Zehnder E. Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities

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VII.4. The functional analysis of the action functional 281

Hölder’s inequality,

j ¤0

j2tJ

j ¤0

jj j

s

jj j



j ¤0

jj j

2s

j ¤0

jj j

D ckxk

;

where c

j ¤0

jj j

2s

< 1converges in view of the assumption 2s > 1. 

Similarly, one proves for s>

C r, where r 2 N is a positive integer, the

implication

x 2 H

H) x 2 C

; R

and, moreover, the existence of a constant c>0such that

kxk

 ckxk

for all x 2 H

In view of Proposition VII.19 the embedding

j W H

! L

Š H

is a compact linear operator and we consider its adjoint operator



W L

! H

;

which satisﬁes

.j.x/; y/

Dhx; j



.y/i

for all x 2 H

and all y 2 L

In order to recall the deﬁnition of the adjoint operator we take an element y 2 L

and observe that the functional f

W x 7! .j.x/; y/

is continuous on the space

, because, by Cauchy–Schwarz,

j.j.x/; y/

jkj.x/k

kyk

kxk

kyk

;

for all x 2 H

, so that kf







kyk

. Thus a well-known theorem of

F. Riesz guarantees a unique element y



2 H

such that the continuous functional

is represented by the scalar product as

.x/ D .j.x/; y/

Dhx; y



for all x 2 H

;

282 Chapter VII. Symplectic invariants

moreover,



Dkf

kkyk

The adjoint operator j



2 L.L

/ is now deﬁned by



.y/ D y



and satisﬁes kj



.y/k

kyk

for all y 2 L

Proposition VII.21. j



/  H

and



.y/k

kyk

for all y 2 L

Proof. We write the deﬁnition of the adjoint operator explicitly and obtain, by the

theorem of Plancherel,

k2Z

iD.j.x/; y/

Dhx; j



.y/i

Dhx



.y/

iC2

k2Z

jkjhx



.y/

for all x 2 H

 L

and y 2 L

. Hence,



.y/

D y

;



.y/

2jkj

;k¤ 0:

Therefore,



.y/.t / D y

k¤0

k2tJ

2jkj

;y2 L

;

and by deﬁnition of the norm in H

we obtain the estimate kj



.y/k

kyk

,as

claimed. 

Proposition VII.22. The adjoint operator j



W L

! H

is a compact linear

operator.

Proof. In view of Proposition VII.21 the map factorizes through the Sobolev space

, so that



W L

! H

! H

;

where, by Proposition VII.19, the embedding I W H

! H

is compact. The

composition of a continuous (hence bounded) operator with a compact operator is

again a compact operator, so that the proposition is proved. 

VII.4. The functional analysis of the action functional 283

Next we extend the second part of the functional ˆ which is given by

b.x/ ´

H.x.t//dt; x 2 :

Since H.z/  0 near z D 0 and H.z/ D Q.z/ for jzj large, we have for a

sufﬁciently large constant M the estimates

jH.z/jM jzj

;

H.z/jM

for all z 2 R

. Therefore, the function b is deﬁned for all x 2 L

; R

/ and,

in particular, for every x 2 E  L

. If we consider b as a function of x 2 L

,we

denote it by

b, so that

b.x/ D

H.x.t//dt; x 2 L

Using the inclusion j W E ! L

, the function b W E ! R is the composition

b.x/ D

b.j.x//; x 2 E:

We distinguish between b and

b, because the compactness of the embedding j will

play a role later on.

Claim. We claim that the function

b W L

! R is differentiable and its deriva-

tive is the bounded linear map d

b.x/.h/ D

hrH.x/;hidt D .rH.x/;h/

Therefore, the L

-gradient is equal to r

b.x/ DrH.x/.

Proof. We start with pointwise identities. Using the chain rule we compute

H.z C /  H.z/ D

H.z C t/dt

hrH.z C t/; idt

DhrH.z/;iC

hrH.z C t/rH.z/;idt;

and hence,

./H.zC / D H.z/ ChrH.z/;iC

hrH.z C t/rH.z/;idt

284 Chapter VII. Symplectic invariants

for all z; 2 R

. Due to the bound jd

H.z/jM on R

it follows by means

of the mean value theorem that

jhrH.z C t/rH.z/;ijDjdH.z C t/ dH.z/j

 sup

a2R

H.a/jjtj

 M jj:

Therefore, the integral in ./ can be estimated for all z 2 R

by M jj

.Ifx 2 L

then also the composition

rH.x/ DrH.x.t//

is an element in L

, because jrH.z/jC jzj for all z 2 R

.Now,ifx; h 2 L

we insert z D x.t/ and  D h.t/ into the identity ./ and integrate over t to obtain

b.x C h/ D

b.x/ C

hrH.x/;hidt C R.x; h/

with a rest term R satisfying the estimate

jR.x; h/jM khk

Recalling the deﬁnition of a derivative, the function

b W L

! R is differentiable at

the point x 2 L

and its derivative d

b.x/ 2 L.L

; R/ is given by the formula

b.x/.h/ D

hrH.x/;hidt

D .rH.x/;h/

From the deﬁnition of the gradient in L

we deduce that r

b.x/ DrH.x/,as

claimed. 

We recall that, using the continuous embedding j W E ! L

, the map b W E ! R

is deﬁned as the composition

b.x/ D

b.j.x//; x 2 E:

By the chain rule its derivative is equal to

db.x/.y/ D d

b.j.x// j.y/



b.j.x//;j.y/





b.j.x//; y

for all x; y 2 E. Hence, by deﬁnition of the gradient in the Hilbert space E,

rb.x/ D j



b.j.x// D j



rH.j.x//:

VII.4. The functional analysis of the action functional 285

Proposition VII.23. The map b W E ! R is continuously differentiable. The gra-

dient rb W E ! E is continuous and compact. Moreover, for all x; y 2 E we have

the estimates

krb.x/ rb.y/k

 M kx  yk

;

jb.x/jM kxk

with a constant M>0.

Proof. From the estimate M  sup

z2R

H.z/j we obtain by the mean value

theorem the estimate

krH.x/ rH.y/k

 M kx  yk

for all x; y 2 L

Therefore, the L

-gradient rH W L

! L

maps bounded sets onto bounded sets.

Since j



W L

! E is compact, the E-gradient rb D j



BrH B j W E ! E is a

compact map. Moreover, using kk

kk

and Proposition VII.21,

we can estimate

krb.x/ rb.y/k

kj



.rH.x/ rH.y//k

kj



.rH.x/ rH.y//k

krH.x/ rH.y/k

 M kx  yk

For M large enough the estimate jH.z/jM jzj

holds true for all z 2 R

,so

that jb.x/j

jH.x.t//jdt  M

jx.t/j

dt D M kxk

, as claimed in the

proposition. 

Summarizing the discussion so far, we have extended the action functional

ˆW  ! R from the space of smooth loops to the function ˆ W E ! R, still

denoted by the same letter, and deﬁned on the Hilbert space E   by

ˆ.x/ D a.x/  b.x/; x 2 E:

The function ˆ W E ! R is continuously differentiable and its E-gradient is equal

rˆ.x/ D x

 x



rb.x/ 2 E

for x D x

C x



2 E. By Proposition VII.23 this gradient is globally

Lipschitz-continuous. We are interested in classical solutions of the Hamiltonian

equation. It is, therefore, crucial to observe that a critical point of the function

ˆW E ! R is not simply an element in E, which might not even be a continuous

function, but is actually a smooth periodic solution of the Hamiltonian vector ﬁeld

having period equal to 1. Indeed, the following regularity theorem holds true.

286 Chapter VII. Symplectic invariants

PropositionVII.24 (Regularity theorem). If x 2E is a critical point of ˆ W E ! R,

so that

rˆ.x/ D 0;

then x belongs to C

; R

/ and solves the Hamiltonian equation

Px.t/ D J rH.x.t//; t 2 R:

Proof [Fourier series, Proposition VII.20]. (1) We ﬁrst prove that x is a continuous

function. If x 2 E satisﬁes rˆ.x/ Dra.x/ rb.x/ D 0, then

hra.x/; viDhrb.x/; vi for all v 2 E;

and hence, using the explicit expressions for the gradients,

h.P

 P



/x; viD



rH.j.x//;v



rH.j.x//;j.v/



Because x D x.t/ and v D v.t/ are elements of L

./ h.P

 P



/x; viD

hrH.x/;vidt:

Next, we develop x 2 L

and rH.x/ 2 L

into their Fourier series

x D

k2Z

k2tJ

; rH.x/ D

k2Z

k2tJ

;

where the Fourier coefﬁcients are deﬁned by

k2tJ

x.t/dt; a

k2tJ

rH.x.t// dt:

If v

are the Fourier coefﬁcients of v, we deduce from ./ the identity

./

k2Z

h2kx

k2Z

by using the deﬁnition of the scalar product h; i

and the theorem of Plancherel.

Inserting the test functions v.t/ ´ e

k2tJ

Nv 2 E for Nv 2 R

and k 2 Z into the

identity ./, we obtain h2kx

; NviDha

; Nvi for all Nv 2 R

and conclude that

2kx

D a

;k2 Z:

In particular,

0 D a

rH.x.t//dt:

VII.4. The functional analysis of the action functional 287

Consequently, by Plancherel’ s theorem,

jkj



DkrH.x//k

< 1;

so that x is an element of H

and hence by PropositionVII.20 a continuous function.

(2) Next, we prove that x is continuously differentiable. In view of step (1) the

function rH.x.t// is continuous, since H is smooth. The continuously differen-

tiable function  W Œ0; 1 ! R

, deﬁned by

.t/ D

J rH.x.s//ds;

is periodic of period 1, hence belongs to the space C

; R

/. Indeed,

.1/ D J

rH.x.s//ds D Ja

D 0

so that .1/ D .0/ D 0. The derivative of  satisﬁes

.t/ D J rH.x.t//; t 2 R:

In order to determine the Fourier coefﬁcients 

of  for k ¤ 0 we compute,

using partial integration and  .1/ D 0 D .0/,



k2tJ

.t/dt

D .k2J/

1

k2tJ

.t/



.k2J /

1

k2tJ

.t/dt

D

.k2J /

1

k2tJ

J rH.x.t// dt

k2

k2tJ

rH.x.t//dt

k2

D x

for all integers k ¤ 0. Due to the uniqueness of the Fourier coefﬁcients, .t/ D

x.t/ C c and so, using .0/ D 0,

.t/ D x.t/  x.0/:

We see that the function x is of class C

288 Chapter VII. Symplectic invariants

(3) Finally, we prove that x is a smooth function. In view of Px D

, the function

x solves the Hamiltonian equation

Px.t/ D J rH.x.t//:

The right-hand side of the Hamiltonian equation is of class C

. Therefore, also the

derivative on the left-hand side is of class C

, so that x is of class C

and hence

the right-hand side is of class C

, so that x is of class C

, and so on. Iteratively

using the Hamiltonian equation it follows that x 2 C

; R

/ is a smooth,

1-periodic solution of the Hamiltonian equation and the proof of Proposition VII.24

is complete. 

Next we show that the extended functional ˆW E ! R satisﬁes the P.S.-con-

dition. Here we shall use the dynamical behavior of the Hamiltonian system X

far away from 0 2 R

whose Hamiltonian function

H.z/ D Q.z/ D . C "/q.z/; jzjR

is a quadratic form, for a large R. The corresponding Hamiltonian vector ﬁeld

Pz D X

.z/ D J rQ.z/ for z 2 R

decomposes into n linear oscillators in the

symplectic planes spanfe

g, so that explicitly, in the symplectic coordinates

z D .x; y/,

D c

;

Dc

for 1  j  n. Recalling

q.z/ D x

C y

nD2

C y

for z D .x; y/ 2 R

 R

, with a large integer N , we read off that the constants

are equal to

D 2. C "/;

D 2. C "/=N

;2 j  n:

The minimal periods of the oscillators are T

D 2=c

. Since " is positive and

small, we see that non-constant periodic solutions of period 1 do not exist! This

fact is one of the reasons for having chosen our special extension of the Hamiltonian

function from the ellipsoid onto the whole space R

. It will be crucial in the proof

of the following proposition.

Proposition VII.25. Every sequence x

in E satisfying rˆ.x

/ ! 0 in E pos-

sesses a convergent subsequence. In particular ˆ W E ! R satisﬁes the P.S.-con-

dition.

VII.4. The functional analysis of the action functional 289

Proof. Explicitly, the condition rˆ.x

/ ! 0 in E reads as

./x

 x



rb.x

/ ! 0 in E:

(1) We ﬁrst assume that the sequence x

is bounded in E.

Since rb is compact (Proposition VII.23), a subsequence of rb.x

/ converges

in E. Due to ./ a subsequence of x

 x



also converges in E. Because the

sequence of constants x

2 R

is bounded, it possesses a convergent subsequence,

too. All in all, a subsequence of the sequence x

D x



converges in E

and the proposition is proved in this case.

(2) If x

is not bounded in E, then there exists a subsequence satisfying

k!1:

We shall show indirectly that this case never happens, so that step (1) demonstrates

the proposition. The sequence

satisﬁes ky

kD1. The following claim leads to the desired contradiction.

Claim. There exists a convergent subsequence

! y in E

whose limit y 2 E satisﬁes kykD1 and

 P



/y rb

.y/ D 0;

where b

.y/ ´

Q.y/ dt.

Postponing the proof of the claim we ﬁrst use it to ﬁnish the proof of Proposi-

tion VII.25. One concludes, as in the proof of Proposition VII.24, that the limit y

belongs to C

; R

/ and solves the Hamiltonian equation

Py.t/ D X

.y.t // D J rQ.y.t //:

Since kykD1, the solution y is a non-constant 1-periodic solution of the Hamil-

tonian vector ﬁeld Pz D X

.z/. However, in view of the above discussion such a

solution cannot exist, so that only case (1) occurs.

Proof of the claim. We multiply the sequence in ./ by

and obtain by means

of the explicit representation of the gradient rb.x

/ the converging sequence

./

rˆ.x

D .P

 P



 j



rH.x



! 0 in E:

290 Chapter VII. Symplectic invariants

Since jrH.z/jM jzj for a constant M>0,

krH.x

 M

 CM

D CM;

where C<1 is the norm of the embedding j W E ! L

. So, this sequence is

bounded in L

. Since the adjoint operator j



W L

! E is a compact operator

(Proposition VII.19), the sequence



rH.x



possesses a convergent subsequence in E. Consequently, due to the convergence

in ./, the sequence .P

P



has a convergent subsequence. Moreover, the

sequence P

of constants is bounded and so possesses a convergent subsequence.

All in all, there is a converging subsequence

! y in E

whose limit satisﬁes kykD1, since all the elements y

have norms equal to 1.

Because the embedding E  L

is continuous, the convergence

! y in L

follows. The functions H and Q agree outside of a ball, so that

jrH.z/ rQ.z/jK for all z 2 R

with a sufﬁciently large constant K>0. In addition, the map y 7!rQ.y/ W L

is linear and continuous, because Q is a quadratic form. Using these ingredients

we obtain the estimate



rH.x

rQ.y/





krH.x

/ rQ.x



rQ.x

rQ.y/





CkrQ.y

/ rQ.y/k

! 0

from which we conclude the convergence

rH.x

!rQ.y/ in L

Since j



W L

! E is continuous, the convergence j





rH.x



! j



.rQ.y//

in E follows. From ./ we now conclude that the limit y solves the equation

 P



/y  j



.rQ.y// D 0:

Now, arguing as in the computation of the gradient rb, we ﬁnd that rb

.y/ D



.rQ.y// is the E-gradient of the function b

. We have conﬁrmed the claim

and so the proposition is proved. 