Zehnder E. Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities

Подождите немного. Документ загружается.

IV.4. Gradient systems on manifolds and Morse theory 181

A. Banyaga and D. Hurtubise. The monograph [100] by M. Schwarz is devoted to

Morse homology theory. We also recommend the article [12] by R. Bott. Morse

theory in inﬁnite dimensions can be found in the books [14] by K. C. Chang, [1]by

S. Abbondandolo and in the articles [76], [77] and [78] by R. S. Palais.

As for the Conley index theory, we refer to the original work [19] by C. Conley,

to [96] by D. Salamon, to [36] by J. Franks and M. Misiurewicz and to the survey

article [69] by K. Mischaikow and M. Mrozek and the references therein. The

Conley index theory in inﬁnite dimensions is developed in [94] by K. Rybakowski.

The Conley Morse theory for continuous ﬂows on locally compact metric spaces

is presented in [20] by C. Conley and E. Zehnder and provides, in particular, an

efﬁcient approach to Morse inequalities.

Chapter V

Hamiltonian vector ﬁelds and symplectic

diffeomorphisms

Chapter V introduces the special class of Hamiltonian vector ﬁelds. They are deter-

mined by a single function called the Hamiltonian function and play an important

role in frictionless systems such as the N -body problem of celestial mechanics.

Hamiltonian vector ﬁelds are deﬁned on symplectic manifolds. These manifolds

are even-dimensional carrying a distinguished 2-form which is closed and nonde-

generate and called a symplectic structure. In contrast to Riemannian structures

which do exist on every manifold, not every even-dimensional manifold admits a

symplectic structure. We shall see that symplectic manifolds of the same dimen-

sion are locally indistinguishable (Darboux’s theorem), again in sharp contrast to

Riemannian manifolds where the curvature is an example of a local invariant. The

ﬂow maps generated by the Hamiltonian vector ﬁelds leave the Hamilton function

(energy conservation) and the symplectic 2-form invariant. They are examples of

symplectic mappings. The transformation theory of Hamiltonian vector ﬁelds, the

Hamiltonian formalism, is treated in detail in the Sections 7 and 8, where we make

use of the exterior differential calculus. It is recalled in the Sections 2–6 and is

presumably familiar to most of our readers. On energy surfaces the Hamiltonian

ﬂow preserves a distinguished volume form (Liouville form), so that in the compact

situation the recurrence theorem of Poincaré and the ergodic theorem of Birkhoff

are applicable. For practical purposes it is good to know that a symplectic map can

be represented by a single function, called a generating function, as is explained in

Section 9. Finally, integrable systems are discussed in Section 10. These systems

are characterized by the property of possessing sufﬁciently many integrals such that

the task of solving the Hamiltonian equation for all times becomes essentially triv-

ial, as will be illustrated by the proof of the theorem of V. I. Arnold and R. Jost about

the existence of action and angle variables. Integrable systems are exceptionally

rare, but there are many natural systems close to integrable systems to which the

KAM perturbation theory is applicable.

V.1 Symplectic vector spaces

We begin with the symplectic linear algebra.

Deﬁnition. A symplectic vector space is a pair .V; !/ consisting of a ﬁnite dimen-

sional real vector space V and a bilinear form ! W V  V ! R which is skew

V.1. Symplectic vector spaces 183

symmetric and nondegenerate. The form ! is called a symplectic structure or sym-

plectic form on V .

Bilinearity refers to the R-linearity in every argument, skew symmetry means

that !.u; v/ D!.v;u/ for all vectors u; v 2 V and nondegeneracy requires

that from !.u; v/ D 0 for all v 2 V it follows that u D 0. Equivalently, the

nondegeneracy requires that the linear mapping !

]

W V ! V



, which associates

with the vector u 2 V the linear form !.u; / on V , is an isomorphism.

Examples. (1) Assuming W to be an n-dimensional vector space and W



to be

its dual space, we deﬁne the vector space V ´ W  W



and the bilinear form

! W V  V ! R by



.h; h



/; .k; k





´ h



.k/  k



.h/:

Then the pair .V; !/ is a symplectic vector space of dimension 2n.

(2) The symplectic standard space .R

/. Here the vector space is equal to

V D R

and the symplectic form is deﬁned by

.u; v/ ´hJu;vi;u;v2 R

;

where h; i denotes the Euclidian scalar product in R

and J denotes the block

matrix

J D



0 1

1 0



with respect to R

D R

 R

. This matrix is nondegenerate, because we have

det J D 1 and it is skew symmetric, J

DJ . In addition, the matrix J has the

properties

D1 and J

DJ D J

1

(3) There is no symplectic vector space of dimension dim V D 1. Indeed,

choosing a non-vanishing vector u ¤ 0 in the 1-dimensional space V , it follows

that u D rv and hence !.u; v/ D 0 for all vectors v in V in view of the skew

symmetry. Therefore the 2-form ! is necessarily degenerate.

As in Euclidian geometry the symplectic orthogonal complement can be deﬁned.

Deﬁnition. The symplectic orthogonal complement of the linear subspace E  V

is the linear subspace

´fv 2 V j !.e; v/ D 0 for all e 2 Eg:

The nondegeneracy of ! is equivalent to V \ V

Df0g.

Proposition V.1. If E  V is a linear subspace, then

dim E C dim E

D dim V

for the symplectic orthogonal complement E

184 Chapter V. Hamiltonian vector ﬁelds and symplectic diffeomorphisms

Proof. Let e

;:::;e

be a basis of E. It then follows from the bilinearity of ! that

the symplectic orthogonal complement is given by

Dfv 2 V j !.e

;v/ D 0; 1  j  rg

1j r

ker !.e

; /:

Therefore E

is the kernel of the linear mapping A W V ! R

, deﬁned by Av D



!.e

;v/;:::;!.e

;v/



. The linear functionals !.e

; / are linearly independent,

since !

]

is an isomorphism. Therefore dim Im A D r and the proposition follows

from the dimension formula dim V D dim Im A C dim ker A. 

From the deﬁnition of E

it follows that E  .E

and applying Propo-

sition V. 1 to the linear subspace E

(replacing E) one concludes that dim E D

dim.E

and so,

E D .E

We make use of the symbolic notation u?v for two vectors satisfying !.u; v/ D 0.

In contrast to the orthogonality in Euclidian geometry we have in general,

E C E

¤ V;

as the following example shows.

Example. If .V; !/ is a symplectic vector space and u ¤ 0 a vector in V ,we

set E ´hui. Due to the skew symmetry, !.u; u/ D 0 and therefore E  E

However, E

¤ V , because ! is nondegenerate.

If E  V is a linear subspace, the restriction of ! onto E  E is still a skew

symmetric bilinear form, but in general it is no longer nondegenerate.

Deﬁnition. A linear subspace E  .V; !/ is called symplectic, if the pair .E; !/

is a symplectic vector space.

Hence, a linear subspace E  V is symplectic if and only if E \ E

Df0g.

Due to Proposition V.1 , this is equivalent to

E ˚ E

D V:

Setting E D .E

, we obtain

E symplectic () E

symplectic :

The following proposition shows that every symplectic vector space .V; !/ looks

in suitable coordinates like the symplectic standard space .R

V.1. Symplectic vector spaces 185

Proposition V.2 (Symplectic basis). If .V; !/ is a symplectic vector space, then

dim V D 2n is an even number and there exists a distinguished basis fe

;:::;e

;

;:::;f

g, satisfying for all i; j D 1;:::;n

!.e

/ D 0; !.f

/ D ı

Such a basis is called a symplectic basis. The coordinates with respect to a sym-

plectic basis are called symplectic coordinates.

Proof. (1) We choose e

¤ 0 in V . Since ! is nondegenerate, there exists a vector

u 2 V satisfying !.u; e

/ ¤ 0. Normalizing we set f

´ !.u; e

1

u and obtain

!.f

/ D 1. Since ! is skew symmetric, the two vectors e

are linearly

independent. According to the construction E ´ spanfe

g is a symplectic

subspace of V of dimension dim E D 2. If dim V D 2, the proposition is proved.

(2) If dim V>2the same argument can be applied to the symplectic subspace

. In view of E ˚ E

D V the proof follows in ﬁnitely many steps. Due to the

nondegeneracy of !,no1-dimensional space can remain after the last step, because,

as we have seen, 1-dimensional spaces cannot be symplectic. 

With respect to symplectic coordinates, the matrix associated with the sym-

plectic form ! is in a normal form which we shall compute next. We choose the

symplectic basis fe

;:::;e

;:::;f

g of .V; !/. With respect to this basis, we

let

u D

j D1

C x

nCj

/ and v D

j D1

C y

nCj

be two vectors belonging to V . Using the deﬁnition of a symplectic basis one

computes

!.u; v/ D

i;j

!.e

/ C

i;j

nCi

!.f

/ C

i;j

nCi

!.e

i;j

nCi

nCj

!.f

i;j

nCi

!.f

/ C

i;j

nCi

!.e

i;j

nCi

 x

nCi

nCj

 x

nCj

DhJx;yi

186 Chapter V. Hamiltonian vector ﬁelds and symplectic diffeomorphisms

where x D .x

;:::;x

/ and y D .y

;:::;y

/, and where J is the matrix

J D



0 1

1 0



with respect to the splitting R

D R

 R

. This is the desired normal form we

shall often meet later on.

Remark. In the literature, another normal form is sometimes used. In view of the

deﬁnition of the symplectic basis, the 2-dimensional subspaces E

´ spanfe

are symplectic subspaces satisfying E

for i ¤ j and

V D E

˚˚E

Representing the vectors u and v in the symplectic coordinates x

;:::;x

with

respect to the above decomposition of V , so that

u D x

C x

CCx

2n1

C x

;

and similarly for v, one veriﬁes that

!.u; v/ Dh

Jx;yi

where

J is the block diagonal matrix

J D

10

Every symplectic structure ! of a symplectic vector space .V; !/ of dimension

2n has the same normal form with respect to a symplectic basis. This is in contrast

to the symmetric bilinear forms of Euclidian geometry where the signature is an in-

variant. In symplectic coordinates we always have !.u; v/ DhJx;yiD!

.x; y/,

so that we are in the symplectic standard space .R

As in Euclidian geometry we obtain the following rules for linear subspaces

E; F  V of a symplectic vector space .V; !/. Some of the rules we have already

veriﬁed and the remaining ones are obvious.

E  F H) F

 E

;

D E;

.E \ F/

D E

C F

;

.E C F/

D E

\ F

;

dim E

C dim E D dim V:

V.1. Symplectic vector spaces 187

In a symplectic vector space we distinguish the following subspaces.

Deﬁnition. We assume .V; !/ to be a symplectic vector space of dimension 2n.A

linear subspace E  V is called

(i) isotropic ,ifE  E

(H) dim E  n),

(ii) Lagrangian,ifE D E

(H) dim E D n),

(iii) coisotropic ,ifE  E

(H) dim E  n),

(iv) symplectic,ifE \ E

Df0g (H) dim E D 2k even).

The statements about the dimension of the subspaces E are obtained in the

following way.

(i) If E  E

, it follows from Proposition V. 1 that dim E  dim E

dim V  dim E and hence 2 dim E  dim V D 2n.

(ii) Lagrangian means maximal isotropic and is obviously characterized by the

two properties E  E

and dim E D dim E

D n.

(iii) is proved as the statement (i) and

(iv) has already been veriﬁed above.

Examples. In the symplectic standard space .R

/ we set x D .q; p/ and

D .q

/ with respect to R

 R

, so that

.x; x

/ DhJx;x

iDhp; q

ihq; p

The subspace

E Dfx 2 R

j q

DDq

D 0g

is coisotropic for r  n and Lagrangian for r D n. The set

E Dfx 2 R

j q

D p

D 0;:::;q

D p

D 0g

is an example of a symplectic subspace for r<n. The subspace

E Dfx 2 R

j q

DDq

D 0; p

DDp

D 0g

is isotropic.

We now consider the linear maps A W V ! V of .V; !/ which leave the sym-

plectic form ! invariant. To be more precise we introduce the notion of a pullback.

We assume V to be a vector space, b W V  V ! R to be a bilinear form and

A 2 L.V / a linear mapping. Then the pullback of the bilinear form b under the

map A is the bilinear form A



b on V deﬁned by



b/.u; v/ ´ b.Au; Av/ for all u; v 2 V:

Deﬁnition. A linear map A W V ! V of a symplectic vector space .V; !/ is called

symplectic,ifA



! D !, or explicitly if

!.Au; Av/ D !.u; v/ for all u; v 2 V:

188 Chapter V. Hamiltonian vector ﬁelds and symplectic diffeomorphisms

In symplectic coordinates or in the symplectic standard space .R

/,we

obtain a simple criterion for the symplectic character of the map A 2 L.R

/.In

view of !

.u; v/ DhJu;vi, the condition A



D !

is equivalent to

hJAu; AviDhJu;vi;u;v2 R

or hA

JAu; viDhJu;vi, for all vectors u, v and hence

JA D J:

A .2n 2n/-matrix A is called symplectic, if it satisﬁes A

JA D J . In the special

case n D 1 the matrix A is symplectic if and only if det A D 1, because for

A D





we have

JA D



0ad bc

ad C bc 0



In particular, such a matrix A is volume preserving. In the general case it follows

from A

JA D J that .det A/

D det J D 1, so that A is invertible. We shall later

show that actually det A D 1.

The set of symplectic matrices constitutes the so-called symplectic group

Sp.n/ ´fA 2 L.R

/ j A

JA D J g

which is one of the classical Lie groups.

Proposition V.3. The set Sp.n/ is a group; if A; B 2 Sp.n/ then also AB 2 Sp.n/,

1

2 Sp.n/ and 1

2 Sp.n/. In addition, A

2 Sp.n/ and J 2 Sp.n/.

Proof. Let J D A

JA, then .A

1

D J and A

1

2 Sp.n/. By inverting

this equation we obtain J

1

D AJ

1

and using J

1

DJ we obtain J D

, so that indeed A

2 Sp.n/. The remaining statements are also easily

veriﬁed. 

Exercise. (1) We assume A to be a .2n  2n/-block matrix of the form

A D





Then A is symplectic precisely if the .n n/-matrices a

c and b

d are symmetric

and a

d  c

b D 1.

(2) We now set b D 0. Then A 2 Sp.n/, precisely if A is a matrix product of

the form

A D



0.a

1



1 0

s 1



V.1. Symplectic vector spaces 189

where a is invertible and where s D s

is symmetric. Both matrices on the right-

hand side are symplectic.

(3) Polar form. Let U 2 L.R

/ be a matrix satisfying det U ¤ 0. Then U has

a unique polar form U D OP , where O is orthogonal and P is a positive symmetric

matrix. Show that if U is belongs to Sp.n/, then also O and P belong to Sp.n/.

(4) Let O be orthogonal, then O 2 Sp.n/ is equivalent to

O D



ba



where a

b is symmetric and a

a Cb

b D 1. These conditions are also necessary

and sufﬁcient for the complex matrix

a C ib 2 U.n/  L.C

to be a unitary matrix. Here U.n/ denotes the group of the unitary .n n/-matrices.

The above construction establishes an algebraic isomorphism

Sp.n/ \ O.2n/ Š U.n/;

where O.k/ denotes the group of the orthogonal .k  k/-matrices.

Proposition V.4. If A 2 Sp.n/, then det A D 1, so that A is volume preserving.

We already know that .det A/

D 1 and hence det A D˙1. To determine the

sign, it is convenient to use the formalism of differential forms.

A brief excursion into differential forms. Take coordinates z D .z

;:::;z

/ in

. Then the linear form

2 L.R

; R/ D .R



is deﬁned as

.a/ ´ e



.a/ ´ a

for all a D .a

;:::;a

/ 2 R

This is the derivative of the j -th coordinate function f

W R

! R deﬁned by

f.z/ D z

for z D .z

;:::;z

/.Byfe



;:::;e



g we denote the dual basis of the

canonical basis fe

;:::;e

g of R

satisfying e



/ D ı

The 2-form (skew symmetric bilinear form) dz

^ dz

is deﬁned as

.dz

^ dz

/.a; b/ ´ det



.a/ dz

.b/

.a/ dz

.b/



D a

 a

190 Chapter V. Hamiltonian vector ﬁelds and symplectic diffeomorphisms

Obviously, dz

^dz

D.dz

^dz

/. With this notation, the canonical symplectic

structure !

on R

can be written as a differential form. In the coordinates z D

.x; y/ 2 R

 R

D R

it has the representation

j D1

^ dx

;

or explicitly,

.u; v/ DhJu;viD

j D1

.dy

^ dx

/.u; v/ for all u; v 2 R

At this point, we recall the concept of a k-form on R

. We assume 1  k  d

and x D .x

;:::;x

/ 2 R

. The special k-form dx

^^dx

is deﬁned by

.dx

^^dx

/.v

;:::;v

/ ´ det

/  dx

for v

;:::;v

2 R

. The k-forms

fdx

^^dx

;1 i

< <i

 d g

constitute a basis of the vector space

/ of all the alternating k-linear forms

˛ on R

, that is, of the continuous maps ˛ W R

R

! R (the product taken

k times), linear in every argument and satisfying

˛.:::;v

;:::;v

;:::/D˛.:::;v

;:::;v

;:::/

for all i ¤ j . The integer k is called the degree of the form ˛ and denoted by deg ˛.

We now assume 1  p; q  d .For˛ 2

and ˇ 2

we can introduce the

exterior product ˛ ^ ˇ 2

pCq

. The .p Cq/-form ˛ ^ˇ is deﬁned by

.˛ ^ ˇ/.v

;:::;v

pCq

pŠqŠ

2S

pCq

"./ ˛.v

.1/

;:::;v

.p/

/ˇ.v

.pC1/

;:::;v

.pCq/

where S

is the group of all permutations of n elements and "./ is the sign of the

permutation  . This product is associative and satisﬁes

ˇ ^ ˛ D .1/

deg ˛ deg ˇ

.˛ ^ ˇ/: