Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Considering the fact that the orbit we have described

goes close to double resonances, this is the best

estimate one may hope for in view of the improved

Nekhoroshev stability estimates at resonances.

The idea is now well spread that the time of

diffusion is exponentially large. However, we point

out that, if it is indeed exponentially small as a

function of the parameter , it is only polynomially

small as a function of the second parameter ,aswas

first understood by P Lochak and proved in Bernard

(1996) using the variational method of U Bessi.

Conclusion

The theories of instability are developing in several

directions. One of them is to try to understand the

limits of stability, and to test to what extent the

stability results obtained so far are optimal. This

aspect has quickly developed recently, for example,

the optimal stability exponent a for convex systems

is almost known. Another direction is to try to give

a description of unstable orbits in typical systems.

This remains a widely open question.

Let us finally mention that the application of the

theories we have presented to concrete systems is

very difficult. One of the reasons is that the

estimates of the threshold 

of validity of Nekhor-

oshev and KAM theorems that can (painfully) be

obtained by inspection in the proofs are very bad,

and it is much too bad, for example, to think about

applications to the solar systems with the physical

values of the parameters.

See also: Averaging Methods; Hyperbolic Billiards; KAM

Theory and Celestial Mechanics; Separatrix Splitting;

Stability Problems in Celestial Mechanics; Stability

Theory and KAM; Weakly Coupled Oscillators.

Further Reading

Arnol’d VI (1964) Instability of dynamical systems with several

degrees of freedom. Soviet Mathematical Doklady 5: 581–585.

Arnol’d VI Mathematical Methods of Classical Mechanics.

Springer.

Bernard P (1996) Perturbation d’un hamiltonien partiellement

hyperbolique. Comptes Rendus de l’Academie des Sciences,

Serie I, 323.

Bessi U (1996) An approach to Arnold’s diffusion through the

calculus of variations. Nonlinear Analysis 26(6):

1115–1135.

Bourgain J and Kaloshin V (2005) On diffusion in high-

dimensional Hamiltonian systems. Journal of Functional

Analysis 229(1): 1–61.

Khinchin AI Mathematical Foundations of Statistical Mechanics.

Dover.

Lochak P (1992) Canonical perturbation theory via simultaneous

approximation. Russian Math. Surveys 47: 57–133.

Lochak P, Marco JP, and Sauzin D (2003) On the splitting of

invariant manifolds in multidimensional near-integrable

Hamiltonian systems. Mem. Amer. Math. Soc. 163(775).

Nekhoroshev NN (1977) An exponential estimate for the time of

stability of nearly integrable Hamiltonian systems. Russian

Math. Surveys 32: 1–65.

Niedreman L Hamiltonian stability and subanalytic geometry.

Ann. Inst. Fourier. (in press).

Po¨ schel J (1993) Nekhoroshev estimates for quasi-convex

Hamiltonian systems. Math. Z. 213: 187–216.

Hamilton–Jacobi Equations and Dynamical

Systems: Variational Aspects

A Siconolfi, Universita

di Roma ‘‘La Sapienza’’,

Rome, Italy

Overview

Given a continuous Hamiltonian H(x, p) defined on

the cotangent bundle of a compact boundaryless

manifold, where x and p are the state and the

momentum variable, respectively, and satisfying

suitable convexity and coercivity assumptions, we

consider the family of Hamilton–Jacobi equations

Hðx; DÞ¼a ½ 1

with a a real parameter. If, in addition, H is

assumed to be smooth, we also consider the

Hamilton’s equations

 ¼ H

ð; Þ;

 ¼H

ð; Þ½2

whose analysis is related to the variational problem

of minimizing the action functional

Lð;

Þdt ½3

among all Lipschitz–continuous or, equivalently,

continuous piecewise C

curves defined on I with

fixed end points. Here I is a compact interval and L,

the Lagrangian, is the Fenchel transform of H.A

‘‘conjugate’’ flow, named after Euler–Lagrange, is

636 Hamilton–Jacobi Equations and Dynamical Systems: Variational Aspects

also defined on the tangent bundle of the underlying

manifold.

A connection between [1] and [2] is provided by

the classical Hamilton–Jacobi method, which shows

that the graph of the differential of any regular, say

, global solution to [1] is an invariant subset for

the Hamiltonian flow. The drawback of this

approach is that such regular solutions do not exist

in general, even for very regular Hamiltonians.

However, for a ny continuous Hamiltonian a

distinguished value of the parameter a can be

detected, denoted by c and qualified, from now on,

as critical, for which there are a.e. subsolutions of

the corresponding Hamilton–Jacobi equations

enjoying some extremality properties. Note that

such functions can be equivalently defined as weak

solutions, in the viscosity sense, of [1] with a = c,or

as fixed points of the associated Lax–Oleinik

semigroup (see Fathi (to appear)). We do not give

these interpretations here to avoid any technicalities.

Even if they are just Lipschitz–continuous on the

whole underlying manifold, these extremal subsolu-

tions become of class C

, when restricted on a

special compact subset, the same for any of them,

say A, and the corresponding differentials coincide

on A. More genera lly, all critical subsolutions, that

is, the a.e. subsolutions to [1] with a = c, are

continuously differentiable on A. This regularity

property holds if H is at least locally Lipschitz–

continuous in both variables. When, in addition, the

Hamiltonian is smooth, so that the Hamiltonian

flow is defined, the graph of this common differ-

ential defined in A, denoted by

A, is an invariant set

for the flow, and is foliated by integral curves of [1]

possessing some global minimiz ing properties with

respect to the action functional.

The aim of this presentation is to give an

explanation of the previously described phe nomena

occurring at the critical level, and of some related

facts, using tools and arguments as simply as

possible. We propose a metric approach to the

subject and consider as central in our analysis a

family of distances, denoted by S

, for any a  c.We

emphasize that such distances can be defined for

only continuous Hamiltonians, and the qualitative

analysis of the critical subsol utions has an interest

independent from the dynamical applications.

Indeed, it can be used in other contexts such as in

homogenization problems, and the large-time beha-

vior of the viscosity solutions to the time-dependent

equation u

þ H(x, Du) ¼ 0.

The discovery of the critical value has a history

that reflects the dual character of the topics, which

has a dynamical as well as a partial differential

equation (PDE) interest.

It was probably Ricardo Man˜e´ who first focused his

attention on it, at the beginning of the 1980s, in

connection with the analysis of integral curves of the

Euler–Lagrange flow with some global minimizing

properties. The set, previously denoted by

A, has been

found and analyzed by Serge Aubry, in a purely

dynamical way, as the union of the supports of such

minimizing curves. On the other hand, John Mather

(1986) independently defined, in a more general

framework, a set, contained in the Aubry set, through

a weak approach that utilizes minimal probability

measures invariant with respect to the Euler–Lagrange

flow. The Mather set is actually the closure of the

union of the supports of such measures. We will follow

the approach of Aubry (see Fathi (2005b)), and will

not introduce the Mather’s measures.

In the viscosity solution theory, the critical value

has instead been introduced in a famous unpub-

lished paper of P L Lions, S R S Varadhan, and

G Papanicolaou (1987), in connect ion with some

periodic homogenization problems for Hamilton–

Jacobi equations. It is worth noticing that they

consider continuous Hamiltonian, defined on the

flat N-dimensional torus, without any convexity

assumption.

They define the critical value, and show the

existence of viscosity solutions to the critical equation

by means of an ergodic approximation, that is, by

considering the equation "u þ H(x, Du) = 0 and then

passing to the limit for " ! 0. The critical viscosity

solutions are used as correctors in the homogeniza-

tion. They do not perform any qualitative analysis,

and if such analysis can be done, and something

similar to the Aubry–Mather sets exists for noncon-

vex Hamiltonian this is still an important open

problem.

The two pieces of the picture were pasted together

by Fathi (1996) with his weak KAM theory (see

Contreras and Iturriaga (1999) and Fathi (2005a)

for a general treatment, where the relevance of the

extremal critical subsolutions ha s first been recog-

nized for the analysis of the dynamics, and the

Aubry–Mather sets have been characterized as a

regularity set for such subsolutions, as described

above). Evans and co-workers have been presently

using more general PDE methods in weak KAM

theory to address some integrability issues and to

find a quantum analog (see Evans and Gomes (2001,

2002) and Evans (2004)).

Critical Value and Extremal Subsolutions

We consider the family of Hamilton–Jacobi equa-

tions [1] defined, for simplicity, on the flat torus

= R

, endowed with the flat Riemannian

Hamilton–Jacobi Equations and Dynamical Systems: Variational Aspects 637

metric induced by the Euclidean metric on R

. The

tangent, as well as the cotangent bundle of T

will

be identified with T

 R

. All the results discussed

in the remainder of the paper are still true in any

compact boundaryless manifold, and some of them

also hold in noncompact manifolds. We require H to

be continuous in both variables, to satisfy the

coercivity assumption,

fðy; pÞ: Hðy; pÞag is compact for any a

and the following (strict) quasiconvexity conditions

for any x 2 T

, a 2 R:

fp: Hðy; pÞag is strictly convex

@fp: Hðy; pÞag¼fp: Hðy; pÞ¼ag

where @, in the above formula, indicates the

boundary. We denote by S

the (possibly empty)

set of the Lipschitz–continuous a.e. subsolutions to

[1]. They will be called in the sequel, for short, just

subsolutions. Due to the convex character of the

Hamiltonian and its continuity, the property of

being a subsolution, for some function u, can be

equivalently expressed by requiring the inequality

H(x, p)  a to hold for any x 2 T

and any p in the

(Clarke) generalized gradient @u(x), defined by

@uðxÞ¼cofp ¼ lim

Duðx

Þ:

differentiability point of u; lim

¼ xg

where co indicates the convex hull. Note that if this

set of weak derivatives reduces to a singleton at some

x, then the function u is strictly differentiable at x,

i.e., it is differentiable and Du is continuous at x.

By a strict subsolution to [1] we mean a

Lipschitz–continuous function w with ess sup

H 

(x,Dw(x)) < a. The property of being a (strict)

subsolution is not affected by addition of constants.

Moreover, the pointwise supremum (resp. infimum)

of any class of equibounded subsolution to [1] is

itself a subsolution, and S

is stable with respect to

the uniform convergence in T

The purpose of this section is to show that there is

a unique value c (the critical value) for which the

corresponding equation

Hðx; DuÞ¼c ½4

possesses subsolutions enjoying some extremality

properties. We, more precisely, call a subsolution u 2

maximal (resp. minimal) if for any open subset 

of T

and any Lipschitz–continuous function  with

u ¼  on @ and ess sup



Hðx; DðxÞÞ < a ½5

one has u   (resp. u  )in.

Any maximal (resp. minimal) subsolution u is

actually an a.e. solution of [1]. If, in fact,

H(x

, Du(x

)) < a for some differentiability point x

of u, then the function (x) = u(x

) þ Du(x

)(x 

)  "jx  x

jþ" (resp. (x) = u(x

) þ Du(x

)(x

) þ "jx  x

j")shouldsatisfy[5] for a suitable

choice of ">0 and of a neighborhood  of x

,and

so should violate the maximality (resp. minimality)

condition for u.

The previous argument can be easily adapted to

show something more general: if u is a maximal

(resp. minimal) subsolution then no subtangents

(resp. supertangents) to u at any y 2 T

can be local

strict subsolutions at y, that is, strict subsolutions in

some neighborhood of y.

The subtangency (resp. supertangency) condition

of a function  to u at a point x

means that x

is a

local minimizer (resp. maximizer) of u  .We

denote by D



u(x

) (resp. D

u(x

)) the sets made up

by the differentials of the C

-subtangent (resp.

supertangent) to u at x

. They are (possibly empty)

closed convex subsets of @u(x

). It is app arent that if

u(x

) 6¼;6¼ D



u(x

) then u is differentiable at x

and D

u(x

) = D



u(x

) = {Du(x

)}.

It is an immediate consequence of the previ ous

fact that no extremal subsolutions can exist in S

whenever [1] admits a strict subsolution, say , since

there are global minimizers and maximizers of u ,

for any u 2S

, because of the compactness of T

The function  is then subtangent and supertangent,

respectively, to u at such points.

The unique value we can look at for finding

extremal subsolutions is therefore

c ¼ inf a 2 R: S

6¼;

½6

The set on the right-hand side of [6] is nonempty

since the null function belongs to S

when a >

max

H(x, 0), and bounded from below by

min

H(x, 0). The value c is consequently well

defined by [6].

Moreover, any sequence u

,witha

decreasing and convergent to c, i s equi-Lipschitz–

continuous because of the coercivity of H,and

equibounded, up to addition of suitable constants.

It is therefore uniformly convergent, up to a

subsequence, to some u, which belongs to S

for any n, since these classes are stable for the

uniform convergence. This implies that u is a

subsolution to [4] ,sothatS

6¼;. The critical

value c is then characterized by the property that

the corresponding eqn [4] admits subsolutions

but not strict subsolutions. Our aim is to show

that extremal subsolutions do exist for the critical

eqn [4].

638 Hamilton–Jacobi Equations and Dynamical Systems: Variational Aspects

For any supercritical value a,thatis,a  c,wecan

define the functional nonsymmetric semidistance:

ðy; xÞ¼supfuðxÞuðyÞ: u 2S

¼ supfuð xÞ: u 2S

; uðyÞ¼0g

for any x, y in T

. It is immediate that S

satisfies

the triangle inequality and S

(y, y) = 0 for any y. But

it fails, in general, to be symm etric and positive if

x 6¼ y. We will nevertheless call it a distance, in the

sequel, to ease terminology. The function

x 7!S

(y, x) is itself a subsolution to [1], for any y,

being the pointwise supremum of a family of

equibounded subsolutions. Taking into account the

inequality

uðxÞuðyÞS

ðx; yÞ

which holds for any u 2S

, and the fact that it

becomes an equality by setting u = S

(x,  ), we also get

S

ðx; yÞ¼inffuðxÞuðyÞ: u 2S

¼ inffuðxÞ: u 2S

; uðyÞ¼0g

and S

(  , y) is, as well, a subsolution to [1]. Note

that

ðx; yÞþS

ðy; xÞ0 for any y; x ½7

The interest of intro ducing the distance S

in the

present context is that, for any a  c and y 2 T

the function x 7!S

(y, x) (resp. x 7!S

(x, y)) satis-

fies the maximality (resp. minimality) condition for

subsolutions of [1] in any open set not containing y.

If, by contradiction, the maximality property of

(y,  ) were violated in some open set  with y 62 

by a  satisfying [5] then one could make the set

{x: (x) > S

(y, x)} nonempty and compactly con-

tained in , by adding a suitable constant. Hence,

the formula

u ¼

maxf; S

ðy; Þg in 

ðy; Þ otherwise



½8

could provide a subsolution to [1] with

u(y) = S

(y, y) = 0 and u > S

(y,  ) at some point of

, which is in contrast with the very definition of S

One can similarly prove the minim ality condition

for S

(  , y).

We now focus our attention on the critical case.

We derive from the previous considerations that if a

maximal subsolution to [4] does not exist then, for

any y, we can find a neighborhood 

of y where

(y,  ) fails to be maximal. We can thus construct,

through a formula like [8],au

with

ess sup



Hð; Du

ðÞÞ < c ½9

in some neighborhood 

of y contain ed in 

Thanks to the compactness of T

, we can extract

from {

} a finite subcover {

}, i = 1, ..., m, for

some m 2 N, and define

u ¼



where 

are positive constants with



= 1. The

convex character of the Hamiltonian and [9] imply

that u is a strict critical subsolution, which cannot

be. We therefore conclude that there is a nonempty

subset of y, denoted henceforth by A, for which

(y,  ) is indeed a maximal critical subsolution. It

can also be proved, by exploiting some stability

properties of the maximal subsolutions, that A is

closed. Similarly, S

(  , y) must be a minimal

critical subsolution for some y. We denote by



the closed set made up by such points.

The previous covering argument shows that if

y 62A (resp. y 62



A) then there is a local strict

critical subsolution at y. The converse is also true:

let in fact  be such a s trict subsolution satisfying

(y) = S

(y, y) = 0; then  is subtangent to S

(y,  )

(resp. supertangent to S

(  , y)) at y,bythevery

definition of the distance S

. This shows that S

(y,  )

(resp. S

(  , y)) is not a maximal (resp. minimal)

critical subsolution, and so y 62A(resp. y 62



A). Since

the previous characterization holds for both A and



A,itfollowsthatA=



A. This set is a generalization

of the (projected) Aubry set. We will come back on

this point later on.

We also see from the covering argument that there

is a critical subsolution , which is strict outside A,

that is, such that ess sup



H(x, D(x)) < c for any

open set  compactly contained in T

nA.

This implies that any y such that {p: H(y, p)  c}

has empty interior, belongs to A. The empty interior

condition in fact implies, thanks to the strict

quasiconvexity of H, that the sublevel set reduces

to a singleton, say {p

}. We know that @u(y) 

{p: H(y, p)  c}, for any u 2S

; therefore, @u(y)isa

singleton and so any critical subsolution u is strictly

differentiable at y with H(y, Du(y)) = H(y, p

) = c.

Hence, there cannot be critical subsolutions which

are strict around y.

The previously described points will be called, in

the sequel, equilibria, and the (possibly empty)

closed set made up by them will be denoted by E.

The reason of this terminology will be explained

later. The differentiability property of the critical

subsolutions at equilibria, can be extended, quite

surprisingly, to any point of A, under more stringent

assumptions on H. We will discuss this issue in the

next section.

Hamilton–Jacobi Equations and Dynamical Systems: Variational Aspects 639

Qualitative Properties of Generalized

Aubry Set

We introduce some dynamical aspects in the picture

by showing that the distances S

, defined in the

previous section for an y a  c, are actually of length

type, in the sense that S

(y, x) equals, for any pair y,

x, the infimum of the intrinsic length of absolutely

continuous, or equivalently Lipschitz-continuous,

curves joining y to x. By intrinsic length, we mean

the total variati on of S

on the curve. It will be

denoted by ‘

, while ‘ will indicate the natural (i.e.,

Euclidean) length.

For this purpose, we proceed to give a line-

integral representation formula of S

. To start with,

we consider a C

subsolution u to [1], some x, y T

and a (Lipschitz-continuous) curve , defined in

some compact interval I, joining y to x. We have

uðxÞuðyÞ¼

 dt 



ð;

Þdt ½10

where, for any (x, v) 2 T

 R

, 

(x, v):=

max

p2Z

(x)

pv and

ðxÞ :¼fp: Hðx; pÞag

Inequality [10] also holds for a Lipschitz-continuous

subsolution to [1] through suitable replacement of

the differential by the gen eralized gradient. The set-

valued map Z

is compact convex valued, by the

coercivity and quasiconvexity assumptions on H,

and continuous with respect to the Hausdorff

metric. The function 

is accordingly continuous

in the first variable, and convex and positively

homogeneous in the second, being a support func-

tion. This implies in particular that the integral on

the right-hand side of [10] is invariant under change

of parameter preserving the orientation. We derive,

from [10],

ðy; xÞinf





ð;

Þdt:  de fined

in ½0; 1 and joining y to x



½11

for any y, x. We denote by

(y, x) the quantity on

the right-hand side of [11]. It is immediate that the

triangle inequality holds for

. The function

u :=

(y,  ) is, moreover, Lipschitz-continuous

since 

(x, v)=jvj is bounded from above in

(R

n{0}) because of the co ercivity of H. Given

v 2 R

, we exploit the definition of S

, the

continuity of 

, and the triangle inequality for S

to get at any differentiability point x

of u,

Duðx

Þv ¼ lim

h!0

uðx

 hvÞuðx

 lim sup

h!0

ðx

 hv; x

 lim

h!0



ðx

 hvt; hvÞdt

¼ lim

h!0



ðx

 hvt; vÞdt

¼ 

ðx

; vÞ

This imp lies by Hahn–Banach theorem that

Du(x

) 2 Z

(x) or, in other terms, that u = S

(y,  ) 2

. We then derive, from [11] and the very

definition of S

ðy; xÞ¼inf



ð;

Þdt:

 defined in ½0; 1 with ð0Þ¼y;

ð1Þ¼x



Taking into account that the integral functional

appearing in the previous formula is lower semicon-

tinuous for the uniform convergence of equi-

Lipschitz-continuous sequence of curves, by standard

variational results, we in turn infer that it equals the

intrinsic length ‘

. Mathematically,

‘

ðÞ¼



ð;

Þdt

for any co mpact interval I and any curve  defined

in I.

Since S

is just a semidistance, we do not have any

a priori information on the sign of ‘

; however, by

[10], the intrinsic length of any cycle must be non-

negative. Furthermore, while j‘

()j must be small

for any curve  with small natural length, by the

coercivity condition on H, no converse estimates

hold, in general. If a > c, some information in this

direction can be gathered by taking a strict subsolu-

tion  to [1], that it can be assumed smooth, up to

regularization by mollification, then D(x)v 



(x, v)  jvj for any (x, v) 2 T

 R

, and some

>0, and consequently

‘

ðÞ

ð

ð;

ÞDðÞ

Þdt

þ ðxÞðyÞ‘ ðÞS

ðx; yÞ½12

for any pair y, x and any curve , defined in some

interval I, joining y to x. The previous formula says,

in particular, that when jx  yj is small then any

curve whose intrinsic length approxim ates S

(y, x)

must have small natural length. The previous

640 Hamilton–Jacobi Equations and Dynamical Systems: Variational Aspects

argument cannot be extended to the critical case.

This gap suggests the next definition. The main

purpose for introducing it is to get a metric

characterization of the Aubry set A.

We say that S

is localizable at some y if for every

">0 there is 0 <

<"such that

ðy; xÞ¼inff‘

ðÞ:  joins y to x and ‘ðÞ <"g½13

whenever jx yj <

.Ify 62A, we adapt the argu-

ment previously used in the strict subcritical case to

get that S

is indeed localizable at y. In this case we

have, in fact, at our disposal a critical subsolution,

say , which is strict in some neighborhood  of y,

thanks to the characterization of the Aubry set given

in the previous section.

We assume, to simplify,  to be C

; under the

natural condition of Lipschitz-continuity, general-

ized gradients should be used in place of differ-

entials. We have D(x)v  (x, v)  jvj for any

x 2 , any v 2 R

, and some >0, and D(x)v 

(x, v), for any x, v. Exploiting these inequalities, we

obtain an estimate analogous to [12] for curves

starting from y, which allows us to prove [13].

Conversely, let y 62E be a point where S

localizable. We claim that Z

(y)  D



u(y), where

u := S

(y,  ). It is enough to show that any p

in the

interior of Z

(y) belongs to D



u(y), since D



u(y)is

closed. Note that the interior of Z

(y) is nonempty

since we are assuming that y is not an equilibrium.

Such a p

belongs to the interior of Z

(x) for x

sufficiently close to y, thanks to the continuity of Z

;

consequently, p(x  y) <‘

() for any x close to y

and any curve  joining y to x with ‘() sufficiently

small. Taking into account [13], we then deduce

pðx  yÞS

ðy; xÞ for x close to y

and so the linear function (x):= p

(x  y)is

subtangent to u at y. This in turn implies that y is

out of A since  is a local strict critical subsolution at

y,andsoS

(y,  ) cannot be a maximal subsolution by

the characterization given in the previous section.

The fact that S

is not localizable at any point of

y 2AnEleads to the announced metric characteriza-

tion of A.Ify is such a point, there is an ">0, a

point x, with jx  yj, and so jS

(y, x)j, as small as

desired, and a curve  joining y to x with ‘

( ) 

(y, x) and ‘() >". We construct a cycle , passing

through y, by juxtaposition of  and the Euclidean

segment joining x to y. We obtain, in this way, a

sequence of cycles 

, passing through y, with length

‘

(

) ! 0 and ‘(

)  ", for any n.

The same result can also be obtained for y 2E.In

this case we select ">0 and v

2 R

with



(y, v

) = 0, and denote by B

a sequence of

Euclidean balls, centered at y, satisfying 

(  , v

) <

1=n in B

. We construct a sequence of cycles,

passing through y, by going up and down on the

line {y þ sv} in such a way that 

(t) 2 B

, for every

t, and "<‘(

) < 2"; therefore 0  ‘

(

) < 2"=n.

Conversely, such a sequence of cycles cannot exist

at any y 62Abecause S

is localizable at y.

We emphasize that the previous definition of A

through cycles and the fact that S

is not localizable

at any point y 2A with intZ

(y) 6¼; shows that,

apart for the special case of equilibria, the property of

being a point of A is definitively not of local nature.

As pointed out already, if y 62A, and so S

localizable at y, then Z

(y)  D



u(y), where

u := S

(y,  ); on the other hand, we know that



u(y)  @u(y) and @u(y)  Z

(y), where the latter

inclusion holds since u is a critical subsolution. We

then derive



uðyÞ¼@uðyÞ¼Z

ðyÞ

We interpret these inequalities as a convexity–type

property, or, to use a more appropriate terminology,

a semiconvexity property of the distance function

(y,  )aty. The same property holds for the

Euclidean distance function jxj at 0.

A contrasting phenomenon takes place if y 2A,

namely S

(y,  ) is semiconcave at y, which means

that D

u(y) = @u(y). This is more complicated to

prove (see Fa thi 2005b), and requires, in addition, H

to be strictly convex in p and locally Lipschitz-

continuous in (x, p). Under these assumptions one

can, more generally, show that S

(y,  ) is semicon-

cave in T

,ify 2A, while it is semiconcave in

n{y} and semiconvex in y,ify 62A. Some

important consequences can be deduced.

First, thanks to the semiconcavity property there

are C

supertangents to u := S

(y,  )aty, whenever

y 2A. Such a function, say , is also supertangent to

S

(  , y), which is a minimal critical subsolution, at

the same point. We know from the previous section

that no supertangents to S

(  , y)aty can be strict

critical subsolution locally at y, and so

H(y, D(y)) = c. This implies that D

u(y) is con-

tained in the boundary of Z

(y). We then see, taking

into account that D

u(y) is convex and Z

(y) strictly

convex, that D

u(y) reduces to a singleton, and so,

by the semiconcavity property, @u(y) reduces to a

singleton. Therefore, S

(y,  ) is strictly differentiable

at y, for any y 2A. One can similarly show that

S

(  , y) is strictly differentiable at y.

Second, given y 2Aand a critical subsolution w,

which can be assumed, up to addition of a constant,

to vanish at y, we see that S

(y,  ) (resp. S

(  , y)) is

supertangent (resp. subtangent) at y because of

its extremality properties. Since both these

Hamilton–Jacobi Equations and Dynamical Systems: Variational Aspects 641

super(sub)-tangents are differentiable, by the previous

point, we deduce that w itself is differentiable at y.

Moreover, the differentials at y of all three functions

under consideration, namely S

(y,  ), S

(  , y), and

w, coincide. In particular, H(y, Dw(y)) = c,and

y 7!Dw(y) is continuous on A,sinceS

(y,  )hasbeen

proved to be strictly differentiable at y,whenevery 2

A. Any critical subsolution, restricted to A,isconse-

quently a continuously differentiable solution to [4].

Summing up, we have discovered (under the

assumption of strict convexity and Lipschitz-

continuity for H) that every critical subsolution is

differentiable on A, and the differential on A is the

same for every critical subsolution. A continuous

map G : A!R

is then defined by taking G(y)

equal to the common differential of any critical

subsolution at y. We denote by

A the graph of G,

which is a subset of the cotangent bundle of T

identified with T

 R

As we have already pointed out, the existence of a

subsolution to [1] is obvious when a > c, and

such a subsolution can be obtained through a

suitable regularization by mollification of any strict

subsolution. The same construction cannot be

performed at the critical level, since no strict critical

subsolutions are available to start the regularization

procedure. We can nevertheless show the existence

of C

critical subsolutions by exploiting the infor-

mation gathered on the Aubry set. We start by

considering a countable locally finite open cover of

nA,{

}; we know from the previous section that

there is a critical subsolution, say w

, which is strict

on 

, for any i. Loosely speaking, we have some

space, also in this case, for regularizing w

in such a

way that the regularized funct ion is still a critical

subsolution, at least on 

We can glue together, with some precautions,

these regularized local critical subsolutions through

a C

partition of the unity, to produce a critical

subsolution which is C

outside A. Using the fact

that any critical subsolution is differentiable on A,

we can further adjust the previous construction so

that the critical subsolution is C

on the whole T

We state this result in the following way: if the

equation [1] has a subsolution then it also has a C

subsolution. It is worth noticing that it holds even if

the underlying manifold is noncompact (see Fathi

(2004, 2005b)).

The Intrinsic Lengths and the Action

Functional

Here we assume H to satisfy all the usual assump-

tions in order to define the Hamilton’s equations [2]

and to have the completeness of the associated

Hamiltonian flow. Namely, we require H to be C

in both variables, C

-strictly convex, that is, H

0inT

 R

and superlinear, in the sense that

lim

jpj!þ1

Hðx; pÞ

jpj

¼þ1 uniformly in x

We define the Lagrangian L as the Fenchel

transform of H. It takes finite values thanks to the

superlinearity condition, and, in addition, inherits,

from H, C

regularity, C

-strict convexity and

superlinearity. In our setting , the Fenchel transform

is involutive.

We call a vector v

and a covector p

conjugate at

a point x if v

= H

(x, p

), and so L(x, v

) = p



H(x, p

). This also implies the relations p

= L

(x, v

)

and H(x, p

) = p

 L(x, v

). If H(x, p

) = a,for

some a,thenp

= 

(x, v

), and p

is the unique

element of Z

) for which such a relation holds.

Since the function y 7!p

 H(y, p

)issubtangent

to L(  , v

)atx, we see that L

(x, v

) = H

(x, p

We introduce, for any (Lipschitz-continuous)

curve  defined in [a, b], for some a < b, the action

functional A() through

AðÞ¼

Lð;

Þdt

We say that the curve  is a minimizer of the action

if A()  A() for any  defined [a, b] and with the

same end points of . It is a classical result in

calculus of variations that an y of such minimizers 

is of class C

and satisfies the Euler–Lagrange

equation

ð;

Þ¼L

ð;

Þ in a; b½

Consequently,  and the conjugate curve

 = L

(,

) satisfy the Hamilton’s equations [2].

Note that all the integral curves of [2] lie in a fixed

level of the Hamiltonian, which is compact by the

superlinearity condition. The corresponding Hamil-

tonian flow is consequently complete.

We show that if x

2E, and Z

) = {G(x

)},

then (x

, G(x

)) is a steady state of the Hamiltoni an

flow. In this case, in fact, c = min

H(x

, p) and so

L(x

,0)= c and H

, G(x

)) = 0, or equivalently

G(x

) and 0 are conjugate at x

. Taking into

account that c is the critical value, we have that

Lðx; 0Þ¼min

Hðx; pÞc for any x 2 T

so that x

is a minimizer of x 7!L(x,0) and

,0)= H

, G(x

) = 0. It is easy to see that,

642 Hamilton–Jacobi Equations and Dynamical Systems: Variational Aspects

conversely, if (x

, p

) is a steady state of the

Hamiltonian flow and H( x

, p

) = c then x

and p

= G(x

We want to establish a relation between A(  ) and

the length functionals ‘

defined in the previous

section for a  c. This will allow, among other

things, to show that the Aubry set

A is invariant for

the Hamiltonian flow and to analyze the properties

of the integral curves lying on it. To this aim, we

consider the minimal geodesics for S

, a  c, that is,

the curves, defined on compact inte rvals, whose

intrinsic lengths ‘

equal the distance S

between

their end points.

If a > c, we claim that, given any pair of points in

, there is a minimal geodesics joining them.

Recalling the formula [12], whose validity depends

on the fact that in the strict supercritical case there is

a smooth strict subsolution to [1], we have

‘

ðÞ!þ1 whenever ‘ðÞ!þ1

The claim is then proved by using the Ascoli

theorem and the lower-semicontinuity property of

‘

. In the critical case, given y 62A, we can use the

same argument to deduce the existence of minimiz-

ing geodesics for S

between y and any point x

sufficiently close to y (in the Euclidean sense). This

comes from the fact that S

is localizable at y, and so

any sequence of curves 

with ‘

(

) ! S

(y, x) has

bounded natural length. For a general pair of points,

we will show, on the contrary, that existence of a

minimal geodesic is not guaranteed in the critical

case.

We consider a minimizing geodesics  for S

between a pair of points y and x. We assume a > c

or a = c and  \E= ;. We want to show that  is a

minimizing curve for the action, up to a change of

parameter. We choose the new parameter in such a

way that

Lð

;

Þþa ¼ 

;

Þ½14

where we have denoted by

 the reparametrized

curve. Since

 stays away from E, the velocities j

j

are bounded from below by a positive constant and

so the domain of definition of

, denoted by [0, T], is

a compact interval. Note that ‘

() = ‘

(

), since the

intrinsic length is invariant under change of para-

meter. We take into account that  is a minimal

geodesic and the inequality L(x, v) þ a  

(x, v),

which holds for any x, v, to get

Að

Þ¼‘

ðÞaT  ‘

ðÞaT  AðÞ

for any  defined in [0, T] with (0) = y, (T) = x.

This proves the announced minimality propert y of

.

Furthermore, we show that the function

u := S

(y,  ) is strictly differentiable at

(s), for s 2

]0, T[, and

Duð

Þ¼L

;

Þ¼:  ½15

in [0, T]. Hence, (

, Du(

)) is a solution of the

Hamilton’s equations in ]0, T[. To see this, we start

from the relations

uð

ðsÞÞds ¼ uð

ðtÞÞ ¼



;

Þds



 ds ½16

which hold in [0, T] because u is Lipschitz-contin-

uous,

 is a minimizing geodesic, and (s)is

conjugate to

(s)at

(s) for any s 2 [0, T]. We

know that

uð

ðsÞÞ ¼ p

ðsÞ

for a:e: s and some p 2 @uð

ðsÞÞ

We have that p 2 Z

(

(s)), sinc e u is a critical

subsolution, and so

ðsÞ

ðsÞ;

ðsÞÞ ¼ ðsÞ

ðsÞÞ

We see, in the light of [16], that equality must hold

in the previous formula, for a.e. s. Therefore,

uð

ðsÞÞ ¼ ðsÞ

ðsÞ for a:e: s ½17

we derive from the fact that the function (  )

(  )is

continuous that ( u(

(  )) is actually continuously

differentiable in ]0, T[ and that [17] holds for any s.

We finally exploit that u is semiconcave in T

n{y},

as pointed out in the previous section, and so

(s)) = @u(

(s)), for any s.If is a C

supertangent to u at

(s) then

Dð

ðsÞÞ

ðsÞÞ ¼

uð

ðsÞÞ

accordingly,

ðsÞ¼ðsÞ

ðsÞ for any s and p 2 @uð

ðsÞÞ

Since @u(

(s))  Z

(

(s)), this implies that @u(

(s)) =

{(s)}. This actually gives the strict differentiability

function u at

(s), and Du(

(s)) = (s) for any s.

The same argument works, with some adjustment,

also when a = c and  \E6¼;. If, for instance, y 62

E, t

= min {t: (t) 2E}, then by reparametrizing  in

[0, t

], as indicated in [14], we get a curve

 defined

in [0, þ1[ which is a minimizer of the action

functional in any compact interval contained in

[0, þ1[. Moreover, u := S

(y,  ) is strictly

Hamilton–Jacobi Equations and Dynamical Systems: Variational Aspects 643

differentiable in ]0, þ1[ and (

, Du(

)) is a solution

of the Hamilton’s equations.

We proceed to investigate the properties of the

Hamiltonian flow on A.Wetakeay

in AnE,and

consider a sequence 

of cycles passing through y

with ‘

(

)) ! 0, ‘(

)  2, for some positive .Such

a sequence does exist in view of the characterization

of A through cycles given in the previous section.

Moreover, we assume that the 

are parametrized by

the natural arc length in [ T

, T

], for some T

 ,

and satisfy 

(0) = y

for any n. There is then defined

a uniform limit curve  in [, ], up to a

subsequence, thanks to the Ascoli theorem.

The idea is to construct a new sequence of cycles



by replacing the portion of the 

between  and

 by , and pasting this new piece with the

remainder of 

through Euclidean segments at the

end points. The 

are still of infinitesimal intrinsic

length ‘

, which shows, in particular, that  is

contained in A. By exploiting that S

is a length

distance, that the 

are cycles, and the formula [7],

with a = c, we get

‘

ð

ÞS

ððÞ;ðÞÞ þ S

ððÞ;ðÞÞ

 0

for any n, and we at last derive

‘

ðÞ¼S

ððÞ;ðÞÞ¼  S

ððÞ;ðÞÞ

Note that the second equality is actually redundant.

By reparametrizing ,asin[14], with a = c, in some

open interval con taining 0 as interio r point and

contained in [, ], we get a curve contai ned in

AnE, denoted by , defined on some open interval I

and satisfying

Aðj

½s;t

Þþcðt  sÞÞ ¼ ‘

ðj

½s;t

¼S

ððtÞ;ðsÞÞ for any t > s ½18

This, in particular, shows that  is a minimizer of the

action functional in any [s, t]  I. If we denote, as

usual, by  the curve conjugate to

, we have,

arguing as above, that (t) is the differential of the

function S

( (s),  )at(t), but, since the differentials

of all critical subsolutions coincide on A, we finally

get that (t) = G((t) for every t 2 I. Therefore,

(, G()) is a solution of the Hamilton’s equation in

I and is contained in

A. The same properties can be

extended on the whole R.

Taking into account that if y 2Ethen (y, G(y)) is

a steady state of the Hamiltonian flow, we in the

end see that

A is foliated by integral curves of

the Hamiltonian flow (, G()), with  enjoying the

variational property [18]. This is indeed a

characterization since if, conversely, a curve 

satisfies [18] then it must be contained in A.

As an application, we finally show that there

cannot be minimal geodesics, for the critical metric

, joining a point of A, say y, to some x 62A,at

least when E = ;. If such a geodesic, say , ex ists,

and is defined in [0, T], for some T > 0, then

(, Du()) is a solution of the Hamilton’s equations,

up to a change of parameter, where u := S(y,  ),

satisfying the initial conditions (0) = y

, (0) =

lim

t!0

Du((t)).

The last relation tells us that (0) 2 @u(y) and,

since u is differentiable at y 2Awith Du(y) = G(y),

we conclude that (0) = G(y). Therefore, (, Du()) is

a part of the integral curve of the Hamiltonian flow

starting at (y, G(y)) that we know, by the above

reasoning, to be contained in

A, which is in

contradiction with (T) = x 62A.

See also: Control Problems in Mathematical Physics;

Dynamical Systems in Mathematical Physics: An

Illustratrion form Water Waves; KAM Theory and Celestial

Mechanics; Minimax Principle in the Calculus of Variations;

Optimal Transportation; Stability Theory and KAM.