Zehnder E. Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities
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V.9. Generating functions of symplectic maps in R
2n
221
Proposition V.35. If U 2 Sp.n/ is a linear symplectic map of .R
2n
;!
0
/, there
exists an elementary symplectic transformation E such that
UE D
AB
CD
with A 2 R
nn
invertible:
Proof. We write the matrix U in block form as U D
AB
CD
with respect to R
n
˚R
n
.
Since U is regular, the .2n n/-matrix .A; B/, which is generated by writing the
two matrices side by side, has rank n so that the matrix has n linearly independent
column vectors fv
1
;:::;v
n
gμF. We would like to move them to the left by
means of elementary transformations E, so that
UE D
A
0
B
0
C
0
D
0
with A
0
consisting of linearly independent column vectors, so that it is invertible.
We permute the columns of U by multiplying from the right with matrices E.
We denote by a
j
and b
j
the j -th column vector of A respectively of B. In view
of the definition of the group of elementary transformations we can reach our goal
if F can be chosen in such a way that
a
j
2 F H) b
j
… F and b
j
2 F H) a
j
… F
for all 1 j n. Because with U also U
T
is symplectic, it follows that AB
T
is
a symmetric matrix. Therefore the following lemma due to C. de Boor implies the
proposition.
Lemma V.36. Let A; B 2 R
nn
be matrices having the column vectors a
j
respec-
tively b
j
for 1 j n.IfAB
T
D BA
T
and if r is the rank of the .2n n/-matrix
.A; B/, then there exists a set F Dfv
1
;:::;v
r
g of r linearly independent column
vectors of .A; B/ satisfying
a
j
2 F H) b
j
… F
and
b
j
2 F H) a
j
… F
for all 1 j n, so that every index j appears at the most once in F.
Proof. If r
A
is the rank of A we choose r
A
linearly independent column vectors
a
j
of A which span the image of A. Let J f1;:::;ng be the set of the cor-
responding indices j , so that Im A D spanfa
j
j j 2 J g. We shall show that
Im.A; B/ spanfa
j
;b
s
j j 2 J;s … J g. Then
dim spanfa
j
;b
s
j j 2 J;s … J gdim Im.A; B/ D r
222 Chapter V. Hamiltonian vector fields and symplectic diffeomorphisms
and the lemma follows. Due to
Im.A; B/ D Im A C Im B D spanfa
j
j j 2 J gCIm B
we only have to show that
Im B spanfa
j
;b
s
j j 2 J;s … J g:
In order to do this we choose a basis c
1
;:::;c
n
of R
n
satisfying ha
j
;c
k
iDı
jk
for
all j; k 2 J .Ifj 2 J , the vector A
T
c
j
has the components
.A
T
c
j
/
s
D
´
1; s D j 2 J;
0; s 2 J; s ¤ j;
and hence
B.A
T
c
j
/ D b
j
C
X
s…J
.A
T
c
j
/
s
b
s
:
We see that b
j
2 spanfb
s
j s … J gCIm.BA
T
/ for j 2 J . From the assumption
AB
T
D BA
T
we conclude Im.BA
T
/ Im A and hence,
b
j
2 spanfb
s
j s … J gCIm A; j 2 J:
Consequently Im B D spanfb
j
j 1 j ng is contained in spanfa
j
;b
s
j j 2
J;s … J g and the lemma is proved.
By applying the proposition to a map u D .a; b/ whose linearization is equal
to du D U at some point, we achieve for u B E D .a
0
;b
0
/ the desired condition
det a
0
¤ 0 and have proved the following statement.
Corollary V.37. Let u be a symplectic map on R
2n
. Using a suitable elementary
symplectic transformation E, the composition u B E can locally be represented by
a generating function as in Proposition V.34.
As an application we shall determine the most general symplectic map u D
.a; b/ for which the first part
x D a./;
is prescribed, satisfies det a
¤ 0 and a is independent of . We represent the map
u by means of the generating function V.;y/by
x D V
y
.; y/;
D V
.; y/:
From the assumption a./ D x D V
y
.; y/ we deduce that V has the form
V.;y/ Dha./; yiv./;
V.9. Generating functions of symplectic maps in R
2n
223
with an arbitrary function v depending only on . From the second equation we
therefore deduce
D V
D a
T
y v
;
so that we obtain for the symplectic map u the explicit representation
x D a./;
y D .a
T
/
1
C v
./
:
This is the most general symplectic map compatible with the map x D a./ above.
Example. The most general symplectic transformation of R
2n
, which is the identity
on the configuration space, namely a D Id, has the form x D and
y D C v
./;
with a function v W R
n
! R.
We assume next that the map a D .a
1
;:::;a
n
/ from R
2n
into R
n
depends
on and on and look for conditions under which it can be extended to a local
symplectic diffeomorphism u D .a; b/ of R
2n
. That such an extension does exist, if
the functions a
j
are independent and in involution, is the statement of the following
theorem which goes back to Liouville.
Theorem V.38 (Liouville). We consider n functions a
j
D a
j
.; / on .R
2n
;!
0
/
whose Poisson brackets vanish, so that
fa
i
;a
j
gD0; 1 i; j n:
We define the map a D .a
1
;:::;a
n
/ and assume that the matrix da D .a
;a
/ has
rank n. Then there exists a symplectic local diffeomorphism u D .a; b/ of the form
x D a.; /;
y D b.; /;
which extends the map x D a.; /.
Proof. Since .a
;a
/ has full rank, we can, after applying an elementary symplectic
transformation, assume that det a
¤ 0. Consequently, the equation x D a.; /
can be solved for D ˛.x;/ and we shall show next that the matrix ˛
is symmet-
ric. If F;G are real-valued functions, then the Poisson bracket in the symplectic
standard space .R
2n
;!
0
/ looks as follows:
fF;GgDhJX
F
;X
G
iDhJ rF;rGiDG
F
T
G
F
T
:
Thus the condition fa
i
;a
j
gD0 for all i; j becomes
a
a
T
a
a
T
D 0:
224 Chapter V. Hamiltonian vector fields and symplectic diffeomorphisms
Multiplication with a
1
from the left-hand side and with .a
T
/
1
from the right-hand
side leads to
a
T
.a
T
/
1
a
1
a
D 0;
so that the matrix a
1
a
is symmetric. Differentiatingthe identity D ˛
a.; /;
we obtain 1 D ˛
x
a
and 0 D ˛
x
a
C˛
and therefore the matrix ˛
D˛
x
a
D
a
1
a
is indeed symmetric.
Explicitly,
@˛
i
@
j
D
@˛
j
@
i
, and consequently the vector field 7! ˛.x;/ is a
gradient vector field and there exists a function W.x;/ such that
˛.x; / D W
.x; /:
This function satisfies W
x
D ˛
x
D a
1
and hence det W
x
¤ 0. In view of
Proposition V.34 the function W generates a local symplectic map u which extends
the map x D a.; / and the proof is complete.
CorollaryV.39. Assume that the nC1 functions a
0
;a
1
;:::;a
n
on .R
2n
;!
0
/ satisfy
fa
i
;a
j
gD0; 0 i; j n;
and assume that the matrix .a
;a
/ has rank n where the map a is defined by
a D .a
1
;:::;a
n
/. Then the function a
0
can be written locally as a function of
a
1
;:::;a
n
, in other words, there exists a function f such that
a
0
.; / D f
a
1
.;/;:::;a
n
.; /
:
In particular, there exist at the most n linearly independent functions whose Poisson
brackets all vanish.
Proof. In view of Liouville’s theorem the map x D a.; / can be extended to
a symplectic map u D .a; b/ W .; / 7! .x; y/. Because its Jacobi-determinant
is equal to 1, we can invert u locally. The inverse map u
1
is again symplectic
and satisfies a B u
1
.x; y/ D x. Next we define the function f by f.x;y/ ´
a
0
B u
1
.x; y/ and show that f D f.x/ is independent of the variable y. It then
follows that
a
0
.; / D a
0
B u
1
.x; y/ D f.x/ D f .a.; //;
as we want to prove. In order to show
@
@y
j
f D 0 for all 1 j n, we take the
coordinate functions x 7! x
j
on R
2n
and obtain for the Poisson brackets ff; x
j
gD
@
@y
j
f . The Poisson brackets vanish, since in view of Proposition V.29,
ff; x
j
gDfa
0
B u
1
;a
j
B u
1
gDfa
0
;a
j
gBu
1
D 0;
because u
1
is symplectic and fa
0
;a
j
gD0, by assumption. We have shown that
f only depends on the variable x and so the proof is complete.
V.10. Integrable systems, action–angle variables 225
V.10 Integrable systems, action–angle variables
Integrable systems are a very special class of Hamiltonian systems. They are charac-
terized by the property of possessing sufficiently many integrals, so that the problem
of solving the differential equation becomes essentially trivial. For general vector
fields on R
m
this would require m 1 integrals, for a Hamiltonian vector field on
R
2n
however, n integrals are sufficient. We begin by recalling the definition of an
integral.
Definition. The smooth function F W M ! R is an integral of the vector field X
on the manifold M ,if
dF .x/ X.x/ D 0; x 2 M
and dF .x/ ¤ 0 for all x 2 M .
The integral F is constant along every orbit of the vector field X, because for
the flow '
t
of X it follows using the chain rule that
d
dt
F.'
t
.x// D dF .'
t
.x//
d
dt
'
t
.x/ D dF .'
t
.x// X.'
t
.x// D 0:
In the case of a Hamiltonian vector field X
H
on a symplectic manifold, the function
F is an integral, if and only if
fH; F g.x/ D 0
and dF .x/ ¤ 0 for all x 2 M .
Definition. A Hamiltonian vector field X
H
is called integrable on the subset D
M of a symplectic manifold .M; !/ of dimension 2n, if it possesses n integrals
F
j
2 C
1
.D/ for 1 j n having the following three properties on D.
(i) dF
1
;:::;dF
n
are linearly independent,
(ii) fH; F
j
gD0,
(iii) fF
j
;F
k
gD0,
for all j D 1;:::;nand k D 1;:::;n.
It is important to observe that we require these properties in the large, since
locally, in the neighborhood of a point p 2 M at which X
H
.p/ ¤ 0 every Hamil-
tonian vector field X
H
is integrable. As a rule one therefore requires of the flow '
t
of X
H
that '
t
.D/ D D holds true for all t.
Example. We consider for an open set U R
n
the symplectic manifold
M D T
n
U
where T
n
is the n-torus R
n
=Z
n
. We make use of the coordinates .x; y/ 2 R
n
R
n
in the covering space. The symplectic form is induced from the standard form !
0
226 Chapter V. Hamiltonian vector fields and symplectic diffeomorphisms
of R
2n
.IfH is a function on M which is independent of the x-coordinates of the
torus so that
H D H.y
1
;:::;y
n
/; y 2 U;
then the n functions F
j
on M , defined by
F
j
.x; y/ D y
j
;1 j n;
satisfy the properties (i) to (iii). The Hamiltonian equations of X
H
can be solved
right away, explicitly, and for all times. Indeed, the system
´
Px D
@
@y
H.y/;
Py D
@
@x
H.y/ D 0
has the solutions
x.t/; y.t/
D
x C t !.y/; y
for all t 2 R and all initial conditions .x; y/ 2 T
n
U , where
!.y/ ´
@
@y
H.y/ 2 R
n
are the so-called frequencies of the torus T
n
fyg. We see that every torus T
n
fyg
is invariant under the flow of the vector field. Moreover, on every torus the flow is
a Kronecker flow.
The example is typical and we shall see that many integrable systems can be
brought into this form by choosing suitable coordinates. In physics the variables
.x; y/ 2 T
n
U are called action–angle variables. The last term is related to the
action integral
Z
n
X
sD1
y
s
dx
s
over suitable closed curves T
n
U . If we choose as a closed curve on the
torus T
n
fyg the special curve .t/ D x.t/ for 0 t 1 having the coordinates
x
j
.t/ D t and x
s
.t/ D 0 for s ¤ j , then
Z
n
X
sD1
y
s
dx
s
D
Z
1
0
y
j
dt D y
j
:
Hence, the variables y
j
are simply obtained as action integrals over suitable closed
curves on the manifolds fF
j
.x; y/ D const.g.
If X
H
is an integrable system on M possessing the n integrals F
j
W M ! R for
j D 1;:::;n, we define the map F W M ! R
n
by
F.x/ D
F
1
.x/;:::;F
n
.x/
2 R
n
:
V.10. Integrable systems, action–angle variables 227
From property (i) it follows that the level sets
N
c
´fx 2 M j F.x/ D cgM
for given constants c 2 R
n
are submanifolds M of dimension n. The Hamiltonian
vector fields X
F
j
are linearly independent and, due to property (iii), that is, due
to dF
j
.X
F
k
/ D 0 for all i and j , they span the tangent space T
x
N
c
at the point
x 2 N
c
,
T
x
N
c
D spanfX
F
j
.x/ j 1 j ng:
From !.X
F
j
;X
F
k
/ D 0 it follows in addition that !.v; w/ D 0 for all pairs of
vectors v; w 2 T
x
N
c
. Therefore T
x
N
c
is a Lagrangian subspace of the symplectic
vector space T
x
M . The manifold N
c
is consequently called a Lagrangian manifold.
The assumption (ii) requires that 0 D !.X
H
;X
F
j
/ D dF
j
.X
H
/ for all j D
1;:::;n, so that X
H
is tangential to all submanifolds N
c
and the flow of X
H
leaves
every submanifold N
c
invariant.
Under an additional topological assumption on N
c
the flow of X
H
can be deter-
mined explicitly and for all times, as the following theorem of Arnold–Jost shows.
This theorem goes back to V. I. Arnold in the special case M D R
2n
. Arnold re-
quired an additional assumption which, however, is superfluous, as R. Jost [57] has
proved in the general case of a symplectic manifold .M; !/.
Theorem V.40 (Arnold–Jost). Let .M; !/ be a symplectic manifold of dimension
2n. We assume, moreover, that there exist n functions F
j
W M ! R satisfying the
properties (i) and (iii) in the definition of an integrable system in every point of M
and that one of the level sets, as for instance N
0
D F
1
.0/ M , is compact and
connected. Then the following holds true:
(a) N
0
is an embedded torus T
n
.
(b) There exists an open neighborhood U of N
0
in M which can be described
by action and angle variables .x; y/ in the following way. There exist two
open neighborhoods D
1
and D
2
of 0 in R
n
having the following properties.
There exists a diffeomorphism W D
2
! D
1
satisfying .0/ D 0.Ifx D
.x
1
;:::;x
n
/ 2 R
n
are the coordinates of the torus T
n
D R
n
=Z
n
and if
y D .y
1
;:::;y
n
/ are the coordinates on D
1
R
n
, then there exists a
diffeomorphism
W T
n
D
1
! U D
[
c2D
2
N
c
\ U M;
which is symplectic,
! D
n
X
j D1
dy
j
^ dx
j
D !
0
;
228 Chapter V. Hamiltonian vector fields and symplectic diffeomorphisms
and which satisfies
./B F B .x; y/ D y:
In particular, induces diffeomorphisms T
n
f0g!N
0
and T
n
fyg!
N
c
\ U , where y D .c/.
CorollaryV.41. An integrable Hamiltonian vector field X
H
possessing the integrals
F
j
for 1 j n, as in Theorem V.40, is, by the symplectic diffeomorphism ,
transformed into the following Hamiltonian system on .T
n
D
1
;!
0
/,
H B .x; y/ D h.y/;
where the Hamiltonian function h only depends on the action variables y and not
on the angle variables x.
Proof. Since is symplectic, it follows from Proposition V.29 for the functions
f
j
D F
j
B W T
n
D
1
! R that
fh; f
j
gDfH B ;F
j
B gDfH; F
j
gB D 0;
so that the functions f
j
are integrals of the Hamiltonian h. Therefore, due to ./
also the coordinate functions y
j
are integrals of h and consequently,
0 Dfh; y
j
gD
@
@x
j
h;
so that the function h is independent of the variables x
j
for all 1 j n,as
claimed.
On
1
.U / D T
n
D
1
the Hamiltonian system looks extremely simple,
X
h
W
´
Px D
@h
@y
.y/;
Py D 0
as in the above example. The solutions are given by
x.t/; y.t/
D
x.0/ C t!.y.0//; y.0/
:
Geometrically every torus T
n
fygT
n
D
2
is invariant under the flow '
t
of
the Hamiltonian vector field X
h
and the flow on the torus T
n
fygis the Kronecker
flow
'
t
j
T
n
fyg
W x 7! x C t!;
having the frequencies ! D
@h
@y
.y/ 2 R
n
depending on the torus T
n
fyg.
V.10. Integrable systems, action–angle variables 229
Proof of Theorem V.40. The proof proceeds in four steps. We first show that N
0
is an n-dimensional torus. In a second step we introduce convenient symplectic
coordinates .x; y/ in the neighborhood of a chosen point p 2 N
0
, using Liouville’s
Theorem V.38. In a third step, these coordinates are extended to an open neigh-
borhood of N
0
by means of the flows of the Hamiltonian vector fields X
F
j
of the
integrals. Finally, the normalization of the periods requires a further symplectic
transformation, and this is the fourth and last step of the proof.
(1) We denote by '
s
j
j
the flow of the Hamiltonian vector field X
F
j
, for 1
j n. Due to ŒX
F
j
;X
F
k
D X
fF
j
;F
k
g
and fF
j
;F
k
gD0 the vector fields X
F
j
commute, hence in view of Proposition V.20 their flows also commute. For s D
.s
1
;:::;s
n
/ 2 R
n
we introduce the family of symplectic mappings '
s
W M ! M
as the composition
'
s
D '
s
1
1
B '
s
2
2
BB'
s
n
n
;
wherever it is defined. It satisfies '
0
D Id and
'
sCt
D '
s
B '
t
for all s; t 2 R
n
in the domain of definition. Moreover, since the functions F
j
are
commuting integrals,
F.'
s
.x// D F.x/; x 2 M
for all s 2 R
n
in the domain of definition, so that the maps '
s
leave the level sets
N
c
invariant. We now choose a point p 2 N
0
. Because N
0
M is compact and
invariant, the symplectic maps '
s
.p/ do exist for all parameters s 2 R
n
and we can
define the action A of R
n
on N
0
by
AW R
n
! N
0
;s7! '
s
.p/:
The tangent map dA.s/W R
n
! T
'
s
.p/
N
0
maps the vectors e
j
2 R
n
onto the
tangent vectors X
F
j
.'
s
.p// 2 T
'
s
.p/
N
0
. These vectors are linearly independent,
because the differentials dF
j
are linearly independent and therefore span the whole
tangent space. Hence the map A is a local diffeomorphism whose image in N
0
is closed as well as open. By assumption, the manifold N
0
is connected, hence
it follows that A.R
n
/ D N
0
, so that the map A is surjective. However, the map
A is not injective, otherwise R
n
would be compact. Consequently, there exists a
non-trivial isotropy group
Dfs 2 R
n
j '
s
.p/ D pg
which is necessarily a discrete subgroup of R
n
, hence a lattice. (More details can be
found, for example, in [21] by L. Conlon.) The lattice is generated by the linearly
independent vectors
1
;:::;
d
2 R
n
, so that
D
˚
D
P
d
j D1
n
j
j
j n
j
2 Z
:
230 Chapter V. Hamiltonian vector fields and symplectic diffeomorphisms
From '
sC
.p/ D '
s
.p/ for all s 2 R
n
and 2 it follows that the map A
induces a diffeomorphism between R
n
= and N
0
. Since due to the assumption,
N
0
is compact, we conclude that d D n so that N
0
Š R
n
= is an n-dimensional
torus, as claimed in the theorem.
(2) In the second step we shall introduce in the neighborhood of the chosen
point p 2 N
0
suitable symplectic coordinates. By adding constants to the integrals
we may assume that F.p/ D 0.
Lemma V.42 (Liouville coordinates). There exists an open neighborhood W
.M; !/ of the distinguished point p and a diffeomorphism W W ! V onto an
open neighborhood of 0 2 R
2n
such that
(i) .p/ D 0,
(ii)
!
0
D !,
(iii) F.
1
.x; y// D y,
where .x; y/ are the symplectic coordinates in .R
2n
;!
0
/.
Proof [Theorem of Darboux, theorem of Liouville]. If
0
W W
0
M ! R
2n
is a
symplectic chart satisfying
0
.p/ D 0, as in Darboux’s theorem and if f.x;y/ D
F.
1
0
.x; y// are the integrals in the chart, then f .0/ D 0 and in view of Liouville’s
Theorem V.38, there exists a symplectic extension ' W R
2n
! R
2n
of the form
'.x; y/ D
g.x;y/;f.x;y/
. By translation we may assume that '.0/ D 0. Then
f.x;y/ D
2
B '.x; y/ where
2
is the projection
2
.x; y/ D y. For the inverse
'
1
it follows that
f.'
1
.x; y// D
2
B ' B '
1
.x; y/ D
2
.x; y/ D y:
Therefore ´ ' B
0
is the desired diffeomorphism.
Corollary V.43. In the Liouville coordinates the flow '
s
looks as follows:
'
s
B
1
.x; y/ D
1
.x C s; y/
for s 2 R
n
,if.x; y/ and .x C s; y/ in V .
Proof. In the Liouville coordinates the Hamiltonian function F
j
is represented by
the function
f
j
.x; y/ D F
j
.
1
.x; y// D y
j
:
Therefore, the flow of the transformed Hamiltonian vector field X
f
j
at time s
j
is
given by
.x; y/ 7! .x C s
j
e
j
;y/;
as claimed.