1.2)

6.1. JETS OF

301

SUBMANIFOLDS

Though J°Z, by definition, is diffeomorphic to Z, let us provide J°Z with the atlas where every chart (U; zA) on a domain U C Z is replaced with the

(

n + m\ _ (n + m)\ m J n\m\

charts on the same domain U which correspond to different partitions of the collec­ tion (z 1 • • • zA) in the collections of n and m coordinates, denoted by

(zV),

A = 1, ,..,!»,

i = 1 , . . . ,m.

(6.1.3)

The transition functions between the coordinate charts (6.1.3) of J°Z, associated with the same coordinate chart (6.1.2) of Z, are reduced simply to an exchange between coordinates xx and yl. Transition functions between arbitrary coordinate charts (6.1.3) of the manifold J°Z read

2 A =ffV*,tf0. x* = g*{x?,yl)1

yi = fi(x",yj),

(6.1.4)

3

f = fi^y ).

Given the coordinate atlas (6.1.3) of the manifold J°Z, the jet manifold J\Z is provided with the adapted coordinates

(x\y\yi),

X = l,...,n,

i =

1,...,»?i.

(6.1.5)

R e m a r k 6.1.2. If S C Z is an imbedded submanifold which belongs to the jet [S]l, there exists an open neighbourhood Uz of z £ Z and a coordinate chart (xx,yl) (6.1.3) which covers Uz so that SfMJz\s given by the coordinate relations y{ = 5 { (x A ) and

(x\y\yi)([S]l) = (xx(z),Si(^(z)),dxSi(x"(z))). • Using the operators of total derivatives d\ = dx + y\di,

CHAPTER

302

6. RELATIVISTIC

MECHANICS

one can obtain the transition functions of the coordinates (6.1.5) under coordinate transformations (6.1.4) in explicit form. Given coordinate transformations (6.1.4), one can easily find that d~x =

[d~,ga{x^^)]dxa.

(6.1.6)

Then we have

t-Kw+K&ypM] ( ^ + ^ ) / w ) -

(6.1.7)

It is readily observed that the transition functions (1.3.1) are a particular case of the coordinate transformations (6.1.7) when the transition functions ga (6.1.4) are independent of coordinates y1. Note that, in contrast with (1.3.1), the coordinate transformations (6.1.7) are not affine. It follows that the fibration J\Z —> Z is not an affine bundle. There is one-to-one correspondence

A : [5]^x^+y-([5U)a,)

(6.1.8)

between the jets [S]\ at a point z € Z and n-dimensional vector subspaces of the tangent space TZZ. It follows that the jet bundle J\Z -> Z has the structure group GL(n, m; M) C GL(m + n, R) of linear transformations of the vector space M m + n which preserve its subspace R n . Its typical fibre is the Grassman manifold Z is a Grassman bundle [45, 70]. In particular, if n = 1, the fibre coordinates y'0 on J\Z —> Z with the transition functions (6.1.7) are precisely the standard coordinates of the projective space R P m . Example 6.1.3. Let Y —> X be an (m + n)-dimensional fibre bundle over an ndimensional base X, and JlY the first order jet manifold of its sections. Let J„Y be the first order jet manifold of n-dimensional subbundles of Y. Then the injection JlY <-> J^Y (6.1.1) is an affine subbundle of the jet bundle J\Y -+ Y. Its fibre at a point y &Y consists of the n-dimensional subspaces of the tangent space TyY whose intersections with the vertical subspace VVY of TyY are the zero vector. •

6.2. RELATIVISTIC

VELOCITY

AND MOMENTUM

PHASE

303

SPACES

Remark 6.1.4. Generalizing the notion of a connection on a bundle, one may think of global sections of the jet bundle J\Z - > Z a s being preconnections on the manifold Z [134]. By virtue of the well-known theorem [80], if such a preconnection T exists, its image T(Z) in the tangent bundle TZ —> Z by the correspondence A (6.1.8) is a vector subbundle of TZ with the structure group GL(n, K). The quotient TZjY(Z) is also a vector subbundle with the structure group GL(m, M). Thus, we have the decomposition TZ = T{Z) e

TZ/T(Z)

of the tangent bundle TZ, which can be treated as a horizontal splitting with respect to the preconnection I\ However, it should be emphasized that, since J\Z —► Z is not an affine bundle, preconnections do not make an affine space. Moreover, it may happen that a preconnection does not exist on a manifold Z. •

6.2

Relativistic velocity and momentum phase spaces

Let us now turn to the case of relativistic mechanics, where Z = Q is an (m + 1)dimensional configuration space which has not necessarily a fibration or a preferable fibration over the time axis E. We call Q a reiativistic configuration space in compar­ ison with a non-relativistic configuration space Q endowed with a fibration Q —> M. This configuration space is provided with the coordinate atlas (6.1.3) together with the transition functions (6.1.4), i.e.,

(q°,q%

i = 1 , . . . , m,

«°-«W),

9W(«V)-

(6.2.1)

The coordinates q° in different charts of Z play the role of a temporal coordinate. Note that, given a coordinate chart (U; q°, q'), we have a local fibre bundle U 5 (q°,qi)^q°£

QCR

(6.2.2)

which can be treated as a configuration space of a local non-relativistic mechanical system. The velocity phase space of relativistic mechanics on the configuration space Q is the first order jet manifold J\Q of 1-dimensional subbundles of Q. It is endowed

CHAPTER 6. RELATIVISTIC

304

MECHANICS

with the coordinates (q°, q\qo) (6.1.5). Their transition functions are obtained as follows. Given coordinate transformations (6.2.1), the total derivative (6.1.6) reads d~0

djb(q°)dqo

[dq°+q°dq«)

dqO.

In accordance with the relation (6.1.7), we have —I

WK°(?)

\d
The solution of this equation is

(8?

I [dqO

9o

>9?\ + 9o

dq" ) ■

(6.2.3)

A glance at this transformation law shows that the velocity phase space J\Z —* Z of relativistic mechanics is really a projective bundle. Example 6.2.1. Let consider the configuration space Q = M4, provided with the Cartesian coordinates

(«V).

i

1,2,3.

This is the case of Special Relativity. For instance, let

9°

g°cha — q1sha,

f
-<7°sha + g'cha,

(6.2.4)

23

q'-

be a Lorentz transformation of the plane (q°, q1). Substituting these expressions in the formula (6.2.3), we obtain

ql

—sha + ggcha cha — q^sha 2,3

^■3

go cha — go sha This is precisely the transformation law of 3-velocities in Special Relativity if we put 9o

cha

1

sha

v

6.2. RELATIVISTIC

VELOCITY

AND MOMENTUM

PHASE

SPACES

305

where v is the velocity of a reference frame moving along the axis q1. • Thus, one can think of the velocity phase space J\Q of relativistic mechanics as the space of 3-velocities v of a relativistic system. For the sake of convenience, we will call J\Q the 3-velocity phase space, though the dimension of Q is not equal necessarily to 3 + 1. Given a coordinate chart {U;q°,ql), 3-velocities v = (vl0) of a relativistic system can be seen as velocities of a local non-relativistic system (6.2.2) with respect to the corresponding local reference frame T = dt on U. However, the notion of a reference frame in non-relativistic mechanics has no meaning in relativis­ tic mechanics since the relativistic transformations ql0 —> <$ and P —> P 1 are not affine and the relative velocity gj — P is not maintained under these transformations. To introduce the relativistic velocities, let us consider the tangent bundle TQ of the configurations space Q. It is equipped with the holonomic coordinates

(9°,g\9°,
A^j^teV^JKTQ

(•6.2.5)

in the tangent space to Q at the point (q°,q')- Conversely, there is the morphism Q

TQ-+JIQ,

Qo°Q

9°'

(6.2.6)

such that go X

IdJlQ.

Indeed, it is readily observed that q'0 and ql/q° have the same transformation laws. It should be emphasized that, though the expressions (6.2.5) and (6.2.6) are singular at 5° = 0, this point belongs to another coordinate chart, and the morphisms A and g are well defined. Thus, one can think of the tangent bundle TQ as being the space of relativistic velocities or 4-velocities of a relativistic system. It is called the 4-velocity phase space.

CHAPTER

306

6. RELATIVISTIC

MECHANICS

Remark 6.2.2. Note that, with respect to the universal unit system, the elements of TQ are not physically dimensionless, in contrast with the physical relativistic velocities. • Since the morphism A of J\Q onto TQ is multivalued and the converse morphism (6.2.6) is a surjection, one may try to find a subbundle W of the tangent bundle TQ such that Q:W-J%

is an injection. Let us assume that Q is oriented and endowed with a pseudoRiemannian metric g with the signature (+, — . . . — ) . The pair (Q,g) is called a relativistic system. This metric g defines the subbundle of veJocity hyperboloids

Wg = {qx 6 TQ : g^q)??

= 1}.

(6.2.7)

Of course, Wg is neither a vector nor an affine subbundle of TQ. Let Q be timeoriented with respect to the pseudo-Riemannian metric g. By definition, this means that Wg is a disjoint union of two connected subbundles W+ and W~ [77, 155]. Then it is readily observed that the restriction of the morphism g (6.2.6) to each of these subbundles is an injection into J\Q. Let us consider the image of this injection in a fibre of J\Q at a point q £ Q. There are local coordinates (q°, q') on a neighbourhood of q € Q such that the pseudo-Riemannian metric g(q) at q comes to the Minkowski metric g(q) =?7 = d i a g ( l , - l , ■ • • , - ! ) .

(6.2.8)

With respect to these coordinates, the velocity hyperboloid Wq C TqQ is given by the equation

(90)2 - E(
This is the union of the subset W+, where q° > 0, and W~, where q° < 0. Restricted to W+, the morphism (6.2.6) takes the familiar form of the relations between 3- and 4-velocities

yfl ~ Ulb)2

\A-n
6.3. KELATTVISTIC

DYNAMICS

307

in Special Relativity. The image of each of the hyperboloids W* in the 3-velocity phase space J{Q by the morphism (6.2.6) is the open ball B * o ) 2 < 1i

(6.2.9)

This relation means that 3-velocities of a relativists system (Q,g) are bounded in accordance with the relativistic principle. Let us turn now to the momentum phase space of relativistic mechanics. Given a chart of the relativistic configuration space Q, the homogeneous Legendre bundle corresponding to the local non-relativistic system (6.2.2) is the cotangent bundle T'Q. This fact motivates us to choose T'Q as the reJativistic momentum phase space, equipped with the holonomic coordinates ($\pA). The cotangent bundle T'Q is endowed with the canonical symplectic form n = 4pMAdgM,

(6.2.10)

called the reJativistic symplectic form. Given a pseudo-Riemannian metric g on the relativistic configuration space Q, we have the corresponding isomorphism between relativistic velocity and momentum phase spaces TQ and T'Q. Prom the physical viewpoint, this isomorphism, however, fails to be correct. With respect to the universal unit system, a metric tensor g is dimensionless, whereas 4-velocities and relativistic momenta have different physical dimension. The latter are of dimension [length]" 1 . Therefore, we should use a mass metric in order to perform the above-mentioned isomorphism.

6.3

Relativistic dynamics

In a straightforward manner, Lagrangian formalism fails to be appropriate to relativistic mechanics because a Lagrangian L = Cdq°,

(6.3.1)

by very definition, is defined only locally on a coordinate chart (6.1.5) of the 3velocity phase space J\Q. For instance, the Lagrangian

Lm = -mil - E(9J) 2 V

(6.3.2)

308

CHAPTER

6. RELATIVISTIC

MECHANICS

of a free relativistic point mass m in Special Relativity, can be defined on each coordinate chart of J\Q, but it is not maintained in a straightforward manner by the Lorentz transformations (6.2.4) because of the term dq°. Given a motion q* = cl(q°) with respect to a coordinate chart (q°, q'), its Lorentz transformation (6.2.4) reads cf> =

q°cha-c1(q0)aha,

c1(9°) = -g 0 (g 0 )sha + c1(
^ V ) = c2'V(9°)). Then we have the Lorentz invariance of the pull-back c*Lm of the Lagrangian (6.3.2), i.e., c Lm — c Lm where dg0 = d?o(5°)dg° and dqo is the total derivative. Therefore, we concentrate our consideration on Hamiltonian relativistic mechan­ ics. Let L be a local regular Lagrangian (6.3.1) on a coordinate chart (q°, qx) of the 3-velocity phase space J\Q- In accordance with Remark 4.8.9, it defines a local Dirac constraint system on the relativistic momentum phase space T*Q in the case of a zero Hamiltonian and the primary constraint space given by dC dq'0

dq'o

(6.3.3)

Since the Lagrangian L is regular, the equations (6.3.3) are solvable for Po = Po(
tf = d0 + Pdi + Ad{ on the primary constraint space (6.3.3), which fulfills the equation

tfjnN = 0, tfi = diPo,

* = d'p0,

where fi^ is the restriction of the relativistic symplectic form (6.2.10) to the con­ straint space (6.3.3). For instance, the Lagrangian (6.3.2) (which is regular on the

6.3. RELATIVISTIC

DYNAMICS

309

ball (6.2.9)) defines a Dirac constraint system on T*Q in the case of a zero Hamiltonian and the primary constraint space 2

V^

2

2

(6.3.4)

i

Therefore, let us describe a relativistic system (Q, g) as an autonomous Hamiltonian system on the symplectic manifold T*Q, characterized by a Hamiltonian (6.3.5)

H : T'Q -» R, called a relativistic

Hamiltonian.

Remark 6.3.1. It should be emphasized that the coordinate —po, but not the relativistic Hamiltonian H, plays the role of a relativistic energy function. • Any relativistic Hamiltonian H (6.3.5) defines the Hamiltonian map H : T*Q -* TQ, (6.3.6)

q» = d*H,

(3.3.4) over Q from the relativistic momentum space T*Q to the 4-velocity phase space TQ. Since the 4-velocities of a relativistic system live in the velocity hyperboloids (6.2.7), we have the constraint subspace N =

WWg,

(6.3.7)

g ^ ^ H ^ H = 1, of the relativistic momentum phase space T*Q. It follows that the relativistic system (Q, g) can be described as an autonomous Dirac constraint system on the primary constraint space N (6.3.7) [154]. Its solutions are integral curves of the Hamiltonian vector field u = u'di + Uid1 on N C T*Q, that obeys the Hamilton equation u\i*NQ =

-i*NdH.

Its solution does not necessarily exist.

(6.3.8)

CHAPTER

310

6. RELATIVISTIC

MECHANICS

Example 6.3.2. The relativistic Hamiltonian of a free relativistic point mass in Special Relativity is

H = -^V> P „

(6.3.9)

where r) is the Minkowski metric (6.2.8), while the constraint space N (6.3.7) is given by the equation (6.3.4). It is readily observed that the restriction of the Hamiltonian (6.3.9) to this constraint space is a constant function, i.e., i"NdH = 0. Then the Hamilton equation (6.3.8) takes the form (6.3.10)

u\i*NSl = 0. We obtain its solution ,t

pi = Ui = 0,


=

u

u°V'kpk

t

=

.

I

=

sjm? - v}kPjPk

and then the familiar expression for 3-velocities mriikpk

i _

Pi = const.

^Jm2 - n>kpjpk

• Example 6.3.3. Let us consider a point electric charge e in the Minkowski space in the presence of an electromagnetic potential Ax- Its relativistic Hamiltonian reads H = - ^ " " ( J V - c4.)(ft "

eA

»)<

while the constraint space TV (6.3.7) is = m2.

v""{Pn - eAll){pv -eAv)

As in the previous Example, the Hamilton equation (6.3.8) takes the form (6.3.10). Its solution is Pk

uk = u^dkAp,

(6.3.11) k

qk

fc==

u°rf (pj - eAj) Jm2 - rfUjpi - eAi){Pj - eAj)

(6.3.12)

6.4. RELATIVISTIC

GEODESIC

EQUATIONS

311

The equality (6.3.12) leads to the usual expression for the 3-velocities Pk

: T -rt-k

I

Substituting this expression in the equality (6.3.11), we obtain the familiar equation of motion of a relativistic charge in an electromagnetic field. • Example 6.3.4. The relativistic Hamiltonian for a point mass m in a gravitational field g on a 4-dimensional manifold Q reads

-L9r{-q)*>*»

H

while the constraint space N (6.3.7) is 9*VVvP» =

m2

-

As in previous Examples, the equation (6.3.8) takes the form (6.3.10). • Examples (6.3.2) - (6.3.4) illustrate the relativistic systems where the restriction of a relativistic Hamiltonian to the constraint space N (6.3.7) is a constant function. In this case, the Hamilton equation takes the form (6.3.10). In accordance with Proposition 3.4.2, this equation has a solution everywhere on the primary constraint space TV (6.3.7). Remark 6.3.5. Given a symplectic manifold (T*Q,Q) and its submanifold N (6.3.7), one may apply the reduction procedure described in Section 2.6. For the relativistic systems in Examples 6.3.2 - 6.3.4, this procedure is not, however, use­ ful since the pull-back Hamiltonian i*NJi on the constraint space N is a constant function and the associated Hamiltonian vector fields belong to the kernel of the presymplectic form i*NQ on N. *

6.4

Relativistic geodesic equations

Let H be a relativistic Hamiltonian on the relativistic momentum phase space T*Q and

in

d'Hdi - diHd*

(6.4.1)

CHAPTER

312

6. RELATIVISTIC

MECHANICS

the corresponding Hamiltonian vector field whose integral curves are solutions of the Hamilton equation

frJfi = -dH. If the Hamiltonian map H (6.3.6) is a diffeomorphism, the Hamiltonian vector field (6.4.1) yields the holonomic vector field

e r H o ^ o H - ^a ^ + f i

e

(dad^H^H

(6.4.2) 1

- <9 d"Hd Q H) o H ,

Dn the 4-velocity space 1 Q, where («*,«*, 4", $ * ) o T H

{
is the tangent morphism to the Hamiltonian map H. The holonomic vector field (6.4.2) defines a conservative second order dynamic equation q»

(dad»ndan

- d Q d"Hd Q H) o H 1 ,

(6.4.3)

called the relativistic dynamic equation, on the relativistic configuration space Q (see Definition 3.1.2). For instance, the relativistic dynamic equation (6.4.3) for a point electric charge in Example 6.3.3 takes the well-known form q» =

rf"qxF^

(6.4.4)

where F is the electromagnetic strength (4.9.10). The relativistic dynamic equation (6.4.3) for a point mass m in a gravitational field g in Example 6.3.4 reads r = {AM,}
(6.4.5)

where {A",}

—^9v'a{Q\9oLV + d*9a\ - dag\u)

are the Christoffel symbols of the pseudo-Riemannian metric g. The equations (6.4.4) and (6.4.5) exemplify relativistic dynamic equations which are geodesic equations


(6.4.6)

6.4. RELATIVISTIC

GEODESIC

EQUATIONS

313

with respect to a connection

K = dqx® {dx + Kfa)

(6.4.7)

on the tangent bundle TQ —* Q (see Definition 3.1.6). We call (6.4.6) the relativistic geodesic equation. For instance, a connection K in the equation (6.4.5), is the LeviCivita connection of the pseudo-Riemannian metric g. In the equation (6.4.4), K is the zero Levi-Civita connection of the Minkowski metric plus the soldering form (6.4.8)

o = rrFvXd(?®dll.

We say that a relativistic geodesic equation (6.4.6) on the 4-velocity phase space TQ describes a relativistic system (Q, g) if its geodesic vector field does not leave the subbundle of velocity hyperboloids (6.2.7). It suffices to require that the condition (dxg^q" + 2g^Kxl)qxq"

= 0

(6.4.9)

holds for all tangent vectors which belong to Wg (6.2.7). Obviously, the Levi-Civita connection {A^I/} of the metric g fulfills the condition (6.4.9). Any connection K on TQ —» Q can be written as ^

= {AM?" + *AV,?A),

where a = a$dqx ® dx is a soldering form. Then the condition (6.4.9) takes the form g ^ g g q V = Q-

(6.4.10)

It is readily observed that the soldering form (6.4.8) in the equation (6.4.4) obeys this condition for the Minkowski metric r). Let now compare relativistic and non-relativistic geodesic equations. In physical applications, one usually thinks of non-relativistic mechanics as being an approxi­ mation of small velocities of relativistic theory. At the same time, the 3-velocities in mathematical formalism of non-relativistic mechanics are not bounded. It has long been recognized that the relation between the mathematical schemes of relativistic and non-relativistic mechanics is not trivial.

314

CHAPTER

6. RELATIVISTIC

MECHANICS

Let a relativistic configuration space Q admit a fibration Q —> R, where M is a time axis. One can think of the fibre bundle Q —> K as being a configuration space of a non-relativistic mechanical system. In order to compare relativistic and non-relativistic dynamics, one should con­ sider pseudo-Riemannian metric on TQ, compatible with the fibration Q —» R. Note that M is a time of non-relativistic mechanics. It is one and the same for all non-relativistic observers. In the framework of a relativistic theory, this time can be seen as a cosmological time. Given a fibration Q —* E, a pseudo-Riemannian metric on the tangent bundle TQ is said to be admissible if it is defined by a pair (gR, T) of a Riemannian metric on Q and a non-relativistic reference frame T, i.e.,

9=

2r®r

9

R

Jf~f~ ' 2

(6.4.11)

I r | = s* FT* = j ^ r , in accordance with the well-known theorem [77]. The vector field T is a time-like vector relative to the pseudo-Riemannian metric g (6.4.11), but not with respect to other admissible pseudo-Riemannian metrics in general. There is the canonical imbedding (4.1.4) of the velocity phase space JlQ of non-relativistic mechanics into the affine subbundle 4° = 1,

** = «&

(6.4.12)

of the 4-velocity phase space TQ. Then one can think of (6.4.12) as the 4-velocities of a non-relativistic system. The relation (6.4.12) differs from the familiar rela­ tion (6.2.6) between 4- and 3-velocities of a relativistic system. In particular, the temporal component q° of 4-velocities of a non-relativistic system equals 1 (rela­ tive to the universal unit system). It follows that the 4-velocities of relativistic and non-relativistic systems occupy different subbundles of the 4-velocity space TQ. Moreover, Theorem 4.4.2 shows that both relativistic and non-relativistic equations of motion can be seen as the geodesic equations on the same tangent bundle TQ. The difference between them lies in the fact that their solutions live in the different subbundles (6.2.7) and (6.4.12) of TQ. At the same time, relativistic equations, expressed in the 3-velocities q'/q° of a relativistic system, tend exactly to the nonrelativistic equations on the subbundle (6.4.12) when q° —» 1, floo —* 1, ie., where non-relativistic mechanics and the non-relativistic approximation of a relativistic theory only coincide.

6.4. RELATIVISTIC

GEODESIC

EQUATIONS

315

Let (q°, q') be a non-relativistic reference frame on Q compatible with the fibration of Q —» R. Given a non-relativistic geodesic equation (4.4.7), we will say that the relativistic geodesic equation (6.4.4) is the relativization of (4.4.7) if the spatial parts of these equations are the same. In accordance with Lemma (4.5.5), any relativistic geodesic equation with respect to a connection K (6.4.7) is a relativization of a non-relativistic geodesic equation with respect to the connection

K = dqx®{dx +

{K\-TiKi)di),

where P = 0 is the connection corresponding to the reference frame (q°,q'). Of course, for different reference frames, we have different non-relativistic limits of the same relativistic equation. The converse procedure is more intricate. Example 4.9.3 shows that Lagrange equations for any non-relativistic quadratic Lagrangian (4.9.11) admit a relativization with respect to a Lagrangian frame con­ nection. It follows that inertial forces do not admit relativization. Conversely, let us consider a geodesic motion (6.4.13)

r = {AM?V

in the presence of a pseudo-Riemannian metric g on a relativistic configuration space Q. Let (q0,^) be local hyperbolic coordinates such that poo = 1, 9ot = 0. These coordinates make up a non-relativistic reference frame for a local fibration Q —* K. Then, by virtue of Lemma (4.5.5), the equation (6.4.13) has the non-relativistic limit (4.9.18) which is the Lagrange equations for the Lagrangian (4.9.16) where $ = 0. This Lagrangian describes a free non-relativistic mechanical system with the mass tensor ffiij = —gij. In view of Proposition 4.4.5, the relativization procedure, however, fails to be unique. Therefore, the relativization (4.9.12) of an arbitrary quadratic Lagrangian (4.9.11) may lead to confusion. In fact, this relativization procedure is appropriate to a gravitational Lagrangian (4.9.15), where <jf is a gravitational potential. An arbitrary quadratic dynamic equation can be written in the form

?So = -(m-1)* Wflfoff + W t f .

9o° = 1-

where {\k^} are the Christoffel symbols of some pseudo-Riemannian metric g, whose spatial part is the mass tensor (—rn^), while

WteS+W)

(6.4.14)

CHAPTER

316

6. RELATIVISTIC

MECHANICS

is an external force. With respect to the coordinates where
9* = { A K V + ^ g \

where the soldering form a must fulfill the condition (6.4.10). It takes place only if QikVj + 9ijVk = 0, i.e., the external force (6.4.14) is the Lorentz-type force plus some potential one. Then, we have ffg = 0,

4 = -&&,

4 = K-

The relativization (6.4.15) exhausts almost all familiar examples. To complete our exposition, we will point out also another relativization procedure. Let a nonrelativistic equation ^'(x^) be a spatial part of a 4-vector £x in the Minkowski space. Then one can write the relativistic equation

ax=e - va0e
(6.4.16)

This is the case, e.g., for the relativistic hydrodynamics in gravitation theory. How­ ever, the non-relativistic limit q° = 1 of (6.4.16) does not coincide with the initial non-relativistic equation. There are also other variants of relativistic hydrodynamic equations [107].

Appendix A Geometry of B R S T mechanics As is well-known, the BRST formalism is one of the corner stones of contemporary quantum field theory. This formalism has been extended to mechanics [5, 65, 66, 140, 141]. One can think of BRST mechanics as being a particular case of supersymmetric mechanics [3, 42, 108], where supercharges are the BRST charges, or mechanics on supermanifolds [4, 33, 41, 95, 165]. We aim to show that Hamiltonian timedependent mechanics on a vertical configuration space VV'Q admits the natural extension to BRST mechanics on graded manifolds. We start from the naive BRST extension of Hamiltonian mechanics where the odd variables are introduced in a formal way. Let the configuration space Q —> K be a vector bundle. Besides the familiar coordinates (t, ql,Pi, ql,pi) on the vertical configuration space VV'Q, let us consider additionally the odd variables (A.l)

£ i Cji ^ » Q

which are assumed to be anticommutative. Following the physical terminology, we will call c and c (A.l) the ghosts and antighosts, respectively. Let & and c1 with respect to the indices i have the same transition functions as the linear coordinates qx, while c, and Cj have the transition functions of the coordinates pi; i.e., c"

c,

dqj

c"

~<

dqi _ dq'iCj'

(A.2)

The formal products of the odd variables (A.l) make up the algebra of polyno­ mials over the ring of complex functions on VV*Q. Provided with the unit element, this is the Z2-graded ring V(c,c) of polynomials of the odd variables. One may say 317

318

APPENDIX

A. GEOMETRY

OF BRST

MECHANICS

that such a polynomial is globally defined if it is invariant under the transformations (A.2). We will follow the convention where the antighosts c are written on the left of the ghosts c. For the sake of convenience, we will use the compact notation yA for both even and odd variables. Put {y} = 0 for the even variables and {y} = 1 for the odd ones. Let us consider the left differentiations d/dy of the graded ring V{c,c), which possess the properties

^ d d dyA dyB

(A.3)

| ( * ) * + (-i)W{w}*|;("). K

'

(A.4)

dyBdyA'

We introduce the exterior forms dy as the dual of the derivatives d/dy such that (A.5)

{dy} = {y}, 8

^J* -*

(A.6)

One may define the exterior product of exterior forms dy by the rule dy A dy1 = {-l)^^+1dy'

A dy.

(A.7)

With this exterior product, the ring V(c, c) is extended to the (Z, Z 2 )-bi-graded ring of polynomials V(c, c, dy) which is defined as the algebra of exterior forms dy over the ring V(c,c), where ydy' = (-1)WMW)V, <M
4>,a e

P{c,c,dy).

(A.8)

Accordingly, the interior product (A.6) obeys the relation ^ J (0 A a) = ( | - j ^ ) A a + (-l)W+«H»>0

A

(

|.jff).

(A.9)

The ring V(c, c, dy) is also provided with the exterior differential

*=? V ^ W l where the derivation

>

4>eV(c,c,dy),

(A. 10)

319

fulfills the conditions (A.3), (A.4) and

> = °' As follows from (A.4) and (A.7), the exterior differential (A. 10) is a nilpotent oper­ ator, i.e., dod

= 0.

It is readily observed that, putting c = 0, c = 0, we restate the familiar exterior algebra of even variables y. The main principle of the BRST extension of mechanics is its invariance under the BRST and anti-BRST transformations whose generators read o id d ..^ ■& = c1—- + Ci— + iql—1 oq' dpi oc dq'

dpi

dcl

.. d + ip;—, oct l

(All) (A.12)

dci

[65, 66]. The generators (A.11) - (A.12) fulfill the nilpotency rule M = Q,

tftf = 0,

&& + && = 0,

i.e., for any function f(y), we have

tfJd(tfJd/) = 0, 0Jd(0|d/) = O,

tfJd(tfJd/)+tfJ(i(tfJd/) = 0. Following the criterion of BRST and anti-BRST invariance, let us consider a Hamiltonian form H (5.2.10) on the momentum phase space V*Q and its vertical extension Hy (5.11.5) on the vertical momentum phase space VV*Q. We aim to find a 1-form He € V(c,c, dy) such that the sum Hs = Hy + Hc is BRST-invariant, i.e., its Lie derivative along the vector field i? (A. 11) vanishes: U{HV

+ Hc) = #\d(Hv + Hc) + d($\(Hv

+ Hc)) = 0.

320

APPENDIX

A. GEOMETRY

OF BRST

MECHANICS

We obtain a desired Hamiltonian form Hs = pidq' — q*dpi + i(cidc' - c'dci) - b\Hdt Uc = i ^

j

:

+ c dj)(cjd ' +

Hcdt,

(A.13)

Jdj)H,

called the BRST extension of the Hamiltonian form H. It is easily justified that this form is also anti-BRST-invariant, i.e., its Lie derivative along the vector field ■d (A. 12) also vanishes. In particular, given a splitting of the Hamiltonian H = plTi]q> + H

(A. 14)

with respect to some reference frame T on the configuration space Q —> K, there is the corresponding splitting of the BRST extended Hamiltonian ns

p ; i V - ^ ( - P A ) + ztfV - ^(-rjCi)

+ dvH +

i{Zjd> + c>dj)(cJdj + d>d3)H with respect to the connection f (5.2.6) on the momentum phase space V*Q —> K. This splitting shows that the Hamiltonian form (A.13) is globally defined, and that the connection is given on the odd variables {c',Ci) as on the elements of the fibre bundle VVQ -> K. We also can write the BRST extension of the canonical 3-form Q (5.1.9) on the momentum phase space V*Q. This extension is deduced from the canonical 3-form flv (5.11.1) on the vertical momentum phase space VV'Q, and reads fls = [dpi A dq' + dpi A dqx + i{dc\ A dc' - dc1 A dci)} A dt.

(A.15)

The form (A.15) is globally defined. By construction, it is BRST- and anti-BRSTinvariant. The Hamiltonian connection 7

= dt + ltd< + ya, + 7
for the Hamiltonian form Hs (A.13) with respect to the canonical form Q$ satisfies the Hamilton equations

7* = q\ = Shi,

(A. 16a)

7i = Pa = - W ,

(A. 16b)

f = ql = dhis,

(A. 16c)

321

7« = Pu =

-9iHs,

(A.16d) j

g* = c\ = (cjdi +

i

c dj)d H,

gi = cti = -(cjcV +

(A.16e)

c>dj)din,

(A.16f)

g* = 4 = (c,0* + Jd^d'H,

(A.16g)

5i = c ti = - ( 0 ^ + ^ ) 9 ^

(A.16h)

for the Hamiltonian form HsIt is readily observed that the equations (A. 16a) - (A. 16b) are the Hamilton equations for the initial Hamiltonian form H. To clarify the meaning of other equations, let us note that, if y(t) is a solution of the Hamilton equations (A. 16a) (A.16h), then the deviations q^t) + eecfit) + ec\t) + ec*(t), Pi(t) + eepi(t) + £Ci(t) + £Ci{t), with the odd parameters e and e are solutions of the Hamilton equations for the Hamiltonian form H extended to the case of even, but not necessarily real variables Ql,Pi-

The following construction plays a prominent role for the BRST quantization procedure. Let H b e a Hamiltonian function in some decomposition (A. 14) of a Ha­ miltonian 7i. Such a Hamiltonian function becomes an operator under quantization. Let us consider the operators n0 = effHodo Up = er

m

e-m

= tf _ p(Cj&> + (Jdj)n,

im

= d + Pfedt + T?dj)H,

o$oe

called the BRST and anti-BRST

*(/) = *!#,

/3eR,

charges, which act on functions f(y) by the law

Hf) = «/■

It easy to see that these operators are nilpotent, i.e., HpoH0

= 0,

H0oHp

= 0.

(A.17)

Let Hs be the BRST extension of the Hamiltonian function H, which is also treated as an operator on functions f(y). Then we obtain the relations "H$ ° 7~Ls — 'Hs ° Up = 0, Hf3oHs-Hs°H0

ipns = H0OH0

= 0, + H 0O

n 0.

322

APPENDIX

A. GEOMETRY

OF BRST

MECHANICS

These relations together with the relations (A. 17) provide the operators Tip, Tip, and Hs with the structure of a Lie superalgebra [35], similar to what we have in supersymmetric mechanics [42, 108]. Now let us restate the above construction in strict mathematical terms. In general, BRST mechanics can be developed as mechanics on supermanifolds [4, 33]. However, its naive variant shows that it suffices to consider the following particular case of mechanics on supermanifolds. • First of all, we should restrict our consideration to vector configuration bun­ dles Q —* K in order to introduce the generators of BRST and anti-BRST transformations. • Secondly, the BRST extension TLS of a Hamiltonian Ti is a polynomial with respect to the odd variables. Therefore, we can narrow the class of superfunctions under consideration. Let us recall a few basic notions. A vector space Q is called a Z2-graded vector space or simply a graded vector space if there is the decomposition Q = So © Qi, where Qo is called its even subspace, while Qi is said to be the odd subspace. An element of a graded vector space is called homogeneous if it belongs to <2o or Q\. The degree of a homogeneous element v e Q is denoted by {v}. A graded vector space is said to be (m, n)-dimensional if dim Q0 = m and dim Qx = n. By a graded commutative algebra B is meant an algebra which is a vector graded space B = B0(& Bi such that aras = (—l^ascv G BT+S,

ar e BT,

as G Bs,

s,r = 0,1.

A graded commutative Banach algebra is a graded commutative algebra if it is a Banach algebra and the condition

||oo + OiH = |K|| + I M is fulfilled. Let V be a vector space, and A=AV=t9AV k

323

its exterior algebra. This is a Z-graded commutative algebra provided with Z 2 graded structure as follows: 2

™

m

•

.

A0 = 1 © A V, m 2m+l

Ai = ® A V. m

It is called the Grassman algebra. Given a basis {cl,i € / } for the vector space V, the elements of the Grassman algebra take the form

E ^i-i.e*1 • • • c**,

a= E

(A. 18)

where the sum is over all the collections of indices (ix • • • ik) such that no two of them are the permutations of each other. For the sake of simplicity, we will omit the symbol of the exterior product of elements of the Grassman algebra A. The Grassman algebra becomes a graded commutative Banach algebra if its elements (A. 18) are endowed with the norm

NI = E E K - i J k

(ti-i f c )

Remark 1.0.1. There is a different definition of a Grassman algebra [89], which is equivalent to the above one only in the case of an infinite-dimensional vector space V [40]. • Let Q be a graded vector space and B a graded commutative algebra with a unit element. If Q is both a left and a right B-module such that vT € Qr,

a3vT = vTas € Qs+r, it is called a graded B-module.

as G Bs,

Each B-graded module is obviously a Bo-module.

Example 1.0.2. Every graded vector space Q can be extended to a graded Bmodule which is its graded B-envelope BQ

(BQ)0 © (BQ)! = (B 0 ® Qo © B\ ® fii) © (Bi ® Qo © B 0 ® Qi).

The graded envelope gm+n

=

(fim

0

jgn)

0

(_gm

ffi

fin)

324

APPENDIX

A. GEOMETRY

OF BRST

MECHANICS

of the (m, n)-dimensional graded vector space Qm,n is said to be a vector superspace. Its Bo-submodule Bm'n

= 5J 1 0 B?

is also called a vector superspace. If B is a Banach algebra, the vector superspace Bm+n is a Banach space provided with the norm

Ikcii

£11*11.

a; € B,

i

where {c1} is a basis for Q m , n . • The main ingredient in a theory of supermanifolds is the sheaf B of graded commutative algebras on a manifold Z. Let (Z,B{U)) be its canonical presheaf where by B{U) is meant the graded commutative algebra of local sections of the sheaf B over an open subset U C Z. If the sheaf B fulfills some conditions, called the Rothstein axioms, it is said to be an R-supermanifold (we refer the reader to [11, 40, 152] for details). The notion of an .R-supermanifold includes Berezin's graded manifolds, the supermanifolds of A.Rogers and infinite-dimensional supermanifolds described by A.Jadczyk and K.Pilch. We restrict our consideration to the following class of .R-supermanifolds [33]. Let E —> Z be a vector bundle with a typical fibre V, and E* —> Z the dual of E —> Z. We consider the Grassman fibre bundle

A£'=M0(8A£*) Z

k Z

(A.19)

over Z. Its typical fibre is the Grassman algebra A* = AV. The sections of the Grassman fibre bundle make up the sheaf of Grassman algebras B on the manifold Z, which belongs to the class of graded manifolds. Its sections, i.e., sections of the fibre bundle (A.19) are called superfunctions. Given bundle coordinates (zx, y') with respect to the bases {CJ} in E —> Z, superfunctions take the form

/ = £ k

£

ail...ikci^---c^,

(A.20)

(ii-tfc)

where {c1} are the dual bases in E*. Let us consider the sheaf der B of graded differentiations of the sheaf B. Let U be an open subset of Z and B(U) the restriction of the sheaf B to U. By a graded

325

differentiation of the sheaf B(U) is meant its endomorphism ■§ such that d : B{U) -* B(U), ■&{ab) = d{a)b + (-l)<">atf(6),

(A.21)

for the homogeneous elements d 6 der 6 and a, b G B(U). The graded differentia­ tions of B(U) constitute the B-module der B(\J), and the presheaf of such a B-module generates the sheaf der B. One can show that der B is isomorphic to the sheaf of sections of the fibre bundle TE ® AE -> Z.

(A.22)

E

On can think of its elements as being A-valued vector fields, called vector superfields, on the fibre bundle E. They read

d = aX

j^+aii-

(A.23)

where ax, a1 are superfunctions (A.20) and _d___d_ dc' dy* are elements of the holonomic bases in TE. Due to the splitting VE = E x E, these elements can be identified with the basis elements c, in E. It follows that the operators d/dcl act on superfunctions (A.20) by the law of the exterior product of vectors and covectors. In accordance with the relation (A.21), they are odd. The operators d/dzx are even, and act on the superfunctions (A.20) as familiar derivatives. The dual of the sheaf der B is the sheaf der*S generated by the linear morphisms (A.24)

der£|u -> B(U). This is the sheaf of sections of the fibre bundle

T*E®AE->Z. E

Its elements take the form
(A.25) 1

x

where a\, a, are superfunctions (A.20), while dc = dy are elements of the holonomic bases in T*E. One can think of these elements as being the A-valued 1-forms on the

326

APPENDIX

A. GEOMETRY

OF BRST

MECHANICS

fibre bundle E. Then the morphisms (A.24) can be treated as the interior product of the vector fields (A.23) and the 1-forms (A.25). One can introduce the exterior differential of superfunctions d : B -> der*£, df{d) = ( _ l ) W W ^ ( / ) = (_!){/} Wtfjdf, which is extended to the exterior algebra of A-valued exterior forms on E, called exterior superforms. These superforms are sections of the fibre bundle AT*E®AE E

-> Z,

(A.26)

where f\T*E is the Grassman fibre bundle induced by T*E, The vector superfields, seen as sections of the fibre bundle (A.22), and the ex­ terior superforms, represented by sections of the fibre bundle (A.26), are exactly those mathematical objects which we have considered above on the naive level. They satisfy the rules (A.3) - (A.10). To obtain the BRST extension of Hamiltonian mechanics in terms of vector superfields and exterior superforms, one can put Z = VV'Q, while E is the vertical tangent bundle of the fibre bundle VVQ -»1 so that V{c,c,dy) = der*B.

Appendix B On quantum time-dependent mechanics Here we are concerned only with a few aspects of quantizing time-dependent mechanics. We will start from the following important remark. Let Q —> M be a configuration space of time-dependent mechanics, V*Q its momentum phase space, and H a Hamiltonian form on V*Q. Let T be a (complete) reference frame together with the adapted coordinates (t, q') on Q. Since V = 0 relative to these coordinates, the energy function with respect to this reference frame coincides with the Hamiltonian H in the decomposition (5.2.10) (see Section 5.7). We have the Hamilton equation (5.2.18a):

4

d{H,

(B.l)

where one can think of q\. = q\ as being the relative velocities with respect to the reference frame T, In another reference frame with the adapted coordinates (£, q11), the Hamiltonian

W(t,9*,pJ) W(*,«*(t,flVi),ft(i,9'',j<)) ft(t,fl",rf)*«'(t, «*,*{) (B.2) H(ti
Hr

n{t,
H'(t,q'\p^ 327

rir*(t,9*,?fl

(B.3)

APPENDIX

328

B. ON QUANTUM TIME-DEPENDENT

(see relation (4.5.2a)). Relative to the coordinates (£, (B.l) reads H_dH'{t,(p,i/i) qt

~

dp[

tne

MECHANICS

Hamilton equation

'

where 4t c a n be treated as the relative velocities with respect to the new reference frame. At the same time, we can rewrite these equations as a

dHr(t,cf,pQ dpi

where gf. are the relative velocities with respect to the initial reference frame. Thus, we arrive at the following conclusion. Let we quantize a Hamiltonian system with a Hamiltonian H, written relative to coordinates (t, q'), and perform a coordinate transformation. In new coordinates, we quantize the Hamiltonian system with re­ spect to the new reference frame if we use the Hamiltonian (B.2), and quantize that with respect to the initial reference frame if we use the Hamiltonian function (B.3). It follows that, in quantum mechanics, a passage from one set of coordinates to another is not a technical procedure. This explains why most quantization for­ mulations lead to correct result only when a Cartesian coordinate system is used [47]. Let us point out the two quantization schemes which are invariant under reference frame transformations. Let Q —* R be a configuration space of time-dependent mechanics and VQ its momentum phase space, provided with the canonical Poisson structure w (5.1.6). Let H be a Hamiltonian form on V'Q. Given a reference frame T together with the adapted coordinates {t,q%) on Q, the Hamiltonian connection 7# (5.2.17) for H can be represented as the sum 1H

= dt + 0n,

where $ « is the Hamiltonian vector field for the Hamiltonian H in the decompo­ sition (5.2.10) with respect to the above-mentioned canonical Poisson structure w. It follows that, given a reference frame T, a time-dependent Hamiltonian system with a Hamiltonian form H can be seen as a Poisson Hamiltonian system (w,H). This Poisson Hamiltonian system can be quantized by the methods of geometric quantization as follows (we refer the reader to [103, 167, 181, 188] for the geomet­ ric quantization technique). We restrict our consideration to the prequantization procedure.

329 Let M be the typical fibre of the configuration space Q and T*M the correspond­ ing typical fibre of V*Q, provided with the canonical symplectic form Q, (2.4.2). This form is exact. Therefore it belongs to the zero integral Chern class 0 € H2(M,Z). Then the bundle product T"M x C -» T*M is the prequa.ntiza.tion bundle over M, provided with the connection A

dpj ® & + dq> 0 (dj -

2mpjdc),

where c is a coordinate on C, such that its curvature is precisely the 2-form R

iA = -2-irin.

(B.4)

A complete reference frame T define a projection 7rr : V*Q ->

T'M.

The pull-back of the canonical symplectic form £3 by this projection is the exact 2-form 7rf fi = dpi A dq' such that IU

wViW-

Then the pull-back bundle ir*r(T*M x C) = V*Q x C -» K*Q

(B.5)

is the prequantization bundle over V*Q, provided with the connection n^A = dp, ® &+ dq1 0 (dt - 2nipidc) such that its curvature obeys the relation similar to (B.4). It follows that the Poisson algebra on the Poisson manifold (V*Q,w) is quantized, i.e., any function / C D°(Vr*(5) defines the operator / : s t-> -djf&s

+ &>f(dj + 2iripj)s

(B.6)

330

APPENDIX

B. ON QUANTUM

TIME-DEPENDENT

MECHANICS

on sections s(t, qj,Pj) of the prequantization bundle (B.5). Since s are simply com­ plex functions on V*Q, we can rewrite the operator (B.6) in the form

f:s>->{f,s}v

+ 2iripj9'

(B.7)

fs.

It is readily observed that this operator is invariant under holonomic transforma­ tions of the momentum phase space V*Q. It follows that prequantization (B.7) is independent of a reference frame. The result is obvious since the Poisson bivector w (5.1.6) belongs to the zero element of the second LP-cohomology group HlP(V*Q, w) (see Example 2.7.5). Another quantization procedure invariant under holonomic time-dependent trans­ formations is based on the Schrodinger equation. Given a reference frame T together with the adapted coordinates (t, q') on Q, let H(t, q^Pj) be a Hamiltonian with re­ spect to this reference frame. Then the corresponding Schrodinger equation reads

H(t,ql,-ihj-)V.

(B.8)

Relative to another reference frame coordinates {t,q"), the Schrodinger equation (B.8) takes the form 5*

H'(t,q'\-ih^-)V,

(B.9)

where H' is the Hamiltonian (B.2). It follows that the Schrodinger equations (B.8) and (B.9) provide quantization with respect to different reference frames. The fol­ lowing example shows that these quantizations are not equivalent in general. Example 2.0.3. Let us consider a 1-dimensional motion described by the Ha.miltonian

H

£+ ^

on the momentum phase space R 3 - » R with respect to some inertial reference frame (t,q,p). The corresponding Schrodinger equation is 9m

V(9) + ™{E-U(q))rl>{q)

0.

Let us consider another reference frame given by a connection T (
H-pT.

331

The corresponding Schrodinger equation is

r(q) + f ? ( £ - U(q)Mq) -

^Y{qW{q) 0.

Let us substitute

^{q) =

p(q)exp(^JT(q)dqSj

(B.10)

in this equation. We obtain

p"(q) + ^[E-

U(q) + ihV(q) + |r 2 (g)] p{q) 0.

(B.H)

For instance, let Y = v = const, be an inertia! reference frame. In this case, the equation (B.ll) takes the form

p"{q) + ^\E

+ ^V*~U(q)]p{q)

0.

It is readily observed that energy levels of the quantum system with respect to the moving inertial reference frames are the shifts

E + -S of those of the initial quantum system. Moreover, the transformation ip —> p (B.10) is not an automorphism of the Hilbert space of states of the initial quantum system. This means that quantum systems with respect to different inertial reference frames are not equivalent. •

Bibliography 1. M.Abbati, R.Cirelli, S. De Santis and E.Ruffini, The second Noether theorem in the formalism of jet-bundles: Symmetries and degeneration, J. Geom. Phys. 17 (1995) 321. 2. R.Abraham and J.Marsden, Foundations of Mechanics, Second Edition (Benjamin/Cummings Publ. Comp., London, 1978). 3. A.Adrianov, F.Cannata, M.Ioffe and D.Nishnianidze, Matrix Hamiltonians: SUSY approach to hidden symmetries, J. Phys. A 30 (1997) 5037. 4. S.Albeverio and Shao-Ming Fei, BRST Structures and Symplectic Geometry on a Class of Supermanifolds, Lett. Math. Phys. 33 (1995) 207. 5. A.Aringazin, BRS and anti-BRS invariant states in path integral approach to Hamiltonian and Birkoffian mechanics, Phys. Lett B 314 (1993) 333. 6. V.Arnold, Mathematical 1978).

Methods of Classical Mechanics (Springer, Berlin,

7. V.Arnold, V.Kozlov and A.Neishtadt, Mathematical aspects of classical and celestial mechanics, in Dynamical Systems III, (Springer, Berlin, 1988) p.l. 8. J. de Azcarraga, A.Perelomov and J.Perez Bueno, The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures, J. Phys. A 29 (1996) 7993. 9. J. de Azcarraga, J.Izquierdo and J. Pfez Buenot, On the higher-order gener­ alizations of Poisson structures, J. Phys. A 30 (1997) L607. 10. F.Barone, R.Grassini and G.Mendella, A generalized Lagrange equation in implicit form for non-conservative mechanics, J. Phys. A 30 (1997) 1575. 332

BIBLIOGRAPHY

333

11. C.Bartocci, U.Bruzzo, D.Hernandez Ruiperez, V.Pestov, Foundations of supermanifold theory: the axiomatic approach, Diff. Geom. and Appl. 3 (1993) 135. 12. M.Bergvelt and E. de Kerf, The Hamiltonian structure of Yang-Mills theories and instantons, Physica 139A (1986) 101. 13. K.Bhaskara and K.Vismanath, Poisson Algebras and Poisson Manifolds, Pithman Research Notes in Mathematics 174 (Longhman Sci., Harlow, 1988). 14. E.Binz, H.Fischer and J.Sniatycki, Geometry of Classical Fields (NorthHolland, Amsterdam, 1988). 15. D.Blair, Contact Manifolds in Riemannian Geometry, Lect. Notes in Mathe­ matics 509 (Springer, Berlin, 1976). 16. G. Bredon, Sheaf theory (McGraw-Hill, N.-Y., 1967). 17. K.Brown, Cohomology of Groups (Springer-Verlag, Berlin, 1982). 18. A.D.Bruno, The normal form of a Hamilton system, Uspehi Matemat. 43 (1988) N l , 23 (in Russian).

Nauk

19. J-L. Brylinski, A differential complex for Poisson manifolds, J. Diff. Geom. 28 (1988) 93. 20. R.Bryant, S.Chern, R.Gardner, H.Goldschmidt, P.Griffiths, Exterior Differ­ ential Systems (Springer, Berlin, 1991). 21. A.Cabras and A.Vinogradov, Extension of the Poisson bracket to differential forms and multi-vectors, J. Geom. Phys. 9 (1992) 75. 22. S.Campbell, An Introduction to Differential Equations and their Applications (Wadsworth Inc., Belmont, 1990). 23. D.Canarutto, Bundle splittings, connections and locally principle fibred man­ ifolds, Bull. U.M.I. Algebra e Geometria Serie VI V - D (1986) 18. 24. F.Cardin and G.Zanzotto, On constrained mechanical systems: DAlembert's and Gauss' princoples, J. Math. Phys. 30 (1989) 1473.

334

BIBLIOGRAPHY

25. F.Cardin and M.Favretti, On nonholonomic and vakonomic dynamics of me­ chanical systems with nonintegrable constraints, J. Geom. Phys. 18 (1996) 295. 26. J.Carinena, J.Gomis, L.Ibort and N.Roman, Canonical transformation theory for presymplectic systems, J. Math. Phys. 26 (1985) 1961. 27. J.Carinena and M.Ranada, Poisson maps and canonical transformations for time-dependent Hamiltonian systems, J. Ma.th.Phys. 30 (1989) 2258. 28. J.Carinena, M.Crampin and L.Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. and Appl. 1 (1991) 345. 29. J.Carinena, J.Fernandez-Nunez and E.Martinez, A geometric approach to Noether's second theorem in time-dependent Lagrangian mechanics, Lett. Math. Phys. 23 (1991) 51. 30. J.Carinena and M.Ranada, Lagrangian systems with constraints. A geometric approach to the method of Lagrange multipliers, J. Phys. A 26 (1993) 1335. 31. J.Carinena and J.Fernandez-Nunez, Geometric theory of time-dependent sin­ gular Lagrangians, Fortschr. Phys. 41 (1993) 517. 32. J.Carinena, L.Ibort, G.Marmo and A.Stern, The Feynman problem and the inverse problem for Poisson dynamics, Phys. Rep. 263 (1995) 153. 33. J.Carinena and H.Figueroa, Hamiltonian versus Lagrangian formulations of supermechanics, J. Phys. A 30 (1997) 2705. 34. G.Caviglia, Helmholtz conditions, covariance, and invariance identities, Int. J. Theor. Phys. 24 (1985) 377. 35. S.Cecotti and C.Vafa, Topological anti-topological fusion, Nucl. Phys. B367 (1991) 359. 36. P.Chernoff and J.Marsden, Properties of Infinite Dimensional Hamiltonian Systems, Lect. Notes in Mathematics 425 (Springer, Berlin, 1974). 37. D.Chinea, M.de Leon, and J.Marrero, The constraint algoritm for timedependent Lagrangians, J. Math. Phys. 35 (1994) 3410.

BIBLIOGRAPHY

335

38. D.Chinea, M.de Leon, and J.Marrero, The canonical double complex for Jacobi manifolds, C. R. Acad. Sci., Paris I 323 (1996) 637. 39. N.Chitaia, S.Goglidze and Yu.Surovtsev, Dynamical systems with first- and second-class constraints, Phys. Rev. D 56 (1997) 1135; 1142. 40. R.Cianci, Introduction to Supermaniifolds (Bibliopolis, Naples, 1990). 41. R.Cianci, M.Prancaviglia and I.Volovich, Variational calculus and PoincareCartan formalism on supermanifolds, J. Phys. A 28 (1995) 723. 42. F.Cooper, A.Khare and U.Sukhatme, Supersymmetry and quantum mechan­ ics, Phys. Rep. 251 (1995) 267. 43. M.Crampin, F.Cantrijn and W.Sarlet, Lifting geometric objects to a cotangent bundle, and the geometry of the cotangent bundle of a tangent bundle, J. Geom. Phys. 4 (1987) 469. 44. P.Dazord, A.Lichnerowicz and C-M.Marie, Structure locale des varietes de Jacobi, J. Math. Pures et Appl. 70 (1991) 101. 45. P.Dedecker, On the generalization of symplectic geometry to multiple integrals in the calculus of variations, in Differential Geometric Methods in Mathemat­ ical Physics, Lect. Notes in Mathematics 570 (Springer, Berlin, 1977), p. 395. 46. P.Dedecker and W.Tulczyjew, Spectral sequences and the inverse problem of the calculus of variations, in Differential Geometric Methods in Mathematical Physics, Lect. Notes in Mathematics 836 (Springer, Berlin, 1980), p. 498. 47. P.A.M.Dirac, The Principles of Quantum Mechanics (Oxford Univ. Press, Oxford, 1976). 48. W.Domitrz and S.Janeczko, Normal forms of symplectic structures on the stratified spaces, Colloquium Mathematicum LXVIII (1995) fasc.l, 101. 49. B.Dubrovin, M.Giordano, G.Marmo and A.Simoni, Poisson brackets on presymplectic manifolds, Int. J. Mod. Phys. 8 (1993) 3747. 50. A.Echeverria Enrfquez, M.Mufioz Lecanda and N.Roman Roy, Geometrical setting of time-dependent regular systems. Alternative models, Rev. Math. Phys. 3 (1991) 301.

336

BIBLIOGRAPHY

51. A.Echeverria Enriquez, M.Murioz Lecanda and N.Roman Roy, Non-standard connections in classical mechanics, J. Phys. A 28 (1995) 5553. 52. F.Fatibene, M.Ferraris and M.Francaviglia, Noether formalism for conserved quantities in classical gauge field theories, J. Math. Phys. 35 (1994) 1644. 53. R.Fulp, J.Lawson and L.Norris, Generalized symplectic geometry as a covering theory for the Hamiltonian theories of classical particles and fields, J. Geom. Phys. 20 (1996) 195. 54. G.Giachetta and L.Mangiarotti, Gauge-invariant and covariant operators in gauge theories, Int. J. Theor. Phys. 29 (1990) 789. 55. G.Giachetta, Jet manifolds in non-holonomic mechanics, J. Math. Phys. 33 (1992) 1652. 56. G.Giachetta and L.Mangiarotti, Constrained Hamiltonian systems and gauge theories, Int. J. Theor.Phys. 34 (1995) 2353. 57. G.Giachetta, L.Mangiarotti and G.Sardanashvily, New Lagrangian and Ha­ miltonian Methods in Field Theory (World Scientific, Singapore, 1997). 58. D.Gitman and I.Tyutin, Canonical Quantization (Nauka, M., 1986) (in Russian).

of Constrained

Fields

59. A.Gomberoff and S.Hojman, Non-standard construction of Hamiltonian struc­ tures, J.Phys. A 30 (1997) 5077. 60. M.Gotay, J.Nester and G.Hinds, Presymplectic manifolds and the DiracBergman theory of constraints, J. Math.Phys. 19 (1978) 2388. 61. M.Gotay and J.Nester, Presymplectic Lagrangian systems, Ann. l'lnst. Henri Poincare 30 (1979) 129; 32 (1980) 1. 62. M.Gotay, On coisotropic imbeddings of presymplectic manifolds, Proc. Amer. Math. Soc. 84 (1982) 111. 63. M.Gotay, A multisymplectic framework for classical field theory and the calcu­ lus of variations. I. Covariant Hamiltonian formalism, in Mechanics, Analysis and Geometry: 200 Years after Lagrange, ed. M.Francaviglia (Elsevier Science Publishers B.V., 1991), p. 203.

337

BIBLIOGRAPHY

64. M.Gotay, A multisymplectic framework for classical field theory and the cal­ culus of variations. II. Space + time decomposition, Diff. Geom. and Appl. 1 (1991) 375. 65. E.Gozzi, M.Reuter and W.Thacker, Hidden BRS invariance in classical me­ chanics, Phys. Rev. D 4 0 (1989) 3363. 66. E.Gozzi, M.Reuter and W.Thacker, Symmetries of the classical path integral on a generalized phase-space manifold, Phys. Rev. D46 (1992) 757. 67. J.Grabowski and PUrbariski, Tangent lifts of Poisson and related structures, J. Phys. A 28 (1995) 6743. 68. P.Griffiths, Exterior Differential Systems (Birkhauser, Boston, 1983).

and the Calculus of Variations

69. D.Grigore and O.Popp, The complete classification of generalized homoge­ neous symplectic manifolds, J. Math. Phys. 30 (1989) 2476. 70. D.Grigore, A generalized Lagrangian formalism in particle mechanics and clas­ sical field theory, Fortschr. d. Phys. 4 1 (1993) 569. 71. F.Guedira and A.Lichnerowicz, Geometrie des algebres de Lie locales de Kirillov, J. Math. Pures et Appl. 6 3 (1984) 407. 72. V.Guillemin, S.Sternberg, Symplectic Univ. Press., Cambridge, 1990).

Techniques in Physics

(Cambridge

73. C. Gunther, The polysimplectic Hamiltonian formalism in field theory and calculus of variations, J. Diff. Geom. 25 (1987) 23. 74. A.Hamoui and A.Lichnerowicz, Geometry of dynamical systems with timedependent constraints and time-dependent Hamiltonians: An approach to­ wards quantization, J. Math. Phys. 25 (1984) 923. 75. J.Harlet, Time and time functions in parametrized non-relativistic quantum mechanics, Class. Quant. Grav. 13 (1996) 361. 76. P.Havas, The connection between conservation laws and invariance groups: Folklore, fiction and fact, Acta Phys. Austr. 38 (1973) 145.

338

BIBLIOGRAPHY

77. S.Hawking and G.Ellis, The Large Scale Structure of a Space-Time Univ. Press, Cambridge, 1973).

(Cambr.

78. M.Henneaux, Equation of motion, commutation relations and ambiguities in the Lagrangian formalism, Ann. Phys. 140 (1982) 1. 79. J.Hietarinta, Nambu tensors and commuting vector fields, J. Phys. A 30 (1997) L27. 80. F.Hirzebruch, Topological Methods in Algebraic Geometry (Springer, Berlin, 1966). 81. S.Hojman and L.Urrutia, On the inverse problem of the calculus of variations, J. Math. Phys. 22 (1981) 1896. 82. S.Hojman, The construction of a Poisson structure out of a symmetry and a conservation law of a dynamical system, J. Phys. A 29 (1996) 667. 83. R.Ibaiiez, M.de Leon, J.Marrero and D.Martin de Diego, Co-isotropic and Legendre-Lagrangian submanifolds and conformal Jacobi morphisms, J. Phys. A 30 (1997) 5427. 84. R.Ibaiiez, M.de Leon, J.Marrero and D.Martm de Diego, Dynamics of Poisson and Nambu-Poisson brackets, J. Math. Phys. 38 (1997) 2332. 85. R.Ibaiiez, M.de Leon, J.Marrero and E.Padron, Nambu-Jacobi and general­ ized Jacobi manifolds, J. Phys. A 31 (1998) 1267. 86. L.Ibort and J.Marin-Solano, A geometric classification of Lagrangian functions and the reduction of evolution space, J. Phys. A 25 (1992) 3353. 87. L.Ibort, M. de Leon and G.Marmo, Reduction of Jacobi manifolds, J. Phys. A 30 (1997) 2783. 88. A.Ibort, M. de Leon, E.Lacomba, J.Marrero, D.Martin de Diego and P.Pitanga, Geometric formulation of mechanical systems subjected to timedependent one-sided constraints, J. Phys. A 31 (1998) 2655. 89. A.Jadczyk and K.Pilch, Superspaces and Supersymmetries, Commun. Phys. 78 (1981) 391.

Math.

339

BIBLIOGRAPHY

90. F.Kamber and P.Tondeur, Foliated Bundles and Characteristic Classes, Lect. Notes in Mathematics 493 (Springer, Berlin, 1975). 91. I.Kanatchikov, On field theoretical generalization of a Poisson algebra, Rep. Math. Phys. 40 (1997) 225. 92. I.Kanatchikov, Towards to Born-Weyl quantization of fields, Int. J. Theor. Phys. 37 (1998) 333. 93. G.Katzin and J.Levine, Characteristic functional structures of infinitesimal symmetry mappings of dynamical systems. I. Velocity-dependent mappings of second-order differential equations, J. Math. Phys. 26 (1985) 3080. 94. G.Katzin and J.Levine, Characteristic functional structures of infinites­ imal symmetry mappings of dynamical systems. IV. Classical (velocityindependent) mappings of second-order differential equations, J. Math. Phys. 30 (1988) 2039. 95. O.Khudaverdian and A.Nersessian, Canonical Poisson brackets of different gradings and strange superalgebras, J. Math. Phys. 32 (1991) 1938. 96. J.Kijowski and W.Tulczyjew, A Symplectic Framework for Field Theories (Springer, Berlin, 1979). 97. T.Kimura, Generalized classical BRST cohomology and reduction of Poisson manifolds, Commun. Math. Phys. 151 (1993) 155. 98. A.Kirillov, Local Lie algebras, Russian Math. Surveys 31 (1976) 55. 99. A.Kirillov, Geometric quantization, in Dynamical Systems IV, eds V.I.Arnol'd and S.PNovikov (Springer, Berlin, 1990) p.137. 100. S.Kobayashi and K.Nomizu, Foundations of Differential Geometry, (John Wiley, N.Y. - Singapore, 1963). 101. I.Kolaf, P.Michor and J.Slovak, Natural Operations in Differential (Springer, Berlin, 1993).

Vol.1.

Geometry

102. Y.Kosmann-Schwarzbach and F.Magri, Poisson-Nijenhuis structures, Ann. l'lnst. Henri Poincare 53 (1990) 35.

340

BIBLIOGRAPHY

103. B.Kostant, Quantization and unitary representation, in Lectures in Modern Analysis and Applications III, Lect. Notes in Mathematics, 170 (Springer, Berlin, 1970), p.87. 104. I.Krasil'shchik, V.Lychagin and A.Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations (Gordon and Breach, Glasgow, 1985). 105. D.Krupka, Some geometric aspects of variations! problems in fibred manifolds, Folia Fac. Sci. Nat. UJEP Brunensis 14 (1973) 1. 106. O.Krupkova, Mechanical systems with nonholonomic constraints, J. Math. Phys. 38 (1997) 5098. 107. B.Kupershmidt, The Variational Principles of Dynamics (World Scientific, Singapore, 1992). 108. A.Lahiri, P.K.Roy and B.Bagchi, Supersymmetry in quantum mechanics, Int. J. Mod. Phys. A 5 (1990) 1383. 109. S.Lang, Differential Manifolds (Addison-Weslay, Reading, Massachusetts, 1972). 110. M.de Leon and P.Rodrigues, Methods of Differential Geometry in Analytical Mechanics (North-Holland, Amsterdam, 1989). 111. M.de Leon and J. Marrero, Constrained time-dependent Lagrangian systems and Lagrangian submanifolds, J. Math. Phys. 34 (1993) 622. 112. M.de Leon and D. Martin de Diego, On the geometry of non-holonomic La­ grangian systems, J. Math. Phys. 37 (1996) 3389. 113. M.de Leon, J. Marrero and D. Martin de Diego, Mechanical systems with nonlinear constraints, Int. J. Theor. Phys. 36 (1997) 979. 114. M.de Leon, J. Marrero and D. Martin de Diego, Non-holonomic Lagrangian systems in jet manifolds, J. Phys. A 30 (1997) 1167. 115. M.de Leon, J. Marrero and E.Padron, Lichnerowicz-Jacobi cohomology, J. Phys. A 30 (1997) 6029.

BIBLIOGRAPHY

341

116. P.Libermann and C-M.Marie, Symplectic Geometry and Analitical Mechanics (D.Reidel Publishing Company, Dordrecht, 1987). 117. A.Lichnerowicz, Les varietes de Poisson et leurs algebres de Lie associees, J. Diff. Geom. 12 (1977) 253. 118. A.Lichnerowicz, Les varietes de Jacobi et leurs algebres de Lie associees, J. Math. Pures et Appl. 57 (1978) 453. 119. R.Littlejohn and M.Reinsch, Internal or shape coordinates in the n-body prob­ lem, Phys. Rev. A 52 (1995) 2035. 120. R.Littlejohn and M.Reinsch, Gauge fields in the separation of rotations and internal motions in the n-body problem, Rev. Mod. Phys. 69 (1997) 213. 121. L.Lusanna, An enlarged phase space for finite-dimensional constrained sys­ tems, unifying their Lagrangian, phase- and velocity-space descriptions, Phys. Rep. 185 (1990) 1. 122. S.Mac Lane, Homology (Springer, Berlin, 1967). 123. C.-M.Marie, On Jacobi manifolds and Jacobi bundles, in Symplectic Ge­ ometry, Groupoids, and Integrable Systems, ed. PDazord and A.Weinstein (Springer, Berlin, 1989), p. 227. 124. C.-M.Marie, Reduction of constrained mechanical systems and stability of relative equilibria, Commun. Math. Phys. 174 (1995) 295. 125. C.-M. Marie, The Schouten-Nijenhuis bracket and interior products, J. Geom. Phys. 23 (1997) 350. 126. G.Marmo, E.Saletan, A.Simoni and B.Vitale, Dynamical Systems. A Differ­ ential Geometric Approach to Symmetry and Reduction (John Wiley, N.Y., 1985). 127. G.Marmo, G.Mendella and W.Tulczyjew, Constrained Hamiltonian systems as implicit differential equations, J. Phys. A 30 (1997) 277. 128. G.Marmo, G.Vilasi and A.Vinogradov, The local structure of n-Poisson and n-Jacobi manifolds and some applications, J. Geom. Phys (1998) (appear).

342

BIBLIOGRAPHY

129. J.Marsden and T.Ratiu, Reduction of Poisson manifolds, Lett. Math. 11 (1986) 161.

Phys.

130. J.Marsden and T.Ratiu, Introduction to Mechanics and Symmetries (Springer, Berlin, 1994). 131. G.Martin, A Darboux theorem for multi-symplectic manifolds, Lett. Phys. 16 (1988) 133.

Math.

132. E.Massa and E.Pagani, Jet bundle geometry, dynamical connections and the inverse problem of Lagrangian mechanics, Ann. Inst. Henri Poincare 61 (1994) 17. 133. P.Michor, A generalization of Hamiltonian mechanics, J. Geom. Phys. 2 (1985) N2, 67. 134. M.Modugno, A.Vinogradov, Some variations on the notion of connections, Ann. Matem. Pura ed Appl. CLXVII (1994) 33. 135. R. Montgomery, The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case, Commun. Math. Phys. 120 (1988) 269. 136. G.Morandi, C.Ferrario, G.Lo Vecchio, G.Marmo and C.Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Phys. Rep. 188 (1990) 147. 137. M.Munoz-Lecanda, Hamiltonian systems with constraints: A geometric ap­ proach, Int. J. Theor. Phys. 28 (1989) 1405. 138. M.Munoz-Lecanda and N.Roman-Roy, Lagrangian theory for presymplectic systems, Ann. Inst. Henri Poincare 57 (1992) 27. 139. M.Munoz-Lecanda and N.Roman-Roy, Gauge systems: Presymplectic and group action theory, Int. J. Theor. Phys. 32 (1993) 2077. 140. Kh.Nirov and A.Razumov, Equivalence between Lagrangian and Hamiltonian BRST formalisms, J. Math. Phys. 34 (1993) 3933.

BIBLIOGRAPHY

343

141. Kh.Nirov and A.Razumov, Generalized Schrodinger representation in BRST quantization, Nucl. Phys. B429 (1994) 389. 142. L.Norris, Schouten-Nijenhuis brackets, J. Math. Phys. 38 (1997) 2694. 143. F.Pardo, The Helmholtz conditions in terms of constants of motion in classical mechanics, J. Math. Phys. 30 (1989) 2054. 144. P.Pereshogin and P.Pronin, Geometrical treatment of nonholonomic phase in quantum mechanics and applications, Int. J. Theor. Phys. 32 (1993) 219. 145. J.Perez Bueno, Generalized Jacobi structures, J. Phys. A 30 (1997) 6509. 146. V.Perlick, The Hamiltonization problem from a global viewpoint, J. Math. Phys. 33 (1992) 599. 147. G.Pidello and W.Tulczyjew, Derivations of differential forms on jet bundles, Ann. Mat. Pura Appl. 147 (1987) 249. 148. R.De Pietri, L.Lusanna and M.Pauri, Standard and generalized Newtonian gravities as "gauge" theories of the extended Galilei group: I. The standard theory, Class. Quant. Grav. 12 (1995) 219. 149. J.Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups (Gordon and Breach, Glasgow, 1978). 150. B.Reinhart, Differential Geometry and Foliations (Springer, Berlin, 1983). 151. F.Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E53 (1996) 1890. 152. M.Rothstein, The axioms of supermanifolds and a new structure arising from them, Trans. Amer. Math. Society 297 (1986) 159. 153. C.Rovelli, Quantum mechanics without time: A model, Phys. Rev. D42 (1990) 2638. 154. C.Rovelli, Time in quantum gravity: An hypothesis, Phys. Rev. D 4 3 (1991) 442.

BIBLIOGRAPHY

344

155. R.Sachs and H.Wu, General Relativity for Mathematicians 1977).

(Springer, Berlin,

156. R.Santilli, Foundations of Theoretical Mechanics I (Springer, N.Y., 1978). 157. G.Sardanashvily, Gauge Theory in Jet Manifolds (Hadronic Press, Palm Har­ bor, 1993). 158. G.Sardanashvily, Constraint field systems in multimomenturn canonical vari­ ables, J. Math. Phys. 35 (1994) 6584. 159. G.Sardanashvily, Generalized Hamiltonian Formalism for Field Theory. Con­ straint Systems. (World Scientific, Singapore, 1995). 160. G.Sardanashvily, Stress-energy-momentum tensors in constraint field theories, J. Math. Phys. 38 (1997) 847. 161. G.Sardanashvily, Hamiltonian time-dependent mechanics, J. Math. Phys. 39 (1998) 2714. 162. W.Sarlet, F.Cantrijn and D.Saunders, A geometric famework for the study of non-holonomic Lagrangian systems, J. Phys. A 28 (1995) 3253. 163. W.Sarlet, F.Cantrijn and D.Saunders, A geometric famework for the study of non-holonomic Lagrangian systems: II, J. Phys. A 29 (1996) 4265. 164. D.Saunders, The Geometry of Jet Bundles (Cambr. Univ. Press, Cambridge, 1989). 165. A.Schwarz, Geometry of Batalin-Vilkovisky quantization, Commun. Math. Phys. 155 (1993) 249. 166. J.Slawianowski, Geometry of Phase Spaces (Wiley, N.Y.,1991). 167. J.Sniatycki, Geometric Quantization Berlin, 1980)

and Quantum

Mechanics

(Springer,

168. J.Souriau, Structures des Systemes Dynamiques (Dunod, Paris, 1970). 169. J.Stasheff, Homological reduction of constrained Poisson algebras, J. Diff. Geom 45 (1997) 221.

345

BIBLIOGRAPHY

170. N.Steenrod, The Topology of Fibre Bundles (Princeton Univ. Press, Prince­ ton, 1972). 171. P.Stefan, Accessible sets, orbits and foliations with singularities, Proc. London Math. Soc. 29 (1974) 699. 172. K.Sundermeyer, Constrained Dynamics (Springer, Berlin, 1982). 173. H.Sussmann, Orbits of families of vector fields and integrability of distribu­ tions, Thins. Amer. Math. Soc. 180 (1973) 171. 174. F.Takens, Symmetries, conservation laws and variational principles, in Geom­ etry and Topology, Lect. Notes in Mathematics 597 (Springer-Verlag, Berlin, 1977), p. 581. 175. L.Takhtajan, On foundations of the generalized Nambu mechanics, Commun. Math. Phys. 160 (1994) 295. 176. W.Tulczyjew, Lagrangian submanifolds and Hamiltonian dynamics, C. R. Acad. Sci. Paris A 283 (1976) 15. 177. W.Tulczyjew, Lagrangian submanifolds and Lagrangian dynamics, C. R. Acad. Sci. Paris A 283 (1976) 675. 178. W.Tulczyjew, The Legendre transformation, Ann. Inst. Henri Poincare A X X V I I (1977) 101. 179. W.Tulczyjew, The Euler-Lagrange resolution, in Differential Geometric Meth­ ods in Mathematical Physics, Lect. Notes in Mathematics 836 (SpringerVerlag, Berlin, 1980), p. 22. 180. W.Tulczyjew, Geometric Naples, 1989).

Formulation

of Physical

Theories

(Bibliopolis,

181. I.Vaisman, Lectures on the Geometry of Poisson Manifolds (Birkhauser Ver­ lag, Basel, 1994). 182. I.Vaisman, Second order Hamiltonian vector fields on tangent bundles, Diff. Geom. and Appl. 5 (1995) 153.

346

BIBLIOGRAPHY

183. A.Vershik, Classical and non-classical dynamics with constraints, in Global Analysis - Studies and Applications 1, Lect. Notes in Mathematics 1108 (Springer, Berlin, 1984), p. 278. 184. A.Vershik, V.Gershkovich, Nonholonomic dynamical systems, in Dynamical Systems VII, eds V.I.Arnol'd and S.P.Novikov (Springer, Berlin, 1994) p.l. 185. F.Warner, Foundations of Differential Manifolds and Lie Groups (Springer, Berlin, 1983). 186. A.Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983) 523. 187. A.Weinstein, Co-isotropic calculus and Poisson gruppoids, J. Math. Japan 40 (1988) 705.

Soc.

188. N.Woodhouse, Geometric Quantization (Clarendon Press, Oxford, 1980). 189. K.Yano and S.Ishihara, Tangent and Cotangent Bundles, Pure and Applied Mathematics Ser. 16 (Marcel Dekker, N.Y., 1973). 190. O. Zakharov, Hamiltonian formalism for nonregular Lagrangian theories in fibered manifolds, J. Math. Phys. 33 (1992) 607. 191. Q.Zhao, The representation of Lie groups and geometric quantizations, Commun. Math. Phys. 194 (1998) 135. 192. R.Zulanke and P.Wintgen, Differentialgeometrie und Faserbiindel, Hochschulbucher fur Mathematik, Band 75 (VEB Deutscher Verlag der Wissenschaften, Berlin, 1972).

Index acceleration absolute, 184 relative, 184 adjoint of a form, 91 adjoint representation, 46 admissible metric, 314 almost tangent structure, 32 annihilator of a distribution, 33 antighost, 317 anti-BRST charge, 321 anti-BRST transformations, 319 atlas, 10

BRST transformations, 319 bundle, 9 affine, 14 associated, 51 of principal connections, 51 principal, 48 vector, 12 dual, 13 canonical canonical canonical canonical canonical canonical canonical canonical canonical

holonomic, 15 of constant local trivializations, 45 automorphism of fibre bundles, 12 holonomic, 53 vertical, 12

automorphism, 242 complex, 91 form on a frame bundle, 53 homology, 91 horizontal splitting, 38 Liouville form, 74 symplectic form, 74 vector field, 67,74 3-form, 230

jet prolongation, 283 Cartan equations, 116,191 Casimir function, 67 chain complex, 89 characteristic distribution, 69 of a presymplectic form, 81 characteristic foliation of a presymplec­ tic form, 81 coadjoint representation, 47 coboundary on a group, 136 cochain complex, 90

base of a fibre bundle, 10 basic De Rham cohomology, 95 basic exterior form, 95 Berry connection, 281 Berry oscillator, 281 bivector field, 29 BRST charge, 321 BRST extension of the canonical -form, 320 BRST extension of a Hamiltonian form, 320 347

348

cocycle associated with an action of a group, 136 cocycle on a group, 136 codistribution, 33 cofiag of a codistribution, 34 cohomology group, 90 cohomology of a group, 136 coisotropic ideal, 89 coisotropic imbedding, 83 coisotropic submanifold of a Jacobi manifold, 60 of a Poisson manifold, 70 of a symplectic manifold, 78 complete family of Hamiltonian forms, 256 composite connection, 55 composite fibre bundle, 53 configuration bundle, 6 configuration space, 3 connection, 42 affine, 44 dual, 44 dynamic, 164 symmetric, 166 complete, 157 curvature-free, 45 flat, 45 holonomic, 158 Lagrangian, 191 linear , 44 principal, 50 associated, 52 connection form, 51 local, 51 conservation law, 219

INDEX

constraint non-holonomic, 216 constraint reaction acceleration, 209 constraint reaction force, 217 constraint reference frame, 214 constraint space, 124 final, 125,130 Lagrangian, 249 primary, 124 constraints admissible, 212 complete, 125 holonomic, 206 ideal, 209 first-class, 126,132 primary, 124 second class, 126,132 secondary, 125 tertiary, 125 contact automorphism, 64 infinitesimal, 64 contact form, 37,61 contact manifold, 61 contact transformation, 63 contraction, 28 coordinates affine, 14 adapted, 176 canonical, 70,140,243 canonically conjugate, 70 fibred, 10 holonomic , 16 induced, 16 copresymplectic structure, 193 Coriolis theorem, 186 cosymplectic structure, 193

INDEX

cotangent bundle, 16 covariant differential, 43 vertical, 55 current, 219 curvature, 44 D'Alembert principle, 209 generalized, 216 Darboux coordinates, 61 De Donder form, 194 democracy group, 202 De Rham cohomology group, 90 De Rham complex, 90 derivative linear, 15 total, 36,40 variational, 187 vertical, 287 differential equation, 41 differential ideal, 34 differential operator, 41 differential system, 33 Dirac constraint system, 129 Dirac Hamiltonian system, 126 completely integrable, 128 direct product of Poisson structures, 68 distribution, 32 completely integrable, 33 involutive, 32 regular, 33 totally non-holonomic, 33 dynamic equation autonomous, 106 conservative, 161 first order, 160,239

349

autonomous, 107 quadratic, 161 second order, 160 autonomous , 108 endomorphism of fibre bundles, 12 vertical, 155 energy function, 116,222 canonical, 218 Hamiltonian, 271 equilibrium equation, 244 Euler-Lagrange map, 203 Euler-Lagrange operator, 187 Euler-Lagrange-Cartan operator, 191 Euler-Lagrange-type operator, 202 even subspace, 322 evolution equation, 112,247 exterior algebra, 26 exterior differential contravariant, 92 horizontal, 203 vertical, 155 exterior form, 25 exterior product of vector bundles, 14 exterior superform, 326 fibration, 10 fibre basis, 13 first integral of motion, 112 flag of a differential system, 33 flag of a distribution, 33 foliation, 34 of level surfaces, 35 simple, 35 singular, 35 force

350

centrifugal, 181 Coriolis, 181 external, 198 inertial, 179 Lorentz, 198 universal, 199 frame, 13 frame bundle, 52 frame connection, 183 Hamiltonian, 237 Lagrangian, 264 free motion equation, 179 fundamental identity, 99 Galilei group, 182 gauge algebra bundle, 49 gauge conditions, 267 gauge convention, 281 gauge fields, 122 gauge freedom, 122 gauge potentials, 51 gauge transformations, 12 of a principal bundle, 271 Gauss principle, 216 general covariant transformations, 53 generalized almost Jacobi manifold, 99 generalized almost Poisson manifold, 99 generalized invariant transformations, 223 generating function, 246 of a foliation, 35 of a Lagrangian submanifold, 79 generator, 21,47 geodesic equation, 109 geodesic vector field, 109

INDEX

geometric quantization, 328 geometric prequantization, 328 germ of a submanifold, 79 ghost, 317 graded commutative algebra, 322 Banach, 322 graded differentiation, 325 graded Lie algebra, 25 graded manifold, 324 graded space, 322 graded S-envelope, 323 graded B-module, 323 graph of a morphism, 70 Grassman algebra, 323 Grassman bundle, 302 Hamilton equation with respect to a symplectic struc­ ture, 114 Hamilton equations, 147,239 constrained, 259 Hamilton operator, 146,238 Hamilton-De Donder equation, 194 Hamilton-Jacobi equation, 246 Hamiltonian, 234 admissible, 126 autonomous, 111 Hamiltonian action, 134 Hamiltonian connection, 147,238 locally, 144,232 Hamiltonian density, 142 Hamiltonian form, 141,234 associated with a Lagrangian, 249 constrained, 259 locally, 233 polysymplectic, 141

INDEX

Hamiltonian function, 235 Hamiltonian map, 115,142,235 Hamiltonian system completely integrable, 115 gauge invariant, 122 generalized, 129 presymplectic, 119 pull-back, 121 Hamiltonian vector field with respect to a Jacobi structure, 59 with respect to a Poisson structure, 68 with respect to a presymplectic struc­ ture, 82 locally, 82 with respect to a symplectic struc­ ture, 76 locally, 76 Helmholtz-Sonin map, 203 homology group, 89 horizontal density, 27 horizontal distribution, 45 horizontal foliation, 45 horizontal form, 27 horizontal vector field, 156 infinitesimal symmetry, 112 integral curve, 20 admissible, 32 integral invariant of Poincare-Cartan, 234 integral manifold, 32 maximal, 33 of maximal dimension, 33 integral of motion, 219

351

integral section, 43 interior product, 28 left, 28 of vector bundles, 13 right, 28 inverse problem, 107,202 involution, 67 isomorphism of fibre bundles, 12 isotropic submanifold, 78 Jacobi bracket, 57 generalized, 99 Jacobi bundle, 72 Jacobi field, 289 Jacobi identity, 57 generalized, 99 Jacobi manifold, 57 generalized, 99 Jacobi morphism, 59 conformal, 60 infinitesimal, 59 Jacobi structure, 57 conformaUy equivalent, 60 Jacobi vectors, 202 mass-weighted, 202 jet bundle, 36 affine, 36 jet manifold, 36 higher order, 40 repeated, 38 second order, 39 sesquiholonomic, 39 jet of sections, 36 jet of submanifolds, 299 jet prolongation of morphisms, 37

INDEX

352

of sections, 37 second order, 40 of vector fields, 37 kernel of a differential operator, 41 of a fibred morphism, 11 of a form, 30 kinetic energy, 201 Koszul-Brylinski-Poisson homology, 91 Lagrange equations, 187 autonomous, 116 Lagrangian, 186 almost regular, 258 autonomous, 115 generalized, 81 hyperregular, 117,189 non-degenerate, 188 quadratic, 262 regular, 115,188 semiregular, 253 Lagrangian function, 186 Lagrangian submanifold, 78 of a Jacobi manifold, 60 of a Poisson manifold, 70 Lagrangian system, 186 left-invariant form, 47 canonical, 47 Legendre bundle, 102,140 homogeneous, 101 Legendre map, 117,188 Legendre morphism, 193 Legendre submanifold, 63 Legendre vector bundle, 140 Leibniz rule, 66

Lichnerowicz-Jacobi cohomology, 94 Lichnerowicz-Poisson cochain complex, 92 Lichnerowicz-Poisson cohomology, 92 Lie algebra dual, 47 left, 46 right, 46 Lie bracket, 20 Lie coalgebra, 47 Lie derivative, 31 Lie-Poisson structure, 72 lift of a vector field canonical, 22 horizontal, 43 vertical, 22 Liouville vector field, 22 mass metric, 196 mass tensor, 196 Maurer-Cartan equation, 47 momentum mapping, 134 equivariant, 135 momentum phase space, 4,108 morphism of fibre bundles, 11 affine, 15 linear, 13 motion, 160 multisymplectic form, 101,102 multivector field, 22 simple, 23 Nambu-Jacobi bracket, 99 Nambu-Jacobi manifold, 99 Nambu-Poisson bracket, 99 Nambu-Poisson manifold, 99

INDEX

353

Newtonian system, 197 standard, 201 Noether conservation law, 221 Noether current, 221 normalizer, 88

presymplectic form, 81 presymplectic manifold, 81 projection, 10 pull-back fibre bundle, 12 pull-back form, 26

odd subspace, 322

rank of a bi vector field, 29 rank of a fibred morphism, 11 reduction leafwise, 87 reduction of a Poisson map, 86 reductive structure, 85 of a Poisson manifold, 85 Reeb vector field, 62 reference frame, 175 complete, 176 relativistic configuration space, 303 relativistic dynamic equation, 312 relativistic geodesic equation, 313 relativistic Hamiltonian, 309 relativistic momentum phase space, 307 relativistic principle, 307 relativistic symplectic form, 307 relativistic system, 306 relativistic velocity, 305 restriction of an exterior form, 26 restriction of a fibre bundle, 12 right-invariant form, 47 canonical, 48

parameter bundle, 274 Pfaffian system, 34 physical phase space, 122 Poincare-Cartan form, 115,187 Poisson action, 137 Poisson algebra, 66 Poisson automorphism, 67 Poisson bivector field, 58 Poisson bracket, 66 generalized, 99 Poisson bundle, 72 Poisson codifferential, 90 Poisson manifold, 66 generalized, 99 reduced, 85 Poisson morphism, 67 Poisson reduction, 85 Poisson structure, 66 coinduced, 68 generating, 69 non-degenerate, 66 regular, 66 zero, 66 polysymplectic form, 103 preconnection, 303 prequantization bundle, 329 presheaf, 18 canonical, 19

Schouten-Nijenhuis bracket, 24 of exterior forms, 97 section, 10 holonomic, 37 shape coordinates, 281 shape space, 281 sheaf, 18

INDEX

354

of continuous functions, 19 soldering form, 43 solution of a differential equation, 41 of a first order dynamic equation, 107 of a Hamiltonian system, 111 of a second order dynamic equa­ tion, 108 spray, 110 stalk, 18 standard vector field, 3 standard 1-form, 3 star operation, 91 subbundle, 11 superfunction, 324 symmetry transformations, 225 symplectic action, 133 symplectic automorphism, 74 symplectic bundle, 72 symplectic foliation, 69 symplectic form, 73 symplectic manifold, 73 reduced, 84 symplectic morphism, 73 symplectic realization, 75 symplectic reduction, 84 symplectic submanifold, 78 symplectomorphism, 74 tangent bundle, 15 tangent lift of a function, 27 of an exterior form, 27 of a multivector field, 23 tangent map, 16

tangent-valued form, 31 canonical, 32 tensor product of vector bundles, 14 torsion of a dynamic connection, 166 total space, 10 transition functions, 10 translation-reduced configuration space, 202 trivialization, 10 typical fibre, 10 unified phase space, 283 universal unit system, 8 variational operator, 203 variational sequence, 203 vector field, 20 complete, 21 holonomic, 108 left-invariant, 46 principal, 272 projectable, 21 right-invariant, 46 subordinate, 32 vertical, 22 vector superfield, 325 vector superspace, 324 velocity absolute, 176 relative, 176 velocity hyperboloids, 306 velocity phase space, 3,108 vertical configuration space, 285 vertical cotangent bundle, 17 vertical extension of a Hamiltonian form, 288

355

INDEX

vertical vertical vertical vertical vertical vertical

extension of a Lagrangian, 291 momentum phase space, 286 splitting, 17 tangent bundle, 17 tangent map, 17 velocity phase space, 286

Whitney sum of affine bundles, 15 Whitney sum of vector bundles, 14 n-body, 281 n-symplectic form, 102 B-supermanifold, 324 1-parameter group of diffeomorphisms, 21 1-paxameter group of local diffeomor­ phisms, 20 3-velocity, 305 3-velocity phase space, 305 4-velocity, 305 4-velocity phase space, 305