<<

. 2
( 5)



>>

„¦0 : W0 ’’ U —
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


  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)


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

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

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