Zehnder E. Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities
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V.8. Symplectic maps 211
Every manifold can be equipped with a Riemannian metric. In sharp contrast,
not every manifold of evendimension admits a symplectic structure, as the following
example of the sphere S
4
shows.
Example. The 4-dimensional sphere S
4
does not admit a symplectic structure.
Proof. Arguing by contradiction we assume that ! is a symplectic structure on S
4
.
Since ! is nondegenerate, the 4-form ´ ! ^ ! ¤ 0 is a volume form on S
4
and hence
Z
S
4
! ^ ! D
Z
S
4
¤ 0:
On the other hand ! is closed and since the second cohomology group of the
sphere S
4
vanishes, H
2
.S
4
/ D 0, there exists according to the de Rham theorem
a 1-form ˛ on S
4
satisfying ! D d˛. Using d! D 0 we compute
d.! ^ ˛/ D d! ^ ˛ C ! ^ d˛ D ! ^ d˛ D ! ^ !
and due to @S
4
D;we arrive, in view of Stoke’s theorem, at the contradiction
Z
S
4
! ^ ! D
Z
S
4
d.! ^ ˛/ D 0:
Analogously one sees that every compact manifold M whose second cohomol-
ogy group H
2
.M / D 0 vanishes, does not admit a symplectic structure. Examples
are all spheres S
2n
for n 2, while in dimension 2n D 2 the volume forms on S
2
and on the oriented Riemann surfaces are symplectic structures.
Important examples of symplectic diffeomorphisms are the flow maps of Hamil-
tonian vector fields introduced next.
Definition. A smooth function H W M ! R on a symplectic manifold .M; !/ is
called a Hamiltonian function. The unique C
1
-vector field X
H
on M , defined by
i
X
H
! DdH;
or explicitly by
!.x/
X
H
.x/;
DdH.x/ for all x 2 M;
is called the associated Hamiltonian vector field.
The vector field X
H
is uniquely defined, since, in view of the nondegeneracy
of the symplectic form !, the map X.x/ 7! !.x/.X.x/; / is a linear isomorphism
from the tangent space T
x
M onto its dual space .T
x
M/
.
212 Chapter V. Hamiltonian vector fields and symplectic diffeomorphisms
In a Darboux chart, i.e., locally in .R
2n
;!
0
/, the equation i
X
H
!
0
DdH
becomes
hJX
H
.x/; viDdH.x/v
D hrH.x/;vi
for all v 2 R
2n
where the gradient rH of the function H is defined with respect to
the Euclidian scalar product h; iin R
2n
. We conclude that JX
H
.x/ DrH.x/
and multiplying by J we obtain, due to J
2
D1, the local representation
X
H
.x/ D J rH.x/
of a Hamiltonian vector field in symplectic coordinates.
We see that X
H
is a very special vector field on M . Since J is antisymmetric,
the Hamiltonian vector field X
H
has properties that are completely different from
the properties of the gradient vector field rH studied in Chapter IV.
Proposition V.24. The flow maps '
t
of a Hamiltonian vector field X
H
are sym-
plectic maps and hence satisfy .'
t
/
! D !.
Proof [Cartan’s formula]. Abbreviating X D X
H
we compute
d
dt
.'
t
/
! D .'
t
/
L
X
!
D .'
t
/
Œi
X
.d!/ C d.i
X
!/
D 0;
since d! D 0 and d.i
X
!/ D d.dH / D 0. Using '
0
D Id we therefore conclude
.'
t
/
! D .'
0
/
! D ! for all t, as claimed.
Exercise. We assume X to be a vector field on .M; !/. Its flow '
t
is symplectic if
and only if L
X
! D 0. In this case X can be written locally as a Hamiltonian vector
field X
F
. (Hint: Make use of Cartan’s formula and Poincaré’s lemma.)
Proposition V.25. The interior product i
X
˛ of vector fields with forms transforms
naturally, i.e., for a diffeomorphism uW M ! N we have
u
.i
X
˛/ D i
u
X
.u
˛/
for all vector fields X on N and all forms ˛ 2
.N /.
Proof. In view of the definitions, we have on one hand
u
.i
X
˛/.v
1
;:::;v
k1
/ D .˛ B u/.X B u; duv
1
;:::;duv
k1
/;
V.8. Symplectic maps 213
on the other hand,
i
u
X
.u
˛/.v
1
;:::;v
k1
/ D .u
˛/.u
X; v
1
;:::;v
k1
/
D .˛ Bu/.du.u
X/;duv
1
;:::;duv
k1
/
D .˛ Bu/.X B u; duv
1
;:::;duv
k1
/;
where we have used that du.u
X/ D du.du
1
X B u/ D X B u.
We next show that under the symplectic maps the Hamiltonian vector fields are
transformed into Hamiltonian vector fields. All one needs to do is to transform the
Hamiltonian function! We recall that by definition of the pullback, u
H D H B u
is the composition.
Proposition V.26 (Transformation theory). If u W M ! M is a symplectic map,
then
u
X
H
D X
u
H
for every smooth function H W M ! R. Conversely, a local diffeomorphism u is
symplectic if the above formula holds true for all functions H W M ! R.
Proof [Proposition V.25, Proposition V.11]. We assume that u
! D !. By means
of the definition of the Hamiltonian vector field and using the Propositions V.25
and V.11 we compute
i
X
u
H
! Dd.u
H/
Du
.dH /
D u
.i
X
H
!/
D i
u
X
H
.u
!/
D i
u
X
H
!:
Since ! is nondegenerate, we conclude that u
X
H
D X
u
H
, as claimed.
In order to verify the converse we assume that for a given local diffeomorphism
u the formula u
X
H
D X
u
H
holds true for all functions H . By the previous
computation, i
u
X
H
.u
!/ D i
u
X
H
!. And hence, !.u
X
H
; / D u
!.u
X
H
; /
for all functions H . By choosing in symplectic coordinates the functions H.x/ D
x
j
for 1 j 2n D dim M , one sees that !.v
j
; / D u
!.v
j
; / for a basis
fv
1
;:::;v
2n
g of the tangent space. Therefore ! u
! D 0, as we wanted to
prove.
Definition. If .M; !/ is a symplectic manifold and F;GW M ! R are smooth func-
tions, then the Poisson bracket of F and G is the smooth function fF;GgW M ! R,
defined by
fF;Gg.x/ ´ !.x/
X
F
.x/; X
G
.x/
;x2 M:
214 Chapter V. Hamiltonian vector fields and symplectic diffeomorphisms
The Poisson bracket can also be written as
fF;GgD!.X
F
;X
G
/ DdF .X
G
/ D dG.X
F
/:
Proposition V.27. If F;GW M ! R are smooth functions, then
ŒX
F
;X
G
D X
fF;Gg
:
Explicitly, ŒX
F
;X
G
D X
H
where the Hamiltonian function H is defined by
H.x/ D !.x/
X
F
.x/; X
G
.x/
2 R.
Proof [Proposition V.24, Proposition V.26]. If '
t
F
is the flow of X
F
, it followsfrom
Propositions V.24 and V.26 that
.'
t
F
/
X
G
D X
GB'
t
F
:
In view of the definition of the Lie bracket we therefore obtain
ŒX
F
;X
G
D
d
dt
ˇ
ˇ
ˇ
tD0
.'
t
F
/
X
G
D X
d
dt
j
tD0
GB'
t
F
:
For the last term we compute
d
dt
ˇ
ˇ
ˇ
tD0
G B '
t
F
D dGX
F
D!.X
G
;X
F
/
D !.X
F
;X
G
/;
where we have used that !.X
G
; / DdG by the definition of the Hamiltonian
vector field X
G
.
PropositionV.27 states that the linear map H 7! X
H
is a homomorphism of Lie-
algebras from the space of smooth functions C
1
.M; R/ equipped with the Poisson
bracket f; g into the linear space of Hamiltonian vector fields on M equipped with
the Lie bracket Π; . This leads to the Jacobi identity for the Poisson bracket. The
following proposition sums up some properties of the Poisson bracket. The Jacobi
identity (ii) below is equivalent to the closedness of the 2-form !, the proof of
which is left as an exercise to the reader. A proof can be found, for example in [56]
by R. Jost.
Proposition V.28. We consider the smooth functions F;G;H W M ! R on the
symplectic manifold .M; !/. The Poisson bracket f; ghas the following properties:
(i) fF;GgDfG; F g,
(ii) ffF;Gg;HgCffG; Hg;FgCffH; F g;GgDd!.X
F
;X
G
;X
H
/ D 0,
(iii) ŒX
F
;X
G
D X
fF;Gg
.
V.8. Symplectic maps 215
Under a symplectic mapping the Poisson bracket is transformed in a natural
way.
Proposition V.29. A local diffeomorphism u W M ! M is symplectic if and only if
fu
F;u
GgDu
fF;Gg
for all smooth functions F;GW M ! R.
Proof [Proposition V.26]. We assume that u is symplectic. Recalling the transfor-
mation formula of Hamiltonian vector fields under symplectic transformations we
compute
fF;GgBu D dG B u.X
F
B u/ D d.G B u/.du/
1
X
F
B u
D d.G B u/X
F Bu
DfG B u; F B ug
as claimed. In order to prove the converse, we assume that the diffeomorphism u
satisfies fF;GgBu DfG B u; F Bug for all functions F and G and deduce, using
the definition of the Poisson bracket,
d.G B u/X
F Bu
D dG B u.X
F
B u/ D d.G B u/.du/
1
X
F
B u D d.G B u/u
X
F
for all functions G. Consequently, u
X
F
D X
u
F
for all functions F , so that
u
! D ! in view of Proposition V.26. The proof of Proposition V.29 is complete.
Proposition V.30 (Energy conservation). The function H W M ! R is an integral
of the Hamiltonian vector field X
H
, i.e.,
dH.x/X
H
.x/ D 0; x 2 M:
Equivalently, H.'
t
.x// D H.x/for all x and t, where '
t
is the flow of X
H
. Hence
the Hamiltonian function is constant along the orbits of the Hamiltonian vector
field.
Proof [Skew symmetry of !]. In view of the definition of the Hamiltonian vector
field, dH.x/D!.X
H
.x/; /. Hence, due to the skew symmetry of !,
dH.x/X
H
.x/ D!
X
H
.x/; X
H
.x/
D 0:
If '
t
is the flow of X
H
we compute
d
dt
H.'
t
.x// D dH.'
t
.x//
d
dt
'
t
.x/
D dH.'
t
.x//X
H
.'
t
.x//
D!.X
H
;X
H
/ B '
t
.x/
D 0;
so that H.'
t
.x// D H.'
0
.x// D H.x/.
216 Chapter V. Hamiltonian vector fields and symplectic diffeomorphisms
The geometrical significance of the energy conservation is the following. Given
a Hamiltonian function H W M ! R, then the level set
S D S
E
Dfx 2 M j H.x/ D EgM
belonging to the prescribed energy value E 2 R is called an energy surface. From
the energy conservation we deduce the invariance of the energy surface under the
Hamiltonian flow,
'
t
.S/ D S
for all t if S is compact.
S
x
X
H
T
x
S
FigureV.2. The Hamiltonian vector field X
H
belongs to the tangent space of a regular energy
surface.
If dH.x/ ¤ 0 for all x 2 S, the energy surface is called regular. Itisa
submanifold of codimension 1 of M and has at the point x 2 S the tangent space
T
x
S D ker dH.x/ Dfv 2 T
x
M j dH.x/v D 0g:
In view of dH.x/X
H
.x/ D 0 the Hamiltonian vector field is tangent to S,
X
H
.x/ 2 T
x
S; x 2 S;
so that its restriction X
H
j
S
is a vector field on S. We next show that a compact
regular energysurface S possesses a canonical volume form which is invariant under
the Hamiltonian flow. As a consequence, the flow on S enjoys many recurrence
properties in view of Poincaré’s Theorem I.14. This encourages us to search for
periodic orbits on S later on.
Proposition V.31 (Liouville). If S .M; !/ is a compact regular energy surface
of a Hamilton function H , then there exists a finite regular Borel measure on S,
which is invariant under the flow '
t
of the Hamiltonian vector field X
H
.
Proof [Proposition V.24, Proposition V.25, Proposition V.30]. By hypothesis, the
energy surface
S Dfx 2 M j H.x/ D const.g
V.8. Symplectic maps 217
is compact and regular, hence in particular dH.x/ ¤ 0 for x 2 S. We choose a
smooth vector field X transversal to S and satisfying
dH.X/ D 1 on S;
and define the .2n 1/-form on S by
´ j
.i
X
/;
where D ! ^^! is the volume form on M associated with the symplectic
structure !, and where j W S ! M is the injection map. The .2n 1/-form is a
volume form on S and, in view of the following lemma, independent of the choice
of the vector field X .
LemmaV.32. If Y
1
and Y
2
are vector fields on M satisfying dH.Y
1
/ DdH.Y
2
/ D1
on S , then
j
.i
Y
1
Y
2
/ D 0:
Proof. The vector field Y D Y
1
Y
2
satisfies dH.Y / D 0 on S and is therefore
tangent to S. Consequently, it follows for the 2n 1 linear independent tangent
vectors y
2
;:::;y
2n
of S that j
.i
Y
/.y
2
;:::;y
2n
/ D .Y; y
2
;:::;y
2n
/ D 0,
because dim S<2n.
Lemma V.33. The flow '
t
of the Hamiltonian vector field X
H
satisfies '
t
W S ! S,
as we already know. We claim that
.'
t
/
D
for all t. In other words the volume form on S is invariant under the Hamiltonian
flow.
Proof. From '
t
Bj D j B'
t
on S we conclude j
B.'
t
/
D .'
t
/
Bj
. Therefore
in view of .'
t
/
D and using Proposition V.25 we compute
.'
t
/
D j
B .'
t
/
.i
X
/
D j
i
.'
t
/
X
..'
t
/
/
D j
.i
.'
t
/
X
/
D C j
.i
.'
t
/
XX
/:
The second term vanishesdue to LemmaV.32, because on S we have dH..'
t
/
X/ D
1. Indeed, it follows from the energy conservation H B '
t
D H by differentiation
that dH.'
t
.x// d'
t
.x/ D dH.x/ and hence for all x 2 S,
dH.x/
.'
t
/
X.x/
D dH.x/
d'
t
.x/
1
X B '
t
.x/
D dH.'
t
.x//
d'
t
.x/d'
t
.x/
1
X B '
t
.x/
D dH.X/ B '
t
.x/ D 1;
in view of the assumption about the vector field X . This proves the lemma.
218 Chapter V. Hamiltonian vector fields and symplectic diffeomorphisms
The volume form induces a regular Borel measure on S, defined by .A/ D
R
A
jj. This measure is invariant under '
t
, since
.'
t
.A// D
Z
'
t
.A/
jjD
Z
A
.'
t
/
jjD
Z
A
jjD.A/; t 2 R:
In addition .S/ < 1, since S is compact and the proof of Proposition V.31 is
complete.
V.9 Generating functions of symplectic maps in R
2n
For practical purposes it is often useful to know that a symplectic diffeomorphism
can be represented locally by a single function. To see this we consider the symplec-
tic map u W .; / 7!
a.; /; b.; /
D .x; y/ of the symplectic standard space
.R
2n
;!
0
/, that is,
x D a.; /;
y D b.; /:
If
det a
¤ 0
at some point where a
denotes the partial derivative in the variable , then we can
solve, in view of the implicit function theorem, the first equation locally in ,to
obtain a function D ˛.x;/ satisfying in a neighborhood of the reference point
the identities x D a
˛.x; /;
and D ˛
a.; /;
, and moreover a
B ˛
x
D
Id. Inserting the function D ˛.x; / into the second equation we arrive at the
following implicit representation of the map u:
D ˛.x; /;
y D ˇ.x; / ´ b
˛.x; /;
:
The advantage of scrambling the variables in this manner is that the condition for
the map to be symplectic is much easier to express in term of the functions ˛ and ˇ
than in terms of a and b. Indeed, as we shall show next the map is symplectic if and
only if there exists a function W D W.x;/ satisfying D ˛.x; / D W
.x; /
and y D ˇ.x; / D W
x
.x; /. We should emphasize that all our statements are of
local nature.
Proposition V.34. If u D .a; b/ W .; / 7! .x; y/ is a symplectic map on R
2n
satisfying det a
¤ 0 at some point, then there exists locally a function W D
W.x;/ satisfying
det W
x
¤ 0;
V.9. Generating functions of symplectic maps in R
2n
219
which represents u in the neighborhood of this point in the implicit form
D W
.x; /;
y D W
x
.x; /:
Conversely, a function W.x;/ satisfying det W
x
¤ 0 defines in this manner
implicitly a local symplectic diffeomorphism u D .a; b/ satisfying det a
¤ 0.
Proof. Let the symplectic map u.; / satisfy the assumptions of the proposition.
We define the 1-form on R
4n
in the coordinates .;;x;y/ 2 .R
n
/
4
by
D
n
X
j D1
y
j
dx
j
C
j
d
j
:
Then,
d D
n
X
j D1
dy
j
^ dx
j
n
X
j D1
d
j
^ d
j
;
which is the difference of the canonical symplectic forms !
0
on the subspaces
f.x; y/gand f.; /gof dimension 2n. We introduce the injections i; j W R
2n
! R
4n
by
i.x;/ D
˛.x; /; ; x; ˇ.x; /
;
j.;/ D
; ; a.; /; b.; /
;
and the local diffeomorphism ' of R
2n
by
'.x; / D .; / D
˛.x; /;
:
Then i D j B ' D .Id;u/B ' and consequently
d.i
/ D i
d
D '
.j
d/
D '
.u
!
0
!
0
/:
Since u is symplectic, u
!
0
!
0
D 0 and hence d.i
/ D 0 and we find by the
Poincaré lemma a function W D W.x;/ satisfying i
D dW.x;/. Explicitly,
n
X
j D1
ˇ
j
.x;/dx
j
C ˛
j
.x;/d
j
D
n
X
j D1
W
x
j
.x;/dx
j
C W
j
.x;/d
j
:
Consequently, ˛ D W
and ˇ D W
x
as claimed. Moreover, W
x
D ˛
x
D a
1
is
invertible.
The converse follows by reversing all the steps. If the given function W satisfies
det W
x
¤ 0 and if ˛ D W
and ˇ D W
x
then i
D dW and hence d.i
/ D
d
2
W D 0 and therefore u
!
0
!
0
D 0, so that the map u is symplectic.
220 Chapter V. Hamiltonian vector fields and symplectic diffeomorphisms
The function W.x;/ is called a generating function of the symplectic map u.
Example. The identity mapping u D Id W .; / 7! .x; y/ D .; / satisfies a
D
Id and is generated by the function W.x;/ Dhx; i.
Thus, any function W.x; Dhx; iCw.x;/ with a function w which is small
along with its first and second derivatives defines a symplectic local diffeomor-
phism near the identity map. We can use the function w to parametrize symplectic
diffeomorphisms near the identity. The symplectic rotation
x D ;
y D;
satisfies a
D 0, so that PropositionV.34 is not applicable. However, using a
D Id,
we can construct another generating function V D V.;x/, which represents the
rotation implicitly by
DV
.; x/;
y D V
x
.; x/;
and satisfies det V
x
¤ 0. This is proved analogously to Proposition V.34 by
using instead of the 1-form
0
D
P
n
j D1
y
j
dx
j
j
d
j
. For every symplectic
diffeomorphism there is some generating function representing it implicitly as we
shall show next.
Definition. The group of elementary symplectic matrices is the subgroup of Sp.n/
generated by the following symplectic linear maps .; / 7! .x; y/.
(i) The rotation
x
1
D
1
;
x
j
D
k
;2 j n;
y
1
D
1
;
y
j
D
k
;2 j n;
of the symplectic .e
1
;f
1
/ plane.
(ii) The permutations of the coordinates that leave the splitting R
n
˚R
n
invariant,
x
j
D
.j/
;
y
j
D
.j/
;
where 2 S
n
is a permutation.
All these mappings are symplectic and can be represented by symplectic matrices
which we abbreviate by E.