0

W0 — U ’’ R

give isomorphisms

„¦

„¦1 : W1 ’’ U — .

1

W1 — U ’’ R

Consider the diagram

„¦

U—

0

’’

W0

B“ “ id

„¦

U—

1

’’

W1

where the linear map B satis¬es „¦1 —¦ B = „¦0 , i.e., „¦0 (ω0 , u) = „¦1 (Bω0 , u), ∀ω0 ∈

W0 , ∀u ∈ U . Extend B to the rest of V by setting it to be the identity on U :

L := IdU • B : U • W0 ’’ U • W1 .

25

2.7. LAGRANGIAN NEIGHBORHOOD THEOREM

Finally, we check that L— „¦1 = „¦0 :

(L— „¦1 )(u • w0 , u • w0 ) „¦1 (u • Bω0 , u • Bω0 )

=

= „¦1 (u, Bω0 ) + „¦1 (Bω0 , u )

= „¦0 (u, ω0 ) + „¦0 (ω0 , u )

„¦0 (u • w0 , u • w0 ) .

=

2.7 Lagrangian Neighborhood Theorem

Theorem 2.11 (Weinstein Lagrangian Neighborhood Theorem [44]) Let

M be a 2n-dimensional manifold, X an n-dimensional submanifold, i : X ’ M the

inclusion map, and ω0 and ω1 symplectic forms on M such that i— ω0 = i— ω1 = 0,

i.e., X is a lagrangian submanifold of both (M, ω0 ) and (M, ω1 ). Then there exist

neighborhoods U0 and U1 of X in M and a di¬eomorphism • : U0 ’ U1 such that

• E U1

U0

d

s

d

• — ω1 = ω 0 .

d commutes and

id i

d

X

Proof. The proof of the Weinstein theorem uses the Whitney extension theorem

(see Appendix A).

Put a riemannian metric g on M ; at each p ∈ M , gp (·, ·) is a positive-de¬nite

inner product. Fix p ∈ X, and let V = Tp M , U = Tp X and W = U ⊥ = ortho-

complement of U in V relative to gp (·, ·).

Since i— ω0 = i— ω1 = 0, U is a lagrangian subspace of both (V, ω0 |p ) and

(V, ω1 |p ). By symplectic linear algebra, we canonically get from U ⊥ a linear iso-

morphism Lp : Tp M ’ Tp M , such that Lp |Tp X = IdTp X and L— ω1 |p = ω0 |p . Lp

p

varies smoothly with respect to p since our recipe is canonical.

By the Whitney theorem (Theorem A.11), there are a neighborhood N of X

and an embedding h : N ’ M with h|X = idX and dhp = Lp for p ∈ X. Hence,

at any p ∈ X,

(h— ω1 )p = (dhp )— ω1 |p = L— ω1 |p = ω0 |p .

p

Applying the Moser local theorem (Theorem 1.11) to ω0 and h— ω1 , we ¬nd a

neighborhood U0 of X and an embedding f : U0 ’ N such that f |X = idX and

f — (h— ω1 ) = ω0 on Uo . Set • = h —¦ f .

Theorem 2.11 has the following generalization; see, for instance, either of [22,

27, 46].

26 LECTURE 2. COTANGENT BUNDLES

Theorem 2.12 (Coisotropic Embedding Theorem) Let M be a manifold

of dimension 2n, X a submanifold of dimension k ≥ n, i : X ’ M the inclusion

map, and ω0 and ω1 symplectic forms on M , such that i— ω0 = i— ω1 and X is

coisotropic for both (M, ω0 ) and (M, ω1 ). Then there exist neighborhoods U0 and

U1 of X in M and a di¬eomorphism • : U0 ’ U1 such that

• E U1

U0

d

s

d

• — ω1 = ω 0 .

d commutes and

id i

d

X

2.8 Weinstein Tubular Neighborhood Theorem

Let (V, „¦) be a symplectic linear space, and let U be a lagrangian subspace.

Claim. There is a canonical nondegenerate bilinear pairing „¦ : V /U — U ’ R.

Proof. De¬ne „¦ ([v], u) = „¦(v, u) where [v] is the equivalence class of v in V /U .

Consequently, we get that „¦ : V /U ’ U — de¬ned by „¦ ([v]) = „¦ ([v], ·) is

an isomorphism, so that V /U U — are canonically identi¬ed.

In particular, if (M, ω) is a symplectic manifold, and X is a lagrangian sub-

manifold, then Tx X is a lagrangian subspace of (Tx M, ωx ) for each x ∈ X. The

space Nx X := Tx M/TxX is called the normal space of X at x. Since we have a

—

canonical identi¬cation Nx X Tx X, we get:

Proposition 2.13 The vector bundles N X and T — X are canonically identi¬ed.

Putting this observation together with the lagrangian neighborhood theorem,

we arrive at:

Theorem 2.14 (Weinstein Tubular Neighborhood Theorem) Let (M, ω)

be a symplectic manifold, X a lagrangian submanifold, ω0 canonical symplectic

form on T — X, i0 : X ’ T — X the lagrangian embedding as the zero section, and

i : X ’ M lagrangian embedding given by inclusion. Then there are neighborhoods

U0 of X in T — X, U of X in M , and a di¬eomorphism • : U0 ’ U such that

• EU

U0

d

s

d

• — ω = ω0 .

d commutes and

i0 d i

d

X

27

2.8. WEINSTEIN TUBULAR NEIGHBORHOOD THEOREM

Proof. This proof relies on (1) the standard tubular neighborhood theorem (see

Appendix A), and (2) the Weinstein lagrangian neighborhood theorem.

T — X, we can ¬nd a neighborhood N0 of X in T — X, a neigh-

1. Since N X

borhood N of X in M , and a di¬eomorphism ψ : N0 ’ N such that

ψ EN

N0

d

s

d

d commutes .

i0 d i

d

X

ω0 = canonical form on T — X

symplectic forms on N0 .

Let

ω1 = ψ — ω

X is lagrangian for both ω0 and ω1 .

2. There exist neighborhoods U0 and U1 of X in N0 and a di¬eomorphism

θ : U0 ’ U1 such that

θ E U1

U0

d

s

d θ — ω1 = ω 0 .

d commutes and

i0 d i0

d

X

Take • = ψ —¦ θ and U = •(U0 ). Check that •— ω = θ— ψ — ω = ω0 .

ω1

Remark. Theorem 2.14 classi¬es lagrangian embeddings: up to symplectomor-

phism, the set of lagrangian embeddings is the set of embeddings of manifolds into

their cotangent bundles as zero sections.

The classi¬cation of isotropic embeddings was also carried out by Weinstein

in [45, 46]. An isotropic embedding of a manifold X into a symplectic man-

ifold (M, ω) is a closed embedding i : X ’ M such that i— ω = 0. Weinstein

showed that neighbourhood equivalence of isotropic embeddings is in one-to-one

correspondence with isomorphism classes of symplectic vector bundles.

The classi¬cation of coisotropic embeddings is due to Gotay [22]. A coisotro-

pic embedding of a manifold X carrying a closed 2-form ± of constant rank into

a symplectic manifold (M, ω) is an embedding i : X ’ M such that i— ω = ±

and i(X) is coisotropic has a submanifold of M . Let E be the characteristic

28 LECTURE 2. COTANGENT BUNDLES

distribution of a closed form ± of constant rank on X, i.e., Ep is the kernel

of ±p at p ∈ X. Gotay showed that then E — carries a symplectic structure in a

neighbourhood of the zero section, such that X embeds coisotropically onto this

zero section, and, moreover every coisotropic embedding is equivalent to this in

™¦

some neighbourhood of the zero section.

2.9 Symplectomorphisms as Lagrangians

Lagarangian submanifolds are important to study symplectomorphisms, as will be

explored in the next lecture.

Let (M1 , ω1 ) and (M2 , ω2 ) be two 2n-dimensional symplectic manifolds. Given

a di¬eomorphism • : M1 ’’ M2 , when is it a symplectomorphism? (I.e., when is

•— ω2 = ω1 ?) Consider the diagram of projection maps

M1 — M 2

(p1 , p2 ) (p1 , p2 )

d

d pr2

pr1

d

d

© ‚

d

c c

p1 M1 M2 p2

Then ω = (pr1 )— ω1 + (pr2 )— ω2 is a 2-form on M1 — M2 which is closed,

dω = (pr1 )— dω1 + (pr2 )— dω2 = 0 ,

0 0

and symplectic,

n n

2n

ω 2n = (pr1 )— ω1 § (pr2 )— ω2 =0.

n

More generally, if »1 , »2 ∈ R\{0}, then »1 (pr1 )— ω1 +»2 (pr2 )— ω2 is also a symplectic

form on M1 — M2 . Take »1 = 1, »2 = ’1 to obtain the twisted product form

on M1 — M2 :

ω = (pr1 )— ω1 ’ (pr2 )— ω2 .

The graph of a di¬eomorphism • : M1 ’’ M2 is the 2n-dimensional sub-

manifold of M1 — M2 :

“• := Graph • = {(p, •(p)) | p ∈ M1 } .

The submanifold “• is an embedded image of M1 in M1 — M2 , the embedding

being the map

γ : M1 ’’ M1 — M2

p ’’ (p, •(p)) .

29

2.9. SYMPLECTOMORPHISMS AS LAGRANGIANS

Proposition 2.15 A di¬eomorphism • is a symplectomorphism if and only if “ •

is a lagrangian submanifold of (M1 — M2 , ω).

Proof. The graph “• is lagrangian if and only if γ — ω = 0.

γ—ω = γ — pr— ω1 ’ γ — pr— ω2

1 2

= (pr1 —¦ γ) ω1 ’ (pr2 —¦ γ)— ω2 .

—

But pr1 —¦ γ is the identity map on M1 and pr2 —¦ γ = •. Therefore,

γ—ω = 0 • — ω2 = ω 1 .

⇐’

Lecture 3

Generating Functions

Generating functions provide a method for producing symplectomorphisms via la-

grangian submanifolds. We will illustrate their use in riemannian geometry and

dynamics. We conclude with an application to the study of the group of sym-

plectomorphisms and to the problem of the existence of ¬xed points, whose ¬rst

instance is the Poincar´-Birkho¬ theorem.

e

3.1 Constructing Symplectomorphisms

Let X1 , X2 be n-dimensional manifolds, with cotangent bundles M1 = T — X1 ,

M2 = T — X2 , tautological 1-forms ±1 , ±2 , and canonical 2-forms ω1 , ω2 .

Under the natural identi¬cation

M 1 — M 2 = T — X1 — T — X2 T — (X1 — X2 ) ,

the tautological 1-form on T — (X1 — X2 ) is

± = (pr1 )— ±1 + (pr2 )— ±2 ,

where pri : M1 — M2 ’ Mi , i = 1, 2 are the two projections. The canonical 2-form

on T — (X1 — X2 ) is

ω = ’d± = ’dpr— ±1 ’ dpr— ±2 = pr— ω1 + pr— ω2 .

1 2 1 2

In order to describe the twisted form ω = pr— ω1 ’ pr— ω2 , we de¬ne an involution

1 2

of M2 = T — X2 by

M2 ’’ M2

σ2 :

(x2 , ξ2 ) ’’ (x2 , ’ξ2 )

—

which yields σ2 ±2 = ’±2 . Let σ = idM1 — σ2 : M1 — M2 ’ M1 — M2 . Then

σ — ω = pr— ω1 + pr— ω2 = ω .

1 2

31

32 LECTURE 3. GENERATING FUNCTIONS

If Y is a lagrangian submanifold of (M1 — M2 , ω), then its “twist” Y σ := σ(Y ) is

a lagrangian submanifold of (M1 — M2 , ω).

Recipe for producing symplectomorphisms M 1 = T — X1 ’ M2 = T — X2 :

1. Start with a lagrangian submanifold Y of (M1 — M2 , ω).

2. Twist it to obtain a lagrangian submanifold Y σ of (M1 — M2 , ω).

3. Check whether Y σ is the graph of some di¬eomorphism • : M1 ’ M2 .

4. If it is, then • is a symplectomorphism by Section 2.9.

Let i : Y σ ’ M1 — M2 be the inclusion map

Yσ

d

pr1 —¦ i d pr2 —¦ i

d

d

© ‚

d

•? E M2

M1

Step 3 amounts to checking whether pr1 —¦ i and pr2 —¦ i are di¬eomorphisms. If yes,

then • := (pr2 —¦ i) —¦ (pr1 —¦ i)’1 is a di¬eomorphism.

T — (X1 — X2 ), we

In order to obtain lagrangian submanifolds of M1 — M2

can use the method of generating functions.

3.2 Method of Generating Functions

For any f ∈ C ∞ (X1 — X2 ), df is a closed 1-form on X1 — X2 . The lagrangian

submanifold generated by f is

Yf := {((x, y), (df )(x,y) ) | (x, y) ∈ X1 — X2 } .

We adopt the notation

—

:= (df )(x,y) projected to Tx X1 — {0},

dx f

—

:= (df )(x,y) projected to {0} — Ty X2 ,

dy f

which enables us to write

Yf = {(x, y, dx f, dy f ) | (x, y) ∈ X1 — X2 }

and

Yfσ = {(x, y, dx f, ’dy f ) | (x, y) ∈ X1 — X2 } .

33

3.2. METHOD OF GENERATING FUNCTIONS

When Yfσ is in fact the graph of a di¬eomorphism • : M1 ’ M2 , we call • the

symplectomorphism generated by f , and call f the generating function,

of • : M1 ’ M2 .

So when is Yfσ the graph of a di¬eomorphism • : M1 ’ M2 ?

Let (U1 , x1 , . . . , xn ), (U2 , y1 , . . . , yn ) be coordinate charts for X1 and X2 ,

with associated charts (T — U1 , x1 , . . . , xn , ξ1 , . . . , ξn ), (T — U2 , y1 , . . . , yn , ·1 , . . . , ·n )

for M1 and M2 . The set

Yfσ = {(x, y, dx f, ’dy f ) | (x, y) ∈ X1 — X2 }

is the graph of • : M1 ’ M2 if and only if, for any (x, ξ) ∈ M1 and (y, ·) ∈ M2 ,

we have

•(x, ξ) = (y, ·) ⇐’ ξ = dx f and · = ’dy f .

Therefore, given a point (x, ξ) ∈ M1 , to ¬nd its image (y, ·) = •(x, ξ) we must

solve the “Hamilton” equations

±

‚f

ξi =

(x, y) ()

‚xi

· = ’ ‚f (x, y)

i ()

‚yi

If there is a solution y = •1 (x, ξ) of ( ), we may feed it to ( ) thus obtaining

· = •2 (x, ξ), so that •(x, ξ) = (•1 (x, ξ), •2 (x, ξ)). Now by the implicit function

theorem, in order to solve ( ) locally for y in terms of x and ξ, we need the

condition

n

‚ ‚f

det =0.

‚yj ‚xi i,j=1

This is a necessary local condition for f to generate a symplectomorphism •.

Locally this is also su¬cient, but globally there is the usual bijectivity issue.

2

Rn , and f (x, y) = ’ |x’y| , the square

Example. Let X1 = U1 Rn , X2 = U2 2

of euclidean distance up to a constant.

The “Hamilton” equations are

± ±

‚f

ξi yi = xi + ξ i

= y i ’ xi

=

‚xi ⇐’

‚f

·

i =’ = y i ’ xi ·i = ξi

‚yi

The symplectomorphism generated by f is

•(x, ξ) = (x + ξ, ξ) .

34 LECTURE 3. GENERATING FUNCTIONS

If we use the euclidean inner product to identify T — Rn with T Rn , and hence

regard • as • : T Rn ’ T Rn and interpret ξ as the velocity vector, then the

symplectomorphism • corresponds to free translational motion in euclidean space.

¨B

¨

¨¨

¨¨

ξ Br

¨ x+ξ

¨¨

¨

r

x

™¦

3.3 Riemannian Distance

Let V be an n-dimensional vector space. A positive inner product G on V is a

bilinear map G : V — V ’ R which is

symmetric : G(v, w) = G(w, v) , and

positive-de¬nite : G(v, v) > 0 when v=0.

De¬nition 3.1 A riemannian metric on a manifold X is a function g which

assigns to each point x ∈ X a positive inner product gx on Tx X.

A riemannian metric g is smooth if for every smooth vector ¬eld v : X ’

T X the real-valued function x ’ gx (vx , vx ) is a smooth function on X.

De¬nition 3.2 A riemannian manifold (X, g) is a manifold X equipped with

a smooth riemannian metric g.

Let (X, g) be a riemannian manifold. The arc-length of a piecewise smooth

curve γ : [a, b] ’ X is

b

dγ dγ dγ dγ

arc-length of γ := dt , where := gγ(t) , .

dt dt dt dt

a

By changing variables in the integral, we see that the arc-length of γ is independent

of the parametrization of γ, i.e., if we reparametrize γ by „ : [a , b ] ’ [a, b], the

new curve γ = γ —¦ „ : [a , b ] ’ X has the same arc-length.

35

3.3. RIEMANNIAN DISTANCE

y = γ(b)

r

γ

E

r

x = γ(a)

dγ

A curve γ is called a curve of constant velocity when is indepen-

dt

dent of t. Given any curve γ : [a, b] ’ X (with dγ never vanishing), there is a

dt

reparametrization „ : [a, b] ’ [a, b] such that γ —¦ „ : [a, b] ’ X is of constant

velocity. The action of a piecewise smooth curve γ : [a, b] ’ X is

2

b

dγ

A(γ) := dt .

dt

a

Exercise 12

Show that, among all curves joining two given points, γ minimizes the action

if and only if γ is of constant velocity and γ minimizes arc-length.

Hint:

(a) Let „ : [a, b] ’ [a, b] be a smooth monotone map taking the endpoints

of [a, b] to the endpoints of [a, b]. Then

2

b d„

dt ≥ b ’ a ,

dt

a

d„

with equality holding if and only if = 1.

dt

(b) Suppose that γ is of constant velocity, and let „ : [a, b] ’ [a, b] be a

reparametrization. Show that A(γ —¦ „ ) ≥ A(γ), with equality only when

„ = identity.

De¬nition 3.3 The riemannian distance between two points x and y of a con-

nected riemannian manifold (X, g) is the in¬mum d(x, y) of the set of all arc-

lengths for piecewise smooth curves joining x to y.

A smooth curve joining x to y is a minimizing geodesic 1 if its arc-length

is the riemannian distance d(x, y).

A riemannian manifold (X, g) is geodesically convex if every point x is

joined to every other point y by a unique (up to reparametrization) minimizing

geodesic.

1 In riemannian geometry, a geodesic is a curve which locally minimizes distance and whose

velocity is constant.

36 LECTURE 3. GENERATING FUNCTIONS

Example. On X = Rn with T X Rn — Rn , let gx (v, w) = v, w , gx (v, v) =

|v|2 , where ·, · is the euclidean inner product, and | · | is the euclidean norm.

Then (Rn , ·, · ) is a geodesically convex riemannian manifold, and the riemannian

distance is the usual euclidean distance d(x, y) = |x ’ y|. ™¦

3.4 Geodesic Flow

Suppose that (X, g) is a geodesically convex riemannian manifold. Assume also

that (X, g) is geodesically complete, that is, every minimizing geodesic can be

extended inde¬nitely. Given (x, v) ∈ T X, let exp(x, v) : R ’ X be the unique

minimizing geodesic of constant velocity with initial conditions exp(x, v)(0) = x

and d exp(x,v) (0) = v.

dt

Consider the function

1

f (x, y) = ’ · d(x, y)2 .

f : X — X ’’ R ,

2

What is the symplectomorphism • : T — X ’ T — X generated by f ?

Proposition 3.4 Under the identi¬cation of T X with T — X by g, the symplecto-

morphism generated by • coincides with the map T X ’ T X, (x, v) ’ exp(x, v)(1).

—

Proof. Let dx f and dy f be the components of df(x,y) with respect to T(x,y)(X —

— —

X) Tx X — Ty X. The metric gx : Tx X — Tx X ’ R induces an identi¬cation

—

’’ Tx X

g x : Tx X

’’ gx (v, ·)

v

Use g to translate • into a map • : T X ’ T X.

Recall that, if

“σ = {(x, y, dx f, ’dy f ) | (x, y) ∈ X — X}

•

is the graph of a di¬eomorphism • : T — X ’ T — X, then • is the symplectomor-

phism generated by f . In this case, •(x, ξ) = (y, ·) if and only if ξ = dx f and

· = ’dy f . We need to show that, given (x, v) ∈ T X, the unique solution of

gx (v) = ξi = dx f (x, y)

= ’dy f (x, y)

gy (w) = ·i

for (y, ·) in terms of (x, ξ) in order to ¬nd •, or, equivalently, for (y, w) in terms

(x, v) in order to ¬nd •.

Let γ be the geodesic with initial conditions γ(0) = x and dγ (0) = v.

dt

37

3.5. PERIODIC POINTS

¢

¢ γ

v¢ E

¢

¢

¢

r

x

By the Gauss lemma (look up [15], for instance), geodesics are orthogonal to

the level sets of the distance function.

To solve the ¬rst equation of the system for y, we evaluate both sides at

d exp(x,v)

v= (0), to conclude that

dt

y = exp(x, v)(1) .

Check that dx f (v ) = 0 for vectors v ∈ Tx X orthogonal to v (that is, gx (v, v ) = 0);

this is a consequence of f (x, y) being the square of the arc-length of a minimizing

geodesic, and it su¬ces to check locally.

The vector w is obtained from the second equation of the system. Com-

pute ’dy f ( d exp(x,v) (1)). Then evaluate ’dy f at vectors w ∈ Ty X orthogonal to

dt

d exp(x,v)

(1); this pairing is again 0 because f (x, y) is the /square of the) arc-length

dt

of a minimizing geodesic. Conclude, using the nondegeneracy of g, that

d exp(x, v)

w= (1) .

dt

For both steps above, recall that, given a function f : X ’ R and a tangent

d

vector v ∈ Tx X, we have dx f (v) = [f (exp(x, v)(u))] u=0 .

du

In summary, the symplectomorphism • corresponds to the map

’’ T X

•: TX

’’ (γ(1), dγ (1)) ,

(x, v) dt

which is called the geodesic ¬‚ow on X.

3.5 Periodic Points

Let X be an n-dimensional manifold. Let M = T — X be its cotangent bundle with

canonical symplectic form ω.

38 LECTURE 3. GENERATING FUNCTIONS

Suppose that we are given a smooth function f : X — X ’ R which gener-

ates a symplectomorphism • : M ’ M , •(x, dx f ) = (y, ’dy f ), by the recipe of

Section sec:method.

What are the ¬xed points of •?

De¬ne ψ : X ’ R by ψ(x) = f (x, x).

Proposition 3.5 There is a one-to-one correspondence between the ¬xed points

of • and the critical points of ψ.

Proof. At x0 ∈ X, dx0 ψ = (dx f + dy f )|(x,y)=(x0 ,x0 ) . Let ξ = dx f |(x,y)=(x0 ,x0 ) .

x0 is a critical point of ψ ⇐’ dx0 ψ = 0 ⇐’ dy f |(x,y)=(x0,x0 ) = ’ξ .

Hence, the point in “σ corresponding to (x, y) = (x0 , x0 ) is (x0 , x0 , ξ, ξ). But “σ

f f

is the graph of •, so •(x0 , ξ) = (x0 , ξ) is a ¬xed point. This argument also works

backwards.

Consider the iterates of •,

•(N ) = • —¦ • —¦ . . . —¦ • : M ’ M , N = 1, 2, . . . ,

N

each of which is a symplectomorphism of M . According to the previous proposition,

if •(N ) : M ’ M is generated by f (N ) , then there is a one-to-one correspondence

critical points of

¬xed points of •(N ) ←’ (N )

: X ’ R , ψ(N ) (x) = f (N ) (x, x)

ψ

Knowing that • is generated by f , does •(2) have a generating function? The

answer is a partial yes:

Fix x, y ∈ X. De¬ne a map

’’ R

X

’’ f (x, z) + f (z, y) .

z

Suppose that this map has a unique critical point z0 , and that z0 is nondegenerate.

Let

f (2) (x, y) := f (x, z0 ) + f (z0 , y) .

Proposition 3.6 The function f (2) : X — X ’ R is smooth and is a generating

function for •(2) .

39

3.6. BILLIARDS

Proof. The point z0 is given implicitly by dy f (x, z0 ) + dx f (z0 , y) = 0. The non-

degeneracy condition is

‚ ‚f ‚f

det (x, z) + (z, y) =0.

‚zi ‚yj ‚xj

By the implicit function theorem, z0 = z0 (x, y) is smooth.

As for the second assertion, f (2) (x, y) is a generating function for •(2) if and

only if

•(2) (x, dx f (2) ) = (y, ’dy f (2) )

(assuming that, for each ξ ∈ Tx X, there is a unique y ∈ X for which dx f (2) = ξ).

—

Since • is generated by f , and z0 is critical, we obtain,

= •(•(x, dx f (2) (x, y)) = •(z0 , ’dy f (x, z0 ))

•(2) (x, dx f (2) (x, y))

=dx f (x,z0 )

= (y, ’dy f (z0 , y) ) .

= •(z0 , dx f (z0 , y))

=’dy f (2) (x,y)

Exercise 13

What is a generating function for •(3) ?

Hint: Suppose that the function

X —X ’’ R

(z, u) ’’ f (x, z) + f (z, u) + f (u, y)

has a unique critical point (z0 , u0 ), and that it is a nondegenerate critical point.

Let ψ (3) (x, y) = f (x, z0 ) + f (z0 , u0 ) + f (u0 , y).

3.6 Billiards

Let χ : R ’ R2 be a smooth plane curve which is 1-periodic, i.e., χ(s + 1) = χ(s),

and parametrized by arc-length, i.e., dχ = 1. Assume that the region Y enclosed

ds

by χ is convex, i.e., for any s ∈ R, the tangent line {χ(s) + t dχ | t ∈ R} intersects

ds

X := ‚Y (= the image of χ) at only the point χ(s).

40 LECTURE 3. GENERATING FUNCTIONS

'

X = ‚Y

r χ(s)

Suppose that we throw a ball into Y rolling with constant velocity and bounc-

ing o¬ the boundary with the usual law of re¬‚ection. This determines a map

• : R/Z — (’1, 1) ’’ R/Z — (’1, 1)

(x, v) ’’ (y, w)

by the rule

when the ball bounces o¬ x with angle θ = arccos v, it will next collide with y and

bounce o¬ with angle ν = arccos w.

4

4

%

4

4

4

xr

˜

˜

˜

˜

˜

j ˜

˜ 4

˜ 4

B

˜ 4

˜4

r

y

Let f : R/Z — R/Z ’ R be de¬ned by f (x, y) = ’|x ’ y|; f is smooth o¬ the

diagonal. Use χ to identify R/Z with the image curve X.

´ 41

3.7. POINCARE RECURRENCE

Suppose that •(x, v) = (y, w), i.e., (x, v) and (y, w) are successive points on

the orbit described by the ball. Then

± df x’y

=’ projected onto Tx X = v

dx

|x ’ y|

df

y’x

=’ = ’w

projected onto Ty X

|x ’ y|

dy

or, equivalently,

±d y ’ x dχ

·

ds f (χ(s), y) = = cos θ = v

|x ’ y| ds

d

x ’ y dχ

· = ’ cos ν = ’w .

f (x, χ(s)) =

|x ’ y| ds

ds

We conclude that f is a generating function for •. Similar approaches work

for higher dimensional billiards problems.

Periodic points are obtained by ¬nding critical points of

X —...—X ’’ R , N >1

N

(x1 , . . . , xN ) ’’ f (x1 , x2 ) + f (x2 , x3 ) + . . . + f (xN ’1 , xN ) + f (xN , x1 )

= |x1 ’ x2 | + . . . + |xN ’1 ’ xN | + |xN ’ x1 | ,

that is, by ¬nding the N -sided (generalized) polygons inscribed in X of critical

perimeter.

Notice that

R/Z — (’1, 1) {(x, v) | x ∈ X, v ∈ Tx X, |v| < 1} A

is the open unit tangent ball bundle of a circle X, that is, an open annulus A. The

map • : A ’ A is area-preserving.

3.7 Poincar´ Recurrence

e

Theorem 3.7 (Poincar´ Recurrence) Suppose that • : A ’ A is an area-

e

preserving di¬eomorphism of a ¬nite-area manifold A. Let p ∈ A, and let U be

a neighborhood of p. Then there is q ∈ U and a positive integer N such that

•(N ) (q) ∈ U.

Proof. Let U0 = U, U1 = •(U), U2 = •(2) (U), . . .. If all of these sets were disjoint,

then, since Area (Ui ) = Area (U) > 0 for all i, we would have

Area A ≥ Area (U0 ∪ U1 ∪ U2 ∪ . . .) = Area (Ui ) = ∞ .

i

42 LECTURE 3. GENERATING FUNCTIONS

To avoid this contradiction we must have •(k) (U) © •(l) (U) = … for some k > l,

which implies •(k’l) (U) © U = ….

Hence, eternal return applies to billiards...

Remark. Theorem 3.7 clearly generalizes to volume-preserving di¬eomorphisms

™¦

in higher dimensions.

Theorem 3.8 (Poincar´™s Last Geometric Theorem) Suppose • : A ’ A

e

is an area-preserving di¬eomorphism of the closed annulus A = R/Z — [’1, 1]

which preserves the two components of the boundary, and twists them in opposite

directions. Then • has at least two ¬xed points.

This theorem was proved in 1925 by Birkho¬, and hence is also called the

Poincar´-Birkho¬ theorem. It has important applications to dynamical sys-

e

tems and celestial mechanics. The Arnold conjecture (1966) on the existence of

¬xed points for symplectomorphisms of compact manifolds (see Section 3.9) may

be regarded as a generalization of the Poincar´-Birkho¬ theorem. This conjecture

e

has motivated a signi¬cant amount of recent research involving a more general

notion of generating function; see, for instance, [18, 20].

3.8 Group of Symplectomorphisms

The symplectomorphisms of a symplectic manifold (M, ω) form the group

Sympl(M, ω) = {f : M ’’ M | f — ω = ω} .

“ What is Tid (Sympl(M, ω))?

(What is the “Lie algebra” of the group of symplectomorphisms?)

“ What does a neighborhood of id in Sympl(M, ω) look like?

We will use notions from the C 1 -topology. Let X and Y be manifolds.

De¬nition 3.9 A sequence of maps fi : X ’ Y converges in the C 0 -topology

to f : X ’ Y if and only if fi converges uniformly on compact sets.

De¬nition 3.10 A sequence of C 1 maps fi : X ’ Y converges in the C 1 -

topology to f : X ’ Y if and only if it and the sequence of derivatives dfi :

T X ’ T Y converge uniformly on compact sets.

Let (M, ω) be a compact symplectic manifold and f ∈ Sympl(M, ω). Then

both Graph f and the diagonal ∆ = Graph id are lagrangian subspaces of (M —

M, pr— ω ’ pr— ω), where pri : M — M ’ M , i = 1, 2, are the projections to each

1 2

factor.

43

3.8. GROUP OF SYMPLECTOMORPHISMS

By the Weinstein tubular neighborhood theorem, there exists a neighbor-

hood U of ∆ ( M ) in (M — M, pr— ω ’ pr— ω) which is symplectomorphic to a

1 2

neighborhood U0 of M in (T — M, ω0 ). Let • : U ’ U0 be the symplectomorphism.

Suppose that f is su¬ciently C 1 -close to id, i.e., f is in some su¬ciently

small neighborhood of id in the C 1 -topology. Then:

1. We can assume that Graph f ⊆ U.

j:M ’U

Let be the embedding as Graph f ,

i:M ’U be the embedding as Graph id = ∆ .

2. The map j is su¬ciently C 1 -close to i.

U0 ⊆ T — M , so the above j and i induce

3. By the Weinstein theorem, U

j0 : M ’ U 0 embedding, where j0 = • —¦ j ,

i0 : M ’ U 0 embedding as 0-section .

Hence, we have

• •

E U0 E U0

U U

d

s

d

s

d d

d d

and

id i0 jd j0

d d

M M

where i(p) = (p, p), i0 (p) = (p, 0), j(p) = (p, f (p)) and j0 (p) = •(p, f (p)) for

p ∈ M.

4. The map j0 is su¬ciently C 1 -close to i0 . Therefore, the image set j0 (M )

—

intersects each Tp M at one point µp depending smoothly on p.

5. The image of j0 is the image of a smooth section µ : M ’ T — M , that is, a

1-form µ = j0 —¦ (π —¦ j0 )’1 .

—

{(p, µp ) | p ∈ M, µp ∈ Tp M }.

We conclude that Graph f

Exercise 14

Vice-versa: show that, if µ is a 1-form su¬ciently C 1 -close to the zero 1-form,

then there is a di¬eomorphism f : M ’ M such that

—

{(p, µp ) | p ∈ M, µp ∈ Tp M } Graph f .

By Section 2.4, we have

Graph f is lagrangian ⇐’ µ is closed .

44 LECTURE 3. GENERATING FUNCTIONS

Conclusion. A small C 1 -neighborhood of id in Sympl(M, ω) is homeomorphic to

a C 1 -neighborhood of zero in the vector space of closed 1-forms on M . So:

{µ ∈ „¦1 (M ) | dµ = 0} .

Tid (Sympl(M, ω))

In particular, Tid (Sympl(M, ω)) contains the space of exact 1-forms

{µ = dh | h ∈ C ∞ (M )} C ∞ (M )/ locally constant functions .

3.9 Fixed Points of Symplectomorphisms

1

Theorem 3.11 Let (M, ω) be a compact symplectic manifold with H deRham (M ) =

0. Then any symplectomorphism of M which is su¬ciently C 1 -close to the identity

has at least two ¬xed points.

Proof. Suppose that f ∈ Sympl(M, ω) is su¬ciently C 1 -close to id. Then the

graph of f corresponds to a closed 1-form µ on M .

dµ = 0

=’ µ = dh for some h ∈ C ∞ (M ) .

1

HdeRham (M ) = 0

If M is compact, then h has at least 2 critical points.

Fixed points of f = critical points of h

Graph f © ∆ = {p : µp = dhp = 0} .

Lagrangian intersection problem:

A submanifold Y of M is C 1 -close to X when there is a di¬eomorphism

X ’ Y which is, as a map into M , C 1 -close to the inclusion X ’ M .

Theorem 3.12 Let (M, ω) be a symplectic manifold. Suppose that X is a com-

1

pact lagrangian submanifold of M with HdeRham (X) = 0. Then every lagrangian

submanifold of M which is C 1 -close to X intersects X in at least two points.

Proof. Exercise.

Arnold conjecture:

Let (M, ω) be a compact symplectic manifold, and f : M ’ M a symplectomor-

phism which is “exactly homotopic to the identity” (see below). Then

#{¬xed points of f } ≥ minimal # of critical points

a smooth function of M can have .

45

3.9. FIXED POINTS OF SYMPLECTOMORPHISMS

Together with Morse theory,2 we obtain3

#{nondegenerate ¬xed points of f } ≥ minimal # of critical points

a Morse function of M can have

2n

dim H i (M ) .

≥

i=0

The Arnold conjecture was proved by Conley-Zehnder, Floer, Hofer-Salamon,

Ono, Futaya-Ono, Lin-Tian using Floer homology (which is an ∞-dimensional

analogue of Morse theory). There are open conjectures for sharper bounds on the

number of ¬xed points.

Meaning of “f is exactly homotopic to the identity:”

Suppose that ht : M ’ R is a smooth family of functions which is 1-periodic,

i.e., ht = ht+1 . Let ρ : M —R ’ M be the isotopy generated by the time-dependent

vector ¬eld vt de¬ned by ω(vt , ·) = dht . Then “f being exactly homotopic to the

identity” means f = ρ1 for some such ht .

In other words, f is exactly homotopic to the identity when f is the

time-1 map of an isotopy generated by some smooth time-dependent 1-periodic

hamiltonian function.

There is a one-to-one correspondence

{ ¬xed points of f } ←’ { period-1 orbits of ρ : M — R ’ M }

because f (p) = p if and only if {ρ(t, p) , t ∈ [0, 1]} is a closed orbit.

Proof of the Arnold conjecture in the case when h : M ’ R is independent of t:

⇐’

p is a critical point of h dhp = 0

⇐’ vp = 0

ρ(t, p) = p , ∀t ∈ R

=’

⇐’ p is a ¬xed point of ρ1 .

Exercise 15

Compute these estimates for the number of ¬xed points on some compact

symplectic manifolds (for instance, S 2 , S 2 — S 2 and T 2 = S 1 — S 1 ).

2A Morse function on M is a function h : M ’ R whose critical points (i.e., points p

where the di¬erential vanishes: dhp = 0) are all nondegenerate (i.e., the hessian at those points

‚2 h

is nonsingular: det = 0).

‚xi ‚xj p

3A ¬xed point p of f : M ’ M is nondegenerate if dfp : Tp M ’ Tp M is nonsingular.

Lecture 4

Hamiltonian Fields

To any real function on a symplectic manifold, a symplectic geometer associates a

vector ¬eld whose ¬‚ow preserves the symplectic form and the given function. The

vector ¬eld is called the hamiltonian vector ¬eld of that (hamiltonian) function.

The concept of a moment map is a generalization of that of a hamiltonian

function, and was introduced by Souriau [40] under the french name application

moment (besides the more standard english translation to moment map, the alter-

native momentum map is also used). The notion of a moment map associated to

a group action on a symplectic manifold formalizes the Noether principle, which

states that to every symmetry (such as a group action) in a mechanical system,

there corresponds a conserved quantity.

4.1 Hamiltonian and Symplectic Vector Fields

Let (M, ω) be a symplectic manifold and let H : M ’ R be a smooth function. Its

di¬erential dH is a 1-form. By nondegeneracy, there is a unique vector ¬eld XH

on M such that ±XH ω = dH. Integrate XH . Supposing that M is compact, or at

least that XH is complete, let ρt : M ’ M , t ∈ R, be the one-parameter family of

di¬eomorphisms generated by XH :

±

ρ0 = idM

dρt

—¦ ρ’1 = XH .

t

dt

47

48 LECTURE 4. HAMILTONIAN FIELDS

Claim. Each di¬eomorphism ρt preserves ω, i.e., ρ— ω = ω, ∀t.

t

d—

= ρ— LXH ω = ρ— (d ±XH ω +±XH dω ) = 0.

Proof. We have dt ρt ω t t

0

dH

Therefore, every function on (M, ω) gives a family of symplectomorphisms.

Notice how the proof involved both the nondegeneracy and the closedness of ω.

De¬nition 4.1 A vector ¬eld XH as above is called the hamiltonian vector

¬eld with hamiltonian function H.

Example. The height function H(θ, h) = h on the sphere (M, ω) = (S 2 , dθ § dh)

has

‚

±XH (dθ § dh) = dh ⇐’ XH = .

‚θ

Thus, ρt (θ, h) = (θ + t, h), which is rotation about the vertical axis; the height

™¦

function H is preserved by this motion.

Exercise 16

Let X be a vector ¬eld on an abstract manifold W . There is a unique vector

¬eld X on the cotangent bundle T — W , whose ¬‚ow is the lift of the ¬‚ow of X.

Let ± be the tautological 1-form on T — W and let ω = ’d± be the canonical

symplectic form on T — W . Show that X is a hamiltonian vector ¬eld with

hamiltonian function H := ±X ±.

Remark. If XH is hamiltonian, then

LXH H = ±XH dH = ±XH ±XH ω = 0 .

Therefore, hamiltonian vector ¬elds preserve their hamiltonian functions, and each

integral curve {ρt (x) | t ∈ R} of XH must be contained in a level set of H:

H(x) = (ρ— H)(x) = H(ρt (x)) , ∀t .

t

™¦

De¬nition 4.2 A vector ¬eld X on M preserving ω (i.e., such that LX ω = 0) is

called a symplectic vector ¬eld.

⇐’

X is symplectic ±X ω is closed ,

⇐’

X is hamiltonian ±X ω is exact .

Locally, on every contractible open set, every symplectic vector ¬eld is hamil-

1

tonian. If HdeRham(M ) = 0, then globally every symplectic vector ¬eld is hamil-

1

tonian. In general, HdeRham (M ) measures the obstruction for symplectic vector

¬elds to be hamiltonian.

49

4.2. HAMILTON EQUATIONS

Example. On the 2-torus (M, ω) = (T 2 , dθ1 § dθ2 ), the vector ¬elds X1 = ‚θ1

‚

‚

™¦

and X2 = ‚θ2 are symplectic but not hamiltonian.

To summarize, vector ¬elds on a symplectic manifold (M, ω) which preserve

ω are called symplectic. The following are equivalent:

• X is a symplectic vector ¬eld;

• the ¬‚ow ρt of X preserves ω, i.e., ρ— ω = ω, for all t;

t

• LX ω = 0;

• ±X ω is closed.

A hamiltonian vector ¬eld is a vector ¬eld X for which

• ±X ω is exact,

i.e., ±X ω = dH for some H ∈ C ∞ (M ). A primitive H of ±X ω is then called a

hamiltonian function of X.

4.2 Hamilton Equations

Consider euclidean space R2n with coordinates (q1 , . . . , qn , p1 , . . . , pn ) and ω0 =

dqj § dpj . The curve ρt = (q(t), p(t)) is an integral curve for XH exactly if

±

dqi (t) = ‚H

dt

‚pi

(Hamilton equations)

dp

‚H

i

(t) = ’

dt ‚qi

n

‚H ‚ ‚H ‚

’

Indeed, let XH = . Then,

‚pi ‚qi ‚qi ‚pi

i=1

n n

±XH (dqj § dpj ) = [(±XH dqj ) § dpj ’ dqj § (±XH dpj )]

± XH ω =

j=1 j=1

n

‚H ‚H

= ‚pj dpj + ‚qj dqj = dH .

j=1

Remark. The gradient vector ¬eld of H relative to the euclidean metric is

n

‚H ‚ ‚H ‚

H := + .

‚qi ‚qi ‚pi ‚pi

i=1

50 LECTURE 4. HAMILTONIAN FIELDS

If J is the standard (almost) complex structure1 so that J( ‚qi ) =

‚ ‚

and

‚pi

‚ ‚

J( ‚pi ) = ’ ‚qi , we have JXH = H. ™¦

The case where n = 3 has a simple physical illustration. Newton™s second

law states that a particle of mass m moving in con¬guration space R3 with

coordinates q = (q1 , q2 , q3 ) under a potential V (q) moves along a curve q(t) such

that

d2 q

m 2 = ’ V (q) .

dt

Introduce the momenta p i = m dqi for i = 1, 2, 3, and energy function H(p, q) =

dt

1

|p|2 +V (q). Let R6 = T — R3 be the corresponding phase space, with coordinates

2m

(q1 , q2 , q3 , p1 , p2 , p3 ). Newton™s second law in R3 is equivalent to the Hamilton

equations in R6 :

± dq 1 ‚H

i

= pi =

dt m ‚pi

d 2 qi

dpi ‚V ‚H

=m 2 =’ =’ .

dt dt ‚qi ‚qi

The energy H is conserved by the physical motion.

4.3 Brackets

Vector ¬elds are di¬erential operators on functions: if X is a vector ¬eld and

f ∈ C ∞ (X), df being the corresponding 1-form, then

X · f := df (X) = LX f .

Given two vector ¬elds X, Y , there is a unique vector ¬eld W such that

LW f = LX (LY f ) ’ LY (LX f ) .

The vector ¬eld W is called the Lie bracket of the vector ¬elds X and Y and

denoted W = [X, Y ], since LW = [LX , LY ] is the commutator.

Exercise 17

Check that, for any form ±,

±[X,Y ] ± = LX ±Y ± ’ ±Y LX ± = [LX , ±Y ]± .

Since each side is an anti-derivation with respect to the wedge product, it

su¬ces to check this formula on local generators of the exterior algebra of

forms, namely functions and exact 1-forms.

1 An

almost complex structure on a manifold M is a vector bundle morphism J : T M ’

T M such that J 2 = ’Id.

51

4.3. BRACKETS

Proposition 4.3 If X and Y are symplectic vector ¬elds on a symplectic manifold

(M, ω), then [X, Y ] is hamiltonian with hamiltonian function ω(Y, X).

Proof.

= L X ±Y ω ’ ± Y L X ω

±[X,Y ] ω

= d±X ±Y ω + ±X d±Y ω ’±Y d±X ω ’±Y ±X dω

0

0 0

= d(ω(Y, X)) .

A (real) Lie algebra is a (real) vector space g together with a Lie bracket

[·, ·], i.e., a bilinear map [·, ·] : g — g ’ g satisfying:

(a) [x, y] = ’[y, x] , ∀x, y ∈ g , (antisymmetry)

∀x, y, z ∈ g .

(b) [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 , (Jacobi identity)

Let

χ(M ) = { vector ¬elds on M }

sympl

(M ) = { symplectic vector ¬elds on M }

χ

ham

(M ) = { hamiltonian vector ¬elds on M } .

χ

The inclusions (χham (M ), [·, ·]) ⊆ (χsympl (M ), [·, ·]) ⊆ (χ(M ), [·, ·]) are inclusions

of Lie algebras.

De¬nition 4.4 The Poisson bracket of two functions f, g ∈ C ∞ (M ; R) is

{f, g} := ω(Xf , Xg ) .

We have X{f,g} = ’[Xf , Xg ] because Xω(Xf ,Xg ) = [Xg , Xf ].

Theorem 4.5 The bracket {·, ·} satis¬es the Jacobi identity, i.e.,

{f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0 .

Proof. Exercise.

De¬nition 4.6 A Poisson algebra (P, {·, ·}) is a commutative associative alge-

bra P with a Lie bracket {·, ·} satisfying the Leibniz rule:

{f, gh} = {f, g}h + g{f, h} .

Exercise 18

Check that the Poisson bracket {·, ·} de¬ned above satis¬es the Leibniz rule.

52 LECTURE 4. HAMILTONIAN FIELDS

We conclude that, if (M, ω) is a symplectic manifold, then (C ∞ (M ), {·, ·}) is

a Poisson algebra. Furthermore, we have a Lie algebra anti-homomorphism

C ∞ (M ) ’’ χ(M )

H ’’ XH

{·, ·} ’[·, ·] .

Exercise 19

Let G be a Lie group, g its Lie algebra and g— the dual vector space of g.

(a) Let g X # be the vector ¬eld generated by X ∈ g for the adjoint repre-

sentation of G on g. Show that

#

g

XY = [X, Y ] ∀Y ∈g.

(b) Let X # be the vector ¬eld generated by X ∈ g for the coadjoint repre-

sentation of G on g— . Show that

#

Xξ , Y = ξ, [Y, X] ∀Y ∈g.

(c) For any ξ ∈ g— , de¬ne a skew-symmetric bilinear form on g by

ωξ (X, Y ) := ξ, [X, Y ] .

Show that the kernel of ωξ is the Lie algebra gξ of the stabilizer of ξ for

the coadjoint representation.

(d) Show that ωξ de¬nes a nondegenerate 2-form on the tangent space at ξ

to the coadjoint orbit through ξ.

(e) Show that ωξ de¬nes a closed 2-form on the orbit of ξ in g — .

Hint: The tangent space to the orbit being generated by the vector ¬elds

X # , this is a consequence of the Jacobi identity in g.

This canonical symplectic form on the coadjoint orbits is also known

as the Lie-Poisson or Kostant-Kirillov symplectic structure.

(f) The Lie algebra structure of g de¬nes a canonical Poisson structure on

g— :

{f, g}(ξ) := ξ, [dfξ , dgξ ]

for f, g ∈ C ∞ (g— ) and ξ ∈ g— . Notice that dfξ : Tξ g— g— ’ R is

identi¬ed with an element of g g—— .

Check that {·, ·} satis¬es the Leibniz rule:

{f, gh} = g{f, h} + h{f, g} .

Example. For the prototype (R2n , ω0 ), where ω0 = dxi § dyi , we have

‚ ‚

Xx i = ’ and X yi =

‚yi ‚xi

so that

1 if i = j

{xi , xj } = {yi , yj } = 0 {xi , yj } = ∀i, j .

and

0 if i = j

53

4.4. INTEGRABLE SYSTEMS

For arbitrary functions f, g ∈ C ∞ (M ) we have hamiltonian vector ¬elds

n

‚f ‚ ‚f ‚

’

Xf = ,

‚yi ‚xi ‚xi ‚yi

i=1

and the classical Poisson bracket

n