<< стр. 2(всего 5)СОДЕРЖАНИЕ >>
в„¦0 : W0 в€’в†’ U в€—
0
W0 Г— U в€’в†’ R
give isomorphisms
в„¦
в„¦1 : W1 в€’в†’ U в€— .
1
W1 Г— U в€’в†’ R

Consider the diagram
в„¦
Uв€—
0
в€’в†’
W0
Bв†“ в†“ id
в„¦
Uв€—
1
в€’в†’
W1

where the linear map B satisп¬Ѓes в„¦1 в—¦ B = в„¦0 , i.e., в„¦0 (П‰0 , u) = в„¦1 (BП‰0 , u), в€ЂП‰0 в€€
W0 , в€Ђu в€€ U . Extend B to the rest of V by setting it to be the identity on U :

L := IdU вЉ• B : U вЉ• W0 в€’в†’ U вЉ• W1 .
25
2.7. LAGRANGIAN NEIGHBORHOOD THEOREM

Finally, we check that Lв€— в„¦1 = в„¦0 :

(Lв€— в„¦1 )(u вЉ• w0 , u вЉ• w0 ) в„¦1 (u вЉ• BП‰0 , u вЉ• BП‰0 )
=
= в„¦1 (u, BП‰0 ) + в„¦1 (BП‰0 , u )
= в„¦0 (u, П‰0 ) + в„¦0 (П‰0 , u )
в„¦0 (u вЉ• w0 , u вЉ• w0 ) .
=

2.7 Lagrangian Neighborhood Theorem
Theorem 2.11 (Weinstein Lagrangian Neighborhood Theorem ) Let
M be a 2n-dimensional manifold, X an n-dimensional submanifold, i : X в†’ M the
inclusion map, and П‰0 and П‰1 symplectic forms on M such that iв€— П‰0 = iв€— П‰1 = 0,
i.e., X is a lagrangian submanifold of both (M, П‰0 ) and (M, П‰1 ). Then there exist
neighborhoods U0 and U1 of X in M and a diп¬Ђeomorphism П• : U0 в†’ U1 such that
П• E U1
U0
d
s В

d В
П• в€— П‰1 = П‰ 0 .
d В  commutes and
id В i
d В
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 в€€ M , gp (В·, В·) is a positive-deп¬Ѓnite
inner product. Fix p в€€ X, and let V = Tp M , U = Tp X and W = U вЉҐ = ortho-
complement of U in V relative to gp (В·, В·).
Since iв€— П‰0 = iв€— П‰1 = 0, U is a lagrangian subspace of both (V, П‰0 |p ) and
(V, П‰1 |p ). By symplectic linear algebra, we canonically get from U вЉҐ a linear iso-
morphism Lp : Tp M в†’ Tp M , such that Lp |Tp X = IdTp X and Lв€— П‰1 |p = П‰0 |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 в†’ M with h|X = idX and dhp = Lp for p в€€ X. Hence,
at any p в€€ X,
(hв€— П‰1 )p = (dhp )в€— П‰1 |p = Lв€— П‰1 |p = П‰0 |p .
p

Applying the Moser local theorem (Theorem 1.11) to П‰0 and hв€— П‰1 , we п¬Ѓnd a
neighborhood U0 of X and an embedding f : U0 в†’ N such that f |X = idX and
f в€— (hв€— П‰1 ) = П‰0 on Uo . Set П• = h в—¦ 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 в‰Ґ n, i : X в†’ M the inclusion
map, and П‰0 and П‰1 symplectic forms on M , such that iв€— П‰0 = iв€— П‰1 and X is
coisotropic for both (M, П‰0 ) and (M, П‰1 ). Then there exist neighborhoods U0 and
U1 of X in M and a diп¬Ђeomorphism П• : U0 в†’ U1 such that
П• E U1
U0
d
s В

d В
П• в€— П‰1 = П‰ 0 .
d В  commutes and
id В i
d В
X

2.8 Weinstein Tubular Neighborhood Theorem
Let (V, в„¦) be a symplectic linear space, and let U be a lagrangian subspace.
Claim. There is a canonical nondegenerate bilinear pairing в„¦ : V /U Г— U в†’ R.
Proof. Deп¬Ѓne в„¦ ([v], u) = в„¦(v, u) where [v] is the equivalence class of v in V /U .

Consequently, we get that в„¦ : V /U в†’ U в€— deп¬Ѓned by в„¦ ([v]) = в„¦ ([v], В·) is
an isomorphism, so that V /U U в€— are canonically identiп¬Ѓed.
In particular, if (M, П‰) is a symplectic manifold, and X is a lagrangian sub-
manifold, then Tx X is a lagrangian subspace of (Tx M, П‰x ) for each x в€€ X. The
space Nx X := Tx M/TxX is called the normal space of X at x. Since we have a
в€—
canonical identiп¬Ѓcation Nx X Tx X, we get:

Proposition 2.13 The vector bundles N X and T в€— X are canonically identiп¬Ѓed.

Putting this observation together with the lagrangian neighborhood theorem,
we arrive at:

Theorem 2.14 (Weinstein Tubular Neighborhood Theorem) Let (M, П‰)
be a symplectic manifold, X a lagrangian submanifold, П‰0 canonical symplectic
form on T в€— X, i0 : X в†’ T в€— X the lagrangian embedding as the zero section, and
i : X в†’ M lagrangian embedding given by inclusion. Then there are neighborhoods
U0 of X in T в€— X, U of X in M , and a diп¬Ђeomorphism П• : U0 в†’ U such that
П• EU
U0
d
s В

d В
П• в€— П‰ = П‰0 .
d В  commutes and
i0 d В i
d В
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 в€— X, we can п¬Ѓnd a neighborhood N0 of X in T в€— X, a neigh-
1. Since N X
borhood N of X in M , and a diп¬Ђeomorphism П€ : N0 в†’ N such that

П€ EN
N0
d
s В

d В
d В  commutes .
i0 d В i
d В
X

П‰0 = canonical form on T в€— X
symplectic forms on N0 .
Let
П‰1 = П€ в€— П‰
X is lagrangian for both П‰0 and П‰1 .
2. There exist neighborhoods U0 and U1 of X in N0 and a diп¬Ђeomorphism
Оё : U0 в†’ U1 such that
Оё E U1
U0
d
s В

d В  Оё в€— П‰1 = П‰ 0 .
d В  commutes and
i0 d В  i0
d В
X

Take П• = П€ в—¦ Оё and U = П•(U0 ). Check that П•в€— П‰ = Оёв€— П€ в€— П‰ = П‰0 .
П‰1

Remark. Theorem 2.14 classiп¬Ѓes 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 classiп¬Ѓcation 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, П‰) is a closed embedding i : X в†’ M such that iв€— П‰ = 0. Weinstein
showed that neighbourhood equivalence of isotropic embeddings is in one-to-one
correspondence with isomorphism classes of symplectic vector bundles.
The classiп¬Ѓcation of coisotropic embeddings is due to Gotay . A coisotro-
pic embedding of a manifold X carrying a closed 2-form О± of constant rank into
a symplectic manifold (M, П‰) is an embedding i : X в†’ M such that iв€— П‰ = О±
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 О± of constant rank on X, i.e., Ep is the kernel
of О±p at p в€€ X. Gotay showed that then E в€— 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
в™¦
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 , П‰1 ) and (M2 , П‰2 ) be two 2n-dimensional symplectic manifolds. Given
a diп¬Ђeomorphism П• : M1 в€’в†’ M2 , when is it a symplectomorphism? (I.e., when is
П•в€— П‰2 = П‰1 ?) Consider the diagram of projection maps
M1 Г— M 2
(p1 , p2 ) (p1 , p2 )
В  d
В  d pr2
pr1
В  d
В  d
В
d
c c
p1 M1 M2 p2
Then П‰ = (pr1 )в€— П‰1 + (pr2 )в€— П‰2 is a 2-form on M1 Г— M2 which is closed,

dП‰ = (pr1 )в€— dП‰1 + (pr2 )в€— dП‰2 = 0 ,
0 0

and symplectic,
n n
2n
П‰ 2n = (pr1 )в€— П‰1 в€§ (pr2 )в€— П‰2 =0.
n

More generally, if О»1 , О»2 в€€ R\{0}, then О»1 (pr1 )в€— П‰1 +О»2 (pr2 )в€— П‰2 is also a symplectic
form on M1 Г— M2 . Take О»1 = 1, О»2 = в€’1 to obtain the twisted product form
on M1 Г— M2 :
П‰ = (pr1 )в€— П‰1 в€’ (pr2 )в€— П‰2 .
The graph of a diп¬Ђeomorphism П• : M1 в€’в†’ M2 is the 2n-dimensional sub-
manifold of M1 Г— M2 :

О“П• := Graph П• = {(p, П•(p)) | p в€€ M1 } .

The submanifold О“П• is an embedded image of M1 in M1 Г— M2 , the embedding
being the map
Оі : M1 в€’в†’ M1 Г— M2
p в€’в†’ (p, П•(p)) .
29
2.9. SYMPLECTOMORPHISMS AS LAGRANGIANS

Proposition 2.15 A diп¬Ђeomorphism П• is a symplectomorphism if and only if О“ П•
is a lagrangian submanifold of (M1 Г— M2 , П‰).

Proof. The graph О“П• is lagrangian if and only if Оі в€— П‰ = 0.

Оів€—П‰ = Оі в€— prв€— П‰1 в€’ Оі в€— prв€— П‰2
1 2
= (pr1 в—¦ Оі) П‰1 в€’ (pr2 в—¦ Оі)в€— П‰2 .
в€—

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

Оів€—П‰ = 0 П• в€— П‰2 = П‰ 1 .
в‡ђв‡’
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 п¬Ѓxed points, whose п¬Ѓrst
instance is the PoincarВґ-Birkhoп¬Ђ theorem.
e

3.1 Constructing Symplectomorphisms
Let X1 , X2 be n-dimensional manifolds, with cotangent bundles M1 = T в€— X1 ,
M2 = T в€— X2 , tautological 1-forms О±1 , О±2 , and canonical 2-forms П‰1 , П‰2 .
Under the natural identiп¬Ѓcation
M 1 Г— M 2 = T в€— X1 Г— T в€— X2 T в€— (X1 Г— X2 ) ,
the tautological 1-form on T в€— (X1 Г— X2 ) is
О± = (pr1 )в€— О±1 + (pr2 )в€— О±2 ,
where pri : M1 Г— M2 в†’ Mi , i = 1, 2 are the two projections. The canonical 2-form
on T в€— (X1 Г— X2 ) is
П‰ = в€’dО± = в€’dprв€— О±1 в€’ dprв€— О±2 = prв€— П‰1 + prв€— П‰2 .
1 2 1 2

In order to describe the twisted form П‰ = prв€— П‰1 в€’ prв€— П‰2 , we deп¬Ѓne an involution
1 2
of M2 = T в€— X2 by
M2 в€’в†’ M2
Пѓ2 :
(x2 , Оѕ2 ) в€’в†’ (x2 , в€’Оѕ2 )
в€—
which yields Пѓ2 О±2 = в€’О±2 . Let Пѓ = idM1 Г— Пѓ2 : M1 Г— M2 в†’ M1 Г— M2 . Then
Пѓ в€— П‰ = prв€— П‰1 + prв€— П‰2 = П‰ .
1 2

31
32 LECTURE 3. GENERATING FUNCTIONS

If Y is a lagrangian submanifold of (M1 Г— M2 , П‰), then its вЂњtwistвЂќ Y Пѓ := Пѓ(Y ) is
a lagrangian submanifold of (M1 Г— M2 , П‰).
Recipe for producing symplectomorphisms M 1 = T в€— X1 в†’ M2 = T в€— X2 :

1. Start with a lagrangian submanifold Y of (M1 Г— M2 , П‰).

2. Twist it to obtain a lagrangian submanifold Y Пѓ of (M1 Г— M2 , П‰).

3. Check whether Y Пѓ is the graph of some diп¬Ђeomorphism П• : M1 в†’ M2 .

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

Let i : Y Пѓ в†’ M1 Г— M2 be the inclusion map

YПѓ
В  d
pr1 в—¦ i В  d pr2 в—¦ i
В  d
В  d
В
d
П•? E M2
M1
Step 3 amounts to checking whether pr1 в—¦ i and pr2 в—¦ i are diп¬Ђeomorphisms. If yes,
then П• := (pr2 в—¦ i) в—¦ (pr1 в—¦ i)в€’1 is a diп¬Ђeomorphism.
T в€— (X1 Г— X2 ), we
In order to obtain lagrangian submanifolds of M1 Г— M2
can use the method of generating functions.

3.2 Method of Generating Functions

For any f в€€ C в€ћ (X1 Г— X2 ), df is a closed 1-form on X1 Г— X2 . The lagrangian
submanifold generated by f is

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

в€—
:= (df )(x,y) projected to Tx X1 Г— {0},
dx f
в€—
:= (df )(x,y) projected to {0} Г— Ty X2 ,
dy f

which enables us to write

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

and
YfПѓ = {(x, y, dx f, в€’dy f ) | (x, y) в€€ X1 Г— X2 } .
33
3.2. METHOD OF GENERATING FUNCTIONS

When YfПѓ is in fact the graph of a diп¬Ђeomorphism П• : M1 в†’ M2 , we call П• the
symplectomorphism generated by f , and call f the generating function,
of П• : M1 в†’ M2 .
So when is YfПѓ the graph of a diп¬Ђeomorphism П• : M1 в†’ M2 ?
Let (U1 , x1 , . . . , xn ), (U2 , y1 , . . . , yn ) be coordinate charts for X1 and X2 ,
with associated charts (T в€— U1 , x1 , . . . , xn , Оѕ1 , . . . , Оѕn ), (T в€— U2 , y1 , . . . , yn , О·1 , . . . , О·n )
for M1 and M2 . The set

YfПѓ = {(x, y, dx f, в€’dy f ) | (x, y) в€€ X1 Г— X2 }

is the graph of П• : M1 в†’ M2 if and only if, for any (x, Оѕ) в€€ M1 and (y, О·) в€€ M2 ,
we have

П•(x, Оѕ) = (y, О·) в‡ђв‡’ Оѕ = dx f and О· = в€’dy f .

Therefore, given a point (x, Оѕ) в€€ M1 , to п¬Ѓnd its image (y, О·) = П•(x, Оѕ) we must
solve the вЂњHamiltonвЂќ equations
пЈ±
в€‚f
пЈґ Оѕi =
пЈІ (x, y) ()
в€‚xi
пЈґ О· = в€’ в€‚f (x, y)
пЈіi ()
в€‚yi

If there is a solution y = П•1 (x, Оѕ) of ( ), we may feed it to ( ) thus obtaining
О· = П•2 (x, Оѕ), so that П•(x, Оѕ) = (П•1 (x, Оѕ), П•2 (x, Оѕ)). Now by the implicit function
theorem, in order to solve ( ) locally for y in terms of x and Оѕ, we need the
condition
n
в€‚ в€‚f
det =0.
в€‚yj в€‚xi i,j=1
This is a necessary local condition for f to generate a symplectomorphism П•.
Locally this is also suп¬ѓcient, but globally there is the usual bijectivity issue.
2
Rn , and f (x, y) = в€’ |xв€’y| , the square
Example. Let X1 = U1 Rn , X2 = U2 2
of euclidean distance up to a constant.
The вЂњHamiltonвЂќ equations are
пЈ± пЈ±
в€‚f
пЈґ Оѕi пЈІ yi = xi + Оѕ i
пЈІ = y i в€’ xi
=
в€‚xi в‡ђв‡’
в€‚f
пЈґО· пЈі
пЈіi =в€’ = y i в€’ xi О·i = Оѕi
в€‚yi

The symplectomorphism generated by f is

П•(x, Оѕ) = (x + Оѕ, Оѕ) .
34 LECTURE 3. GENERATING FUNCTIONS

If we use the euclidean inner product to identify T в€— Rn with T Rn , and hence
regard П• as П• : T Rn в†’ T Rn and interpret Оѕ as the velocity vector, then the
symplectomorphism П• corresponds to free translational motion in euclidean space.

ВЁB
ВЁ
ВЁВЁ
ВЁВЁ
Оѕ Br
ВЁ x+Оѕ
ВЁВЁ
ВЁ
r
x

в™¦

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 Г— V в†’ R which is

symmetric : G(v, w) = G(w, v) , and
positive-deп¬Ѓnite : G(v, v) > 0 when v=0.

Deп¬Ѓnition 3.1 A riemannian metric on a manifold X is a function g which
assigns to each point x в€€ X a positive inner product gx on Tx X.
A riemannian metric g is smooth if for every smooth vector п¬Ѓeld v : X в†’
T X the real-valued function x в†’ gx (vx , vx ) is a smooth function on X.

Deп¬Ѓnition 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 Оі : [a, b] в†’ X is

b
dОі dОі dОі dОі
arc-length of Оі := dt , where := gОі(t) , .
dt dt dt dt
a

By changing variables in the integral, we see that the arc-length of Оі is independent
of the parametrization of Оі, i.e., if we reparametrize Оі by П„ : [a , b ] в†’ [a, b], the
new curve Оі = Оі в—¦ П„ : [a , b ] в†’ X has the same arc-length.
35
3.3. RIEMANNIAN DISTANCE

y = Оі(b)
r

Оі
E

r
x = Оі(a)

dОі
A curve Оі is called a curve of constant velocity when is indepen-
dt
dent of t. Given any curve Оі : [a, b] в†’ X (with dОі never vanishing), there is a
dt
reparametrization П„ : [a, b] в†’ [a, b] such that Оі в—¦ П„ : [a, b] в†’ X is of constant
velocity. The action of a piecewise smooth curve Оі : [a, b] в†’ X is
2
b
dОі
A(Оі) := dt .
dt
a

Exercise 12
Show that, among all curves joining two given points, Оі minimizes the action
if and only if Оі is of constant velocity and Оі minimizes arc-length.
Hint:
(a) Let П„ : [a, b] в†’ [a, b] be a smooth monotone map taking the endpoints
of [a, b] to the endpoints of [a, b]. Then
2
b dП„
dt в‰Ґ b в€’ a ,
dt
a
dП„
with equality holding if and only if = 1.
dt
(b) Suppose that Оі is of constant velocity, and let П„ : [a, b] в†’ [a, b] be a
reparametrization. Show that A(Оі в—¦ П„ ) в‰Ґ A(Оі), with equality only when
П„ = identity.

Deп¬Ѓnition 3.3 The riemannian distance between two points x and y of a con-
nected riemannian manifold (X, g) is the inп¬Ѓmum 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 Г— Rn , let gx (v, w) = v, w , gx (v, v) =
|v|2 , where В·, В· is the euclidean inner product, and | В· | is the euclidean norm.
Then (Rn , В·, В· ) is a geodesically convex riemannian manifold, and the riemannian
distance is the usual euclidean distance d(x, y) = |x в€’ y|. в™¦

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 indeп¬Ѓnitely. Given (x, v) в€€ T X, let exp(x, v) : R в†’ 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) = в€’ В· d(x, y)2 .
f : X Г— X в€’в†’ R ,
2
What is the symplectomorphism П• : T в€— X в†’ T в€— X generated by f ?

Proposition 3.4 Under the identiп¬Ѓcation of T X with T в€— X by g, the symplecto-
morphism generated by П• coincides with the map T X в†’ T X, (x, v) в†’ exp(x, v)(1).

в€—
Proof. Let dx f and dy f be the components of df(x,y) with respect to T(x,y)(X Г—
в€— в€—
X) Tx X Г— Ty X. The metric gx : Tx X Г— Tx X в†’ R induces an identiп¬Ѓcation

в€—
в€’в†’ Tx X
g x : Tx X
в€’в†’ gx (v, В·)
v

Use g to translate П• into a map П• : T X в†’ T X.
Recall that, if

О“Пѓ = {(x, y, dx f, в€’dy f ) | (x, y) в€€ X Г— X}
П•

is the graph of a diп¬Ђeomorphism П• : T в€— X в†’ T в€— X, then П• is the symplectomor-
phism generated by f . In this case, П•(x, Оѕ) = (y, О·) if and only if Оѕ = dx f and
О· = в€’dy f . We need to show that, given (x, v) в€€ T X, the unique solution of

gx (v) = Оѕi = dx f (x, y)
= в€’dy f (x, y)
gy (w) = О·i

for (y, О·) in terms of (x, Оѕ) in order to п¬Ѓnd П•, or, equivalently, for (y, w) in terms
(x, v) in order to п¬Ѓnd П•.
Let Оі be the geodesic with initial conditions Оі(0) = x and dОі (0) = v.
dt
37
3.5. PERIODIC POINTS

Вў
Вў Оі
vВў E
Вў
Вў
Вў
r
x

By the Gauss lemma (look up , for instance), geodesics are orthogonal to
the level sets of the distance function.
To solve the п¬Ѓrst 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 в€€ 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 suп¬ѓces to check locally.
The vector w is obtained from the second equation of the system. Com-
pute в€’dy f ( d exp(x,v) (1)). Then evaluate в€’dy f at vectors w в€€ 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 в†’ R and a tangent
d
vector v в€€ Tx X, we have dx f (v) = [f (exp(x, v)(u))] u=0 .
du
In summary, the symplectomorphism П• corresponds to the map

в€’в†’ T X
П•: TX
в€’в†’ (Оі(1), dОі (1)) ,
(x, v) dt

which is called the geodesic п¬‚ow on X.

3.5 Periodic Points

Let X be an n-dimensional manifold. Let M = T в€— X be its cotangent bundle with
canonical symplectic form П‰.
38 LECTURE 3. GENERATING FUNCTIONS

Suppose that we are given a smooth function f : X Г— X в†’ R which gener-
ates a symplectomorphism П• : M в†’ M , П•(x, dx f ) = (y, в€’dy f ), by the recipe of
Section sec:method.

What are the п¬Ѓxed points of П•?

Deп¬Ѓne П€ : X в†’ R by П€(x) = f (x, x).

Proposition 3.5 There is a one-to-one correspondence between the п¬Ѓxed points
of П• and the critical points of П€.

Proof. At x0 в€€ X, dx0 П€ = (dx f + dy f )|(x,y)=(x0 ,x0 ) . Let Оѕ = dx f |(x,y)=(x0 ,x0 ) .

x0 is a critical point of П€ в‡ђв‡’ dx0 П€ = 0 в‡ђв‡’ dy f |(x,y)=(x0,x0 ) = в€’Оѕ .

Hence, the point in О“Пѓ corresponding to (x, y) = (x0 , x0 ) is (x0 , x0 , Оѕ, Оѕ). But О“Пѓ
f f
is the graph of П•, so П•(x0 , Оѕ) = (x0 , Оѕ) is a п¬Ѓxed point. This argument also works
backwards.

Consider the iterates of П•,

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

each of which is a symplectomorphism of M . According to the previous proposition,
if П•(N ) : M в†’ M is generated by f (N ) , then there is a one-to-one correspondence

critical points of
п¬Ѓxed points of П•(N ) в†ђв†’ (N )
: X в†’ R , П€(N ) (x) = f (N ) (x, x)
П€

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

Fix x, y в€€ X. Deп¬Ѓne a map

в€’в†’ R
X
в€’в†’ 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 Г— X в†’ R is smooth and is a generating
function for П•(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

в€‚ в€‚f в€‚f
det (x, z) + (z, y) =0.
в€‚zi в€‚yj в€‚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 П•(2) if and
only if
П•(2) (x, dx f (2) ) = (y, в€’dy f (2) )

(assuming that, for each Оѕ в€€ Tx X, there is a unique y в€€ X for which dx f (2) = Оѕ).
в€—

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

= П•(П•(x, dx f (2) (x, y)) = П•(z0 , в€’dy f (x, z0 ))
П•(2) (x, dx f (2) (x, y))
=dx f (x,z0 )
= (y, в€’dy f (z0 , y) ) .
= П•(z0 , dx f (z0 , y))
=в€’dy f (2) (x,y)

Exercise 13
What is a generating function for П•(3) ?
Hint: Suppose that the function
X Г—X в€’в†’ R
(z, u) в€’в†’ 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 П€ (3) (x, y) = f (x, z0 ) + f (z0 , u0 ) + f (u0 , y).

3.6 Billiards

Let П‡ : R в†’ R2 be a smooth plane curve which is 1-periodic, i.e., П‡(s + 1) = П‡(s),
and parametrized by arc-length, i.e., dП‡ = 1. Assume that the region Y enclosed
ds
by П‡ is convex, i.e., for any s в€€ R, the tangent line {П‡(s) + t dП‡ | t в€€ R} intersects
ds
X := в€‚Y (= the image of П‡) at only the point П‡(s).
40 LECTURE 3. GENERATING FUNCTIONS

'

X = в€‚Y
r П‡(s)

Suppose that we throw a ball into Y rolling with constant velocity and bounc-
ing oп¬Ђ the boundary with the usual law of reп¬‚ection. This determines a map

П• : R/Z Г— (в€’1, 1) в€’в†’ R/Z Г— (в€’1, 1)
(x, v) в€’в†’ (y, w)

by the rule
when the ball bounces oп¬Ђ x with angle Оё = arccos v, it will next collide with y and
bounce oп¬Ђ with angle ОЅ = arccos w.

4
4
%
4
4
4
xr
В˜
В˜
В˜
В˜
В˜
j В˜
В˜ 4
В˜ 4
B
В˜ 4
В˜4
r
y

Let f : R/Z Г— R/Z в†’ R be deп¬Ѓned by f (x, y) = в€’|x в€’ y|; f is smooth oп¬Ђ the
diagonal. Use П‡ to identify R/Z with the image curve X.
Вґ 41
3.7. POINCARE RECURRENCE

Suppose that П•(x, v) = (y, w), i.e., (x, v) and (y, w) are successive points on
the orbit described by the ball. Then
пЈ± df xв€’y
пЈґ =в€’ projected onto Tx X = v
пЈґ
пЈґ dx
пЈІ |x в€’ y|
пЈґ df
пЈґ yв€’x
пЈґ
пЈі =в€’ = в€’w
projected onto Ty X
|x в€’ y|
dy
or, equivalently,
пЈ±d y в€’ x dП‡
пЈґ В·
пЈґ ds f (П‡(s), y) = = cos Оё = v
пЈґ
пЈІ |x в€’ y| ds
пЈґd
пЈґ x в€’ y dП‡
пЈґ
пЈі В· = в€’ cos ОЅ = в€’w .
f (x, П‡(s)) =
|x в€’ y| ds
ds
We conclude that f is a generating function for П•. Similar approaches work
for higher dimensional billiards problems.
Periodic points are obtained by п¬Ѓnding critical points of
X Г—...Г—X в€’в†’ R , N >1
N
(x1 , . . . , xN ) в€’в†’ f (x1 , x2 ) + f (x2 , x3 ) + . . . + f (xN в€’1 , xN ) + f (xN , x1 )
= |x1 в€’ x2 | + . . . + |xN в€’1 в€’ xN | + |xN в€’ x1 | ,
that is, by п¬Ѓnding the N -sided (generalized) polygons inscribed in X of critical
perimeter.
Notice that
R/Z Г— (в€’1, 1) {(x, v) | x в€€ X, v в€€ Tx X, |v| < 1} A
is the open unit tangent ball bundle of a circle X, that is, an open annulus A. The
map П• : A в†’ A is area-preserving.

3.7 PoincarВґ Recurrence
e
Theorem 3.7 (PoincarВґ Recurrence) Suppose that П• : A в†’ A is an area-
e
preserving diп¬Ђeomorphism of a п¬Ѓnite-area manifold A. Let p в€€ A, and let U be
a neighborhood of p. Then there is q в€€ U and a positive integer N such that
П•(N ) (q) в€€ U.

Proof. Let U0 = U, U1 = П•(U), U2 = П•(2) (U), . . .. If all of these sets were disjoint,
then, since Area (Ui ) = Area (U) > 0 for all i, we would have

Area A в‰Ґ Area (U0 в€Є U1 в€Є U2 в€Є . . .) = Area (Ui ) = в€ћ .
i
42 LECTURE 3. GENERATING FUNCTIONS

To avoid this contradiction we must have П•(k) (U) в€© П•(l) (U) = в€… for some k > l,
which implies П•(kв€’l) (U) в€© U = в€….
Hence, eternal return applies to billiards...
Remark. Theorem 3.7 clearly generalizes to volume-preserving diп¬Ђeomorphisms
в™¦
in higher dimensions.

Theorem 3.8 (PoincarВґвЂ™s Last Geometric Theorem) Suppose П• : A в†’ A
e
is an area-preserving diп¬Ђeomorphism of the closed annulus A = R/Z Г— [в€’1, 1]
which preserves the two components of the boundary, and twists them in opposite
directions. Then П• has at least two п¬Ѓxed points.

This theorem was proved in 1925 by Birkhoп¬Ђ, and hence is also called the
PoincarВґ-Birkhoп¬Ђ theorem. It has important applications to dynamical sys-
e
tems and celestial mechanics. The Arnold conjecture (1966) on the existence of
п¬Ѓxed points for symplectomorphisms of compact manifolds (see Section 3.9) may
be regarded as a generalization of the PoincarВґ-Birkhoп¬Ђ theorem. This conjecture
e
has motivated a signiп¬Ѓcant 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, П‰) form the group

Sympl(M, П‰) = {f : M в€’в†’ M | f в€— П‰ = П‰} .

вЂ“ What is Tid (Sympl(M, П‰))?
(What is the вЂњLie algebraвЂќ of the group of symplectomorphisms?)
вЂ“ What does a neighborhood of id in Sympl(M, П‰) look like?
We will use notions from the C 1 -topology. Let X and Y be manifolds.

Deп¬Ѓnition 3.9 A sequence of maps fi : X в†’ Y converges in the C 0 -topology
to f : X в†’ Y if and only if fi converges uniformly on compact sets.

Deп¬Ѓnition 3.10 A sequence of C 1 maps fi : X в†’ Y converges in the C 1 -
topology to f : X в†’ Y if and only if it and the sequence of derivatives dfi :
T X в†’ T Y converge uniformly on compact sets.

Let (M, П‰) be a compact symplectic manifold and f в€€ Sympl(M, П‰). Then
both Graph f and the diagonal в€† = Graph id are lagrangian subspaces of (M Г—
M, prв€— П‰ в€’ prв€— П‰), where pri : M Г— M в†’ 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 в€† ( M ) in (M Г— M, prв€— П‰ в€’ prв€— П‰) which is symplectomorphic to a
1 2
neighborhood U0 of M in (T в€— M, П‰0 ). Let П• : U в†’ U0 be the symplectomorphism.
Suppose that f is suп¬ѓciently C 1 -close to id, i.e., f is in some suп¬ѓciently
small neighborhood of id in the C 1 -topology. Then:

1. We can assume that Graph f вЉ† U.
j:M в†’U
Let be the embedding as Graph f ,
i:M в†’U be the embedding as Graph id = в€† .

2. The map j is suп¬ѓciently C 1 -close to i.

U0 вЉ† T в€— M , so the above j and i induce
3. By the Weinstein theorem, U
j0 : M в†’ U 0 embedding, where j0 = П• в—¦ j ,
i0 : M в†’ U 0 embedding as 0-section .
Hence, we have

П• П•
E U0 E U0
U U
d
s 
В  d
s 
В
d В  d В
d В  d В
and
id В  i0 jd В  j0
d В  d В
M M
where i(p) = (p, p), i0 (p) = (p, 0), j(p) = (p, f (p)) and j0 (p) = П•(p, f (p)) for
p в€€ M.

4. The map j0 is suп¬ѓciently C 1 -close to i0 . Therefore, the image set j0 (M )
в€—
intersects each Tp M at one point Вµp depending smoothly on p.

5. The image of j0 is the image of a smooth section Вµ : M в†’ T в€— M , that is, a
1-form Вµ = j0 в—¦ (ПЂ в—¦ j0 )в€’1 .
в€—
{(p, Вµp ) | p в€€ M, Вµp в€€ Tp M }.
We conclude that Graph f

Exercise 14
Vice-versa: show that, if Вµ is a 1-form suп¬ѓciently C 1 -close to the zero 1-form,
then there is a diп¬Ђeomorphism f : M в†’ M such that
в€—
{(p, Вµp ) | p в€€ M, Вµp в€€ Tp M } Graph f .

By Section 2.4, we have

Graph f is lagrangian в‡ђв‡’ Вµ is closed .
44 LECTURE 3. GENERATING FUNCTIONS

Conclusion. A small C 1 -neighborhood of id in Sympl(M, П‰) is homeomorphic to
a C 1 -neighborhood of zero in the vector space of closed 1-forms on M . So:

{Вµ в€€ в„¦1 (M ) | dВµ = 0} .
Tid (Sympl(M, П‰))

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

{Вµ = dh | h в€€ C в€ћ (M )} C в€ћ (M )/ locally constant functions .

3.9 Fixed Points of Symplectomorphisms
1
Theorem 3.11 Let (M, П‰) be a compact symplectic manifold with H deRham (M ) =
0. Then any symplectomorphism of M which is suп¬ѓciently C 1 -close to the identity
has at least two п¬Ѓxed points.

Proof. Suppose that f в€€ Sympl(M, П‰) is suп¬ѓciently C 1 -close to id. Then the
graph of f corresponds to a closed 1-form Вµ on M .

dВµ = 0
=в‡’ Вµ = dh for some h в€€ C в€ћ (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 в€© в€† = {p : Вµp = dhp = 0} .

Lagrangian intersection problem:
A submanifold Y of M is C 1 -close to X when there is a diп¬Ђeomorphism
X в†’ Y which is, as a map into M , C 1 -close to the inclusion X в†’ M .

Theorem 3.12 Let (M, П‰) 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, П‰) be a compact symplectic manifold, and f : M в†’ M a symplectomor-
phism which is вЂњexactly homotopic to the identityвЂќ (see below). Then

#{п¬Ѓxed points of f } в‰Ґ 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 п¬Ѓxed points of f } в‰Ґ minimal # of critical points
a Morse function of M can have
2n
dim H i (M ) .
в‰Ґ
i=0

The Arnold conjecture was proved by Conley-Zehnder, Floer, Hofer-Salamon,
Ono, Futaya-Ono, Lin-Tian using Floer homology (which is an в€ћ-dimensional
analogue of Morse theory). There are open conjectures for sharper bounds on the
number of п¬Ѓxed points.
Meaning of вЂњf is exactly homotopic to the identity:вЂќ
Suppose that ht : M в†’ R is a smooth family of functions which is 1-periodic,
i.e., ht = ht+1 . Let ПЃ : M Г—R в†’ M be the isotopy generated by the time-dependent
vector п¬Ѓeld vt deп¬Ѓned by П‰(vt , В·) = dht . Then вЂњf being exactly homotopic to the
identityвЂќ means f = ПЃ1 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

{ п¬Ѓxed points of f } в†ђв†’ { period-1 orbits of ПЃ : M Г— R в†’ M }

because f (p) = p if and only if {ПЃ(t, p) , t в€€ [0, 1]} is a closed orbit.
Proof of the Arnold conjecture in the case when h : M в†’ R is independent of t:

в‡ђв‡’
p is a critical point of h dhp = 0
в‡ђв‡’ vp = 0
ПЃ(t, p) = p , в€Ђt в€€ R
=в‡’
в‡ђв‡’ p is a п¬Ѓxed point of ПЃ1 .

Exercise 15
Compute these estimates for the number of п¬Ѓxed points on some compact
symplectic manifolds (for instance, S 2 , S 2 Г— S 2 and T 2 = S 1 Г— S 1 ).

2A Morse function on M is a function h : M в†’ R whose critical points (i.e., points p
where the diп¬Ђerential vanishes: dhp = 0) are all nondegenerate (i.e., the hessian at those points
в€‚2 h
is nonsingular: det = 0).
в€‚xi в€‚xj p
3A п¬Ѓxed point p of f : M в†’ M is nondegenerate if dfp : Tp M в†’ Tp M is nonsingular.
Lecture 4

Hamiltonian Fields

To any real function on a symplectic manifold, a symplectic geometer associates a
vector п¬Ѓeld whose п¬‚ow preserves the symplectic form and the given function. The
vector п¬Ѓeld is called the hamiltonian vector п¬Ѓeld of that (hamiltonian) function.
The concept of a moment map is a generalization of that of a hamiltonian
function, and was introduced by Souriau  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, П‰) be a symplectic manifold and let H : M в†’ R be a smooth function. Its
diп¬Ђerential dH is a 1-form. By nondegeneracy, there is a unique vector п¬Ѓeld XH
on M such that Д±XH П‰ = dH. Integrate XH . Supposing that M is compact, or at
least that XH is complete, let ПЃt : M в†’ M , t в€€ R, be the one-parameter family of
diп¬Ђeomorphisms generated by XH :

пЈ±
пЈґ ПЃ0 = idM
пЈІ
пЈґ dПЃt
пЈі в—¦ ПЃв€’1 = XH .
t
dt

47
48 LECTURE 4. HAMILTONIAN FIELDS

Claim. Each diп¬Ђeomorphism ПЃt preserves П‰, i.e., ПЃв€— П‰ = П‰, в€Ђt.
t

dв€—
= ПЃв€— LXH П‰ = ПЃв€— (d Д±XH П‰ +Д±XH dП‰ ) = 0.
Proof. We have dt ПЃt П‰ t t
0
dH

Therefore, every function on (M, П‰) gives a family of symplectomorphisms.
Notice how the proof involved both the nondegeneracy and the closedness of П‰.

Deп¬Ѓnition 4.1 A vector п¬Ѓeld XH as above is called the hamiltonian vector
п¬Ѓeld with hamiltonian function H.

Example. The height function H(Оё, h) = h on the sphere (M, П‰) = (S 2 , dОё в€§ dh)
has
в€‚
Д±XH (dОё в€§ dh) = dh в‡ђв‡’ XH = .
в€‚Оё
Thus, ПЃt (Оё, h) = (Оё + t, h), which is rotation about the vertical axis; the height
в™¦
function H is preserved by this motion.

Exercise 16
Let X be a vector п¬Ѓeld on an abstract manifold W . There is a unique vector
п¬Ѓeld X on the cotangent bundle T в€— W , whose п¬‚ow is the lift of the п¬‚ow of X.
Let О± be the tautological 1-form on T в€— W and let П‰ = в€’dО± be the canonical
symplectic form on T в€— W . Show that X is a hamiltonian vector п¬Ѓeld with
hamiltonian function H := Д±X О±.

Remark. If XH is hamiltonian, then

LXH H = Д±XH dH = Д±XH Д±XH П‰ = 0 .

Therefore, hamiltonian vector п¬Ѓelds preserve their hamiltonian functions, and each
integral curve {ПЃt (x) | t в€€ R} of XH must be contained in a level set of H:

H(x) = (ПЃв€— H)(x) = H(ПЃt (x)) , в€Ђt .
t

в™¦

Deп¬Ѓnition 4.2 A vector п¬Ѓeld X on M preserving П‰ (i.e., such that LX П‰ = 0) is
called a symplectic vector п¬Ѓeld.

в‡ђв‡’
X is symplectic Д±X П‰ is closed ,
в‡ђв‡’
X is hamiltonian Д±X П‰ is exact .
Locally, on every contractible open set, every symplectic vector п¬Ѓeld is hamil-
1
tonian. If HdeRham(M ) = 0, then globally every symplectic vector п¬Ѓeld is hamil-
1
tonian. In general, HdeRham (M ) measures the obstruction for symplectic vector
п¬Ѓelds to be hamiltonian.
49
4.2. HAMILTON EQUATIONS

Example. On the 2-torus (M, П‰) = (T 2 , dОё1 в€§ dОё2 ), the vector п¬Ѓelds X1 = в€‚Оё1
в€‚

в€‚
в™¦
and X2 = в€‚Оё2 are symplectic but not hamiltonian.
To summarize, vector п¬Ѓelds on a symplectic manifold (M, П‰) which preserve
П‰ are called symplectic. The following are equivalent:

вЂў X is a symplectic vector п¬Ѓeld;

вЂў the п¬‚ow ПЃt of X preserves П‰, i.e., ПЃв€— П‰ = П‰, for all t;
t

вЂў LX П‰ = 0;

вЂў Д±X П‰ is closed.

A hamiltonian vector п¬Ѓeld is a vector п¬Ѓeld X for which

вЂў Д±X П‰ is exact,

i.e., Д±X П‰ = dH for some H в€€ C в€ћ (M ). A primitive H of Д±X П‰ is then called a
hamiltonian function of X.

4.2 Hamilton Equations

Consider euclidean space R2n with coordinates (q1 , . . . , qn , p1 , . . . , pn ) and П‰0 =
dqj в€§ dpj . The curve ПЃt = (q(t), p(t)) is an integral curve for XH exactly if
пЈ±
пЈґ dqi (t) = в€‚H
пЈґ
пЈґ dt
пЈІ в€‚pi
(Hamilton equations)
пЈґ dp
пЈґ в€‚H
пЈґ i
пЈі (t) = в€’
dt в€‚qi
n
в€‚H в€‚ в€‚H в€‚
в€’
Indeed, let XH = . Then,
в€‚pi в€‚qi в€‚qi в€‚pi
i=1

n n
Д±XH (dqj в€§ dpj ) = [(Д±XH dqj ) в€§ dpj в€’ dqj в€§ (Д±XH dpj )]
Д± XH П‰ =
j=1 j=1
n
в€‚H в€‚H
= в€‚pj dpj + в€‚qj dqj = dH .
j=1

Remark. The gradient vector п¬Ѓeld of H relative to the euclidean metric is
n
в€‚H в€‚ в€‚H в€‚
H := + .
в€‚qi в€‚qi в€‚pi в€‚pi
i=1
50 LECTURE 4. HAMILTONIAN FIELDS

If J is the standard (almost) complex structure1 so that J( в€‚qi ) =
в€‚ в€‚
and
в€‚pi
в€‚ в€‚
J( в€‚pi ) = в€’ в€‚qi , we have JXH = H. в™¦
The case where n = 3 has a simple physical illustration. NewtonвЂ™s second
law states that a particle of mass m moving in conп¬Ѓguration 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 = в€’ 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 в€— R3 be the corresponding phase space, with coordinates
2m
(q1 , q2 , q3 , p1 , p2 , p3 ). NewtonвЂ™s second law in R3 is equivalent to the Hamilton
equations in R6 :
пЈ± dq 1 в€‚H
i
пЈґ = pi =
пЈІ
dt m в€‚pi
d 2 qi
dpi в€‚V в€‚H
пЈґ
пЈі =m 2 =в€’ =в€’ .
dt dt в€‚qi в€‚qi
The energy H is conserved by the physical motion.

4.3 Brackets
Vector п¬Ѓelds are diп¬Ђerential operators on functions: if X is a vector п¬Ѓeld and
f в€€ C в€ћ (X), df being the corresponding 1-form, then

X В· f := df (X) = LX f .

Given two vector п¬Ѓelds X, Y , there is a unique vector п¬Ѓeld W such that

LW f = LX (LY f ) в€’ LY (LX f ) .

The vector п¬Ѓeld W is called the Lie bracket of the vector п¬Ѓelds X and Y and
denoted W = [X, Y ], since LW = [LX , LY ] is the commutator.

Exercise 17
Check that, for any form О±,
Д±[X,Y ] О± = LX Д±Y О± в€’ Д±Y LX О± = [LX , Д±Y ]О± .
Since each side is an anti-derivation with respect to the wedge product, it
suп¬ѓces 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 в†’
T M such that J 2 = в€’Id.
51
4.3. BRACKETS

Proposition 4.3 If X and Y are symplectic vector п¬Ѓelds on a symplectic manifold
(M, П‰), then [X, Y ] is hamiltonian with hamiltonian function П‰(Y, X).

Proof.
= L X Д±Y П‰ в€’ Д± Y L X П‰
Д±[X,Y ] П‰
= dД±X Д±Y П‰ + Д±X dД±Y П‰ в€’Д±Y dД±X П‰ в€’Д±Y Д±X dП‰
0
0 0
= d(П‰(Y, X)) .

A (real) Lie algebra is a (real) vector space g together with a Lie bracket
[В·, В·], i.e., a bilinear map [В·, В·] : g Г— g в†’ g satisfying:

(a) [x, y] = в€’[y, x] , в€Ђx, y в€€ g , (antisymmetry)

в€Ђx, y, z в€€ g .
(b) [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 , (Jacobi identity)

Let
П‡(M ) = { vector п¬Ѓelds on M }
sympl
(M ) = { symplectic vector п¬Ѓelds on M }
П‡
ham
(M ) = { hamiltonian vector п¬Ѓelds on M } .
П‡

The inclusions (П‡ham (M ), [В·, В·]) вЉ† (П‡sympl (M ), [В·, В·]) вЉ† (П‡(M ), [В·, В·]) are inclusions
of Lie algebras.

Deп¬Ѓnition 4.4 The Poisson bracket of two functions f, g в€€ C в€ћ (M ; R) is

{f, g} := П‰(Xf , Xg ) .

We have X{f,g} = в€’[Xf , Xg ] because XП‰(Xf ,Xg ) = [Xg , Xf ].

Theorem 4.5 The bracket {В·, В·} satisп¬Ѓes the Jacobi identity, i.e.,

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

Proof. Exercise.

Deп¬Ѓnition 4.6 A Poisson algebra (P, {В·, В·}) is a commutative associative alge-
bra P with a Lie bracket {В·, В·} satisfying the Leibniz rule:

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

Exercise 18
Check that the Poisson bracket {В·, В·} deп¬Ѓned above satisп¬Ѓes the Leibniz rule.
52 LECTURE 4. HAMILTONIAN FIELDS

We conclude that, if (M, П‰) is a symplectic manifold, then (C в€ћ (M ), {В·, В·}) is
a Poisson algebra. Furthermore, we have a Lie algebra anti-homomorphism

C в€ћ (M ) в€’в†’ П‡(M )
H в€’в†’ XH
{В·, В·} в€’[В·, В·] .

Exercise 19
Let G be a Lie group, g its Lie algebra and gв€— the dual vector space of g.
(a) Let g X # be the vector п¬Ѓeld generated by X в€€ g for the adjoint repre-
sentation of G on g. Show that
#
g
XY = [X, Y ] в€ЂY в€€g.

(b) Let X # be the vector п¬Ѓeld generated by X в€€ g for the coadjoint repre-
sentation of G on gв€— . Show that
#
XОѕ , Y = Оѕ, [Y, X] в€ЂY в€€g.

(c) For any Оѕ в€€ gв€— , deп¬Ѓne a skew-symmetric bilinear form on g by
П‰Оѕ (X, Y ) := Оѕ, [X, Y ] .
Show that the kernel of П‰Оѕ is the Lie algebra gОѕ of the stabilizer of Оѕ for
(d) Show that П‰Оѕ deп¬Ѓnes a nondegenerate 2-form on the tangent space at Оѕ
to the coadjoint orbit through Оѕ.
(e) Show that П‰Оѕ deп¬Ѓnes a closed 2-form on the orbit of Оѕ in g в€— .

Hint: The tangent space to the orbit being generated by the vector п¬Ѓelds
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 deп¬Ѓnes a canonical Poisson structure on
gв€— :
{f, g}(Оѕ) := Оѕ, [dfОѕ , dgОѕ ]
for f, g в€€ C в€ћ (gв€— ) and Оѕ в€€ gв€— . Notice that dfОѕ : TОѕ gв€— gв€— в†’ R is
identiп¬Ѓed with an element of g gв€—в€— .
Check that {В·, В·} satisп¬Ѓes the Leibniz rule:
{f, gh} = g{f, h} + h{f, g} .

Example. For the prototype (R2n , П‰0 ), where П‰0 = dxi в€§ dyi , we have
в€‚ в€‚
Xx i = в€’ and X yi =
в€‚yi в€‚xi
so that
1 if i = j
{xi , xj } = {yi , yj } = 0 {xi , yj } = в€Ђi, j .
and
0 if i = j
53
4.4. INTEGRABLE SYSTEMS

For arbitrary functions f, g в€€ C в€ћ (M ) we have hamiltonian vector п¬Ѓelds
n
в€‚f в€‚ в€‚f в€‚
в€’
Xf = ,
в€‚yi в€‚xi в€‚xi в€‚yi
i=1

and the classical Poisson bracket
n
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