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

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(iii) the Hamiltonian flow on each torus M

is linear

and ‘‘quasiperiodic’’ with frequencies !

defined

by d’

=dt = !

, f

, ..., f

); and

(iv) the Hamiltonian equations are integrable by

quadratures.

If a Hamiltonian verifies the assumptions of the

theorem of Liouville, one can prove that it exists,

locally, canonical coordinates j = (’

, ..., ’

) and

I = (I

, ..., I

) such that the Hamiltonian function

depends only on the variables I

. Then

d’

¼

@’

¼ 0

These equations are immediately integrated as

follows:

= constant; and ’

= !

 t þ ’

ð0Þ; with

ðI

; I

; ...; I

Þ¼



I¼cst

Such local coordinates (’

, I

) are called ‘‘action-

angle’’ variables. They were defined for the first time

by Delaunay and they play an important part in the

theory of perturbations.

Remark An invariant torus T

of the theorem of

Liouville is characterized by the constant values of the

actions I

, which determine the frequencies !

on it.

Such a torus is said to be nonresonant if the relation

between the frequencies !

i = 1

= 0(where

, ..., k

are integers) implies that k

= 0, 8i.The

frequencies !

are then rationally independent. If a

torus is nonresonant, the phase trajectories are dense

everywhere and the motion is quasiperiodic on it.

A torus is said to be resonant, if the frequencies !

are rationally dependent: they verify a relatio n

i = 1

 !

= 0, with (k

,..., k

) 6¼ (0, ..., 0). Then

the phase trajectories are not dense on the torus;

they belong to tori of lower dimension.

A consequence of the theorem of Liouville is that,

if a two-degree-of-freedom Hamiltonian system

possesses one first integral F (in addition to H , and

independent of H), it is integrable because F is

necessarily in involution with H:{F, H} = 0.

An example of system with three degrees of

freedom which is integrable is the Lagrangi an

symmetric top with one fixed point (there exists a

cylindric symmetry for the inertia momenta and the

center of mass is on the symmetry axis). This system

possesses three first integrals that are in involution

and independent: H, and the angular momenta M

and M

, which correspond to the (constant)

frequencies of precession and nutation of the top.

The level sets M

are here tori of dimension 3, which

are indexed by the three frequencies (or by the

constant values of the three integrals).

There are other integrable cases for this problem

of a rigid body with a fixed point (see Kozlov

(1983)): the Euler’s case (when the fixed point is the

center of mass); the Kowalevskaya’s case (in which

the inertia momenta verify two relations and the

third coordinate of the center of mass vanishes – see

Kowalevski (1889)); and the Goryachev–Chaplygin’s

case, which is integrable only on a single integral

level.

A fundamental and classical example of integrable

Hamiltonian system is the Kepler’s problem: the

motion of a ponctual mass in the gravitational

(Newtonian) field of a center, for instance, a planet

in the field of attraction of the Sun.

Another example is the problem of two fixed centers:

an infinitesimal mass in the field of two centers, problem

which was integrated by Lagrange (Lagrange, 1810).

Isolated Periodic Orbits

and Nonintegrability

We consider a real Hamiltonian system with n

degrees of freedom and we suppose that there exists

a particular T-periodic solution 

(which is not an

equilibrium). Along 

, we consider the linearized

equations deduced from the Hamiltonian system.

They can be decoupled into the tangential equation

(one degree of freedom) which possesses the first

integral dH and the normal variational system which

can be written as

d

¼ J  Kð 

ðtÞÞ   ½2

where

J ¼

0 I

I 0



is the standard symplectic matrix of order 2(n  1)

and K (

(t)) is a T-periodic matrix depending on

the solution 

The solutions of the linear system [2] form a

vector space. As a definition, the monodromy

matrix M(T) expresses how fundamental solutions

of the linear system [2] are transformed after one

period T, that is, along the periodic closed orbit 

ðt þ TÞ¼MðTÞðtÞ

Poincare´ showed that if one of the eigenvalues of

M(T) is different from 1, then the periodic solution



is isolated. Furthermore, if the number of first

626 Hamiltonian Systems: Obstructions to Integrability

integrals of the Hamiltonian system, independent

along 

, is equal to k, then at least 2k eigenvalues

of M(T) are equal to 1.

Theorem (Poincare´ 1892). If the Hamiltonian

system possesses n integrals in involution, and

independent along a periodic solution 

, then 

is nonisolated.

Then, if the Hamiltonian system possesses a dense

set of isolated periodic orbits, it cannot have n

integrals in involution and independent in an open

domain.

Nearly Integrable Hamiltonian Systems,

Theorem of Poincare´

Consider the Hamiltonian system with n degree s of

freedom, depending on a small real parameter

" 2 ("

, þ"

), defined by the analytic function H:

Hðj; I;"Þ¼H

ðIÞþ"  H

ðj; IÞ½3

where j = (’

, ..., ’

) 2 T

, I = (I

, ..., I

) 2 R

, and

where H

is periodic in the angles ’

This system is called ‘‘nearly integrable’’ because

when " = 0, the ‘‘unperturbed system’’ H

is integr-

able in the action-angle variables j, I:

Hðj; I; 0Þ¼H

ðIÞ

then

¼ 0;

¼ wðIÞ

system which can be integrated by quadratures:

I ¼ I

and j ¼ j

þ wðI

Þt

According to the theorem of Liouville, the motion of

the unperturbed problem takes place on n-dimen-

sional tori (S

)

in the phase space. On these

invariant tori, indexed by the actions I, the motion

is generally quasiperiodic (if the frequencies w(I) are

rationally independent).

We are now interested in studying the perturbed

system [3] with " 6¼ 0, and its integrability which is,

according to Poincare´ (1892), ‘ ‘the fundamental

problem of dynamics.’ ’ This problem of nearly

integrable Hamiltonian systems is directly inspired

by celestial mechanics where the motions in the

solar system are, in a first approximation, described

by the (integrable) Kepler’s problem. In particular,

the ‘ ‘restricted three-body problem’’ is the study of the

motion of a planet in the gravitational field of the Sun,

with the perturbative attraction of Jupiter. It is also the

problem of the Moon in the field of the Earth, with

the perturbative attraction of the Sun (Poincare´ 1892).

Theorem of Poincare´ (Poincare´ 1892). Assume

that, in the Hamiltonian function [3]:

(i) (nondegeneracy condition) the unperturbed

Hamiltonian H

is nondegenerate, that is,

det



¼ det



6¼ 0

in an open domain of the phase space;

(ii) (genericity condition) no coefficient h

(I) in the

Fourier expansion of H

with

ðj; IÞ¼

k2Z

ðIÞe

i k;j

does identically vanish in the nonresonant

domain G 2 R

of the actions defined by

G ¼

(

I 2 R

i¼1

 !

ðIÞ¼0;

iff ðk

; ...; k

Þ¼ð0; ...; 0Þ

)

then, there is no analytic first integral F(j, I, ")

independent of the Hamiltonian function H.

Thus, a perturbation of a nondegenerate integrable

Hamiltonian system is generically nonintegrable.

When one wants to apply this theorem to celestial

mechanics, a peculiarity is that the unperturbed

problem corresponds to the Keplerian system, which

is degenerate, and this is a specific difficulty of these

systems.

Splitting of Separatrices and

Nonintegrability

Consider a Hamiltonian system with n = 2 (degrees

of freedom) defined as in eqn [3] by a perturbation

of an integrable Hamiltonian:

Hð’

;’

; I

;"Þ

¼ H

ðI

; I

Þþ"  H

ð’

;’

; I

Þ½4

The unperturbed problem is integrable and its four-

dimensional phase space is foliated by two-dimensional

invariant tori T

: I = constant. If H

is nondegenerate,

the nonresonant tori are dense and the resonant tori also

are dense in the phase space.

According to Kolmogorov’s theorem and

the Kolomogorov–Ar nol’d–Mo ser (KAM) theory

(Arnol’d 1985), the majority of the nonresonant

tori of the unperturbed problem H

are p reserve d

in the full problem [4]: they are slightly

deformed, and are invariant in the perturbed

Hamiltonian Systems: Obstructions to Integrability 627

system. T he r esonant tori of H

are destroyed in

the perturbed problem.

Now we consid er, in the phase space, a transverse

surface S to the invariant tori T

of the perturbed

system. A trajectory of the system generated by [4],

which crosses S through a point w

, will cross S

again, for the first time, through a point w

: this

defines the ‘‘first return map’’ or ‘‘Poincare´’s map’’

R : w

7!R(w

) = w

. S is called a Poincare´’s sec-

tion. If w

belongs to a preserved invariant torus

of the perturbed system, the successive points

, w

= R(w

), w

= R(w

), w

= R(w

), ... belong

to the intersection of this torus with S; thus, they

belong to a curve diffeomorphic to a circle, which is

an invariant curve of the map R.Ifw

does not

belong to a preserved invariant torus of [4], the

sequence of points w

, w

, ... through the

Poincare´’s map belongs to a curve much more

complicated than a curve diffeomorphic to a

circle (Poincare´ 1890, Arnol’d 1985) and the

‘‘chaotic’’ behavior of this sequence is the mark of

the nonintegrability of the system [4].

The best way of numerically showing the

‘‘evidence’’ of nonintegrability is to study the

example of a system with ‘‘one and a half’’ degree

of freedom, that is, a system with one degree of

freedom whose Hamiltonian depends on time:

H(’, I, t). An example of such a system is the

problem of a mathematical pendulum whose length

l performs periodic oscillations, defined by the

Hamiltonian function

Hð’; p; tÞ¼

 !

ð1 þ "  f ðtÞÞ  cos ’ ½5

where ’ 2 S

, p 2 R, and f is periodic of period T.

The unperturbed system (" = 0) is integrable (one

degree of freedom with a Hamiltoni an independent

of t):

ð’; pÞ¼

 !

 cos ’

The phase portrait of this problem is similar to the

one of the simple mathematical pendulum of

constant length: on the cylinder S

 R there are

two equilibria (stable and unstable) and separatrices

‘‘beginning’’ and ‘‘finishing’’ at the hyperbolic point

. The invariant stable and unstable manifolds

associated to  and represented by these separatrices

were called by Poincare´ as ‘‘homoclinic’’ trajectories,

because each of them, drawn on the phase cylinder,

joins equilibrium  to itself.

If " 6¼ 0, we define a Poincare´ section of the

perturbed system [4] in the follow ing way: from an

initial point w

(’

), we consider the successive

planes perpendicular to the t-axis in the ‘‘extended’’

phase space {(’,p,t)}, defined by : t

þT,

þ2T, t

þ3T,... and we look at the

successive intersection s of the orbit of w

with

these planes: w

,.... If we identify all the

successive planes and if we draw on the same

picture, the points w

,..., we obtain a

phase portrait in which the equilibria of the

unperturbed problem are prese nt, but the separatrix

which ‘‘leaves’’ the point  is not confounded with

the separatrix which ‘‘ends’’ at ,asinthe

unperturbed problem: the two invariant curves are

transversal to each other: they ‘‘split’’ and have an

infinite number of intersections. This splitting is the

traduction of the nonintegrability of the perturbed

system [5].

A method to detect this splitting of separatrices

consists in computing the Melnikov’s function which

gives a measure of the angle between the separatrices

at their first intersection when they split.

Many concrete Hamiltonian systems have been

studied by this method and numerical investigations

on the splitting have permitted detection of their

nonintegrability.

Topological Obstructions to Integrability

We are interested in a natural mechan ical system

with two degrees of freedom and we suppo se that

the state space N is a real analytic surface which is

compact and orientable. Then, N consists of a two-

dimensional sphere with k handles (or a torus with k

holes). The number k is a top ological invariant of

the surface and is called the genus of N.

Let H be the Hamiltonian function associated to

this problem. The Hamiltonian system possesses the

first integral H. It is completely integrable if and

only if another analytic integral F exists, function-

ally independent of H. In this case, the state space N

belongs necessarily to a very restrictive class of

surfaces.

Theorem (Kozlov 1983). If the genus k of the

state manifold N is not equal to 0 or 1(i.e., if N is

neither diffeomorphic to the sphere S

nor to the torus

), then the Hamiltonian system generated by H does

not possess a first integral, analytic on T



Nand

functionally independent of the energy integral H.

Note that this theorem does not apply to first

integrals which are C

only, and examples can be

given which illustrate this case (Kozlov 1983, 1989).

For systems with more than two degrees of

freedom, an open question is to know whether the

complete integrability imposes restrictions to the

topology of the state manifold N.

628 Hamiltonian Systems: Obstructions to Integrability

Singular Point Analysis, Branching

of Solutions and Ziglin’s Theory

If we look at the classical Hamiltonian problems

which have been integrated, their first integrals are

real functions which can be continued in the

complex domain as one-valued holomorphic or

meromorphic functions of the complex time t

(polynomials, rational functions, etc.). This fact

leads to the concept of ‘‘complex integrability.’’

But the nonintegrability of a complex Hamiltonian

system does not imply the nonintegrability of its

restriction to the real domain: it may happen that a

real analytic first integral does not possess a

continuation in the complex domain as a mero-

morphic function.

Adopting this point of view, S Kowalevskaya

(Kowalevski 1889) studied the problem of a top

rotating around a fixed point, and she discovered a

new case of integrability for this classical problem of

Hamiltonian mechanics. She searched for conditions

on the parameters such that the movable singula-

rities of the solutions in the complex plane of time

are poles (as a definition, a singularity is movable if

its location in the complex domain depends on the

initial conditions). Such differential systems are said

to be of ‘‘Painleve´ ’s type.’’ In this case, the solutions

are single valued in the complex t-plane and there is

no branching of these solutions. The leading idea is

the following: a first integral must be constant along

a solution, and an eventual branching would change

its value along a loop around a singularity in the

complex t-plane. However, finite branchings of

solutions can be compatible with integrability.

The main tool in this analysis is the calculation of

the ‘‘Kowalevski’s exponents’’ which determine the

eventual branching of a solution around a

singularity.

In spite of the efficiency of the Painleve´ analysis for

the search of integrable (or nonintegrable) systems, the

relation between the analytic properties of their

solutions (Painleve´) and their integrability in the

sense of Liouville remains mysterious. The most

fundamental result obtained in this field is a theorem

of Adler and van Moerbeke which proves that, if a

system has the Painleve´ property and if it is integrable

in the sense of Arnol’d–Liouville, then it is algebrai-

cally integrable (Adler and van Moerbeke 1989).

The discovery of S Kowalevskaya inspired

Ziglin, who related the existence of m eromorphic

first integrals for a Hamiltonian system, to the

properties of the linearized equations along a

particular periodic solution of this system, espe-

cially to the monodromy group associated to this

linear system. Ziglin used the constraints imposed

to this monodromy group by the existence of first

integrals.

Let us consider a Hamiltoni an system defined on

a complex analytic symplectic manifold of dimen-

sion 2n, and suppose that there exists a family of

periodic solutions . The linearized equati ons

deduced from the Hamiltonian system along  are

decoupled into tangential and normal equations. We

are interested by the normal equations, which are

linear with periodic coefficients.

Ziglin’s Theorem (Ziglin 1983). Assume that a

Hamiltonian system has a family of particular

solutions 

(which are not equilibria) parametrized

by periodic functions of the complex time and

depending analytically on a real parameter h 2

, h

). Let G be the monodromy group of the

normal variational equation associated to the solu-

tion 

. A monodromy matrix g 2 G is said to be

nonresonant if every eigenvalue of g is different

from a root of unity. If the Hamiltonian system has

a meromorphic integral F, functionally independent

of the Hamiltonian H in a neighborhood of 

, and

if the monodromy group G contains a nonresonant

element g

, then for any g

2 G, the commutator



= g

1

 g

1

 g

satisfies either g



= Id or



= (g

)

As a corollary of this theorem, we have sufficient

conditions of nonintegrability: if the necessary con-

ditions of integrability of Zi glin are not satisfied by

a Hamiltonian system, it is not analytic ally integr-

able. For instance, this will happen if we can find

two nonresonant monodromy matrices g

and g

which do not commute. If the periodic solution 

has two complex periods, the monodromy group G

has generators g

and g

, respectively associated to

each of these periods and their commutativity can

be sometimes studied.

These sufficient conditions of nonintegrability

were studied for particular Hamiltonian systems,

first by Ziglin himself.

Several concrete systems with two degrees of free-

dom were proved to be nonintegrable by Ito, Yoshida,

Churchill, Rod, and many other mathematicians who

applied this ‘‘Ziglin’s method’’: for instance, the

He´non–Heiles system, the Yang–Mills system, and

Hamiltonian systems with a homogeneous potential.

Nonintegrability and Differential Galois

Theory (Morales-Ruiz 1999)

Recently, the integrability of Hamiltonian systems

was studied with algebraic tools from the differen-

tial Galois theory, applied to linear differential

Hamiltonian Systems: Obstructions to Integrability 629

systems. As in Ziglin’s theory, we consider a

particular solution  (not necessarily periodic) of a

differential system generated by an analytic Hamil-

tonian H with n degrees of freedom, and the (linear)

variational equations along . The idea is that, if the

Hamiltonian system is integrable, we can assume

that the linearized equations along  must also have

a ‘‘regular behavior.’’ If the Hamiltonian system is

integrable, it will also be the case for the variational

equations.

The normal variational system (of order 2n  2)

can be written as

d

¼ J  KððtÞÞ   ½6

with

J =

0 I

I 0



K((t )) is a matrix depending on the particular

solution .

We have to define the ‘‘Galois group’’ of the linear

equation [6]. Recall that in the classical Galois

theory of algebraic equations, the Galois group is

defined by the automorphisms which map roots

onto roots of the equation. In an analogous way, in

the differential Galois theory, we consider the maps

which send a fundamental solution of eqn [6] on a

fundamental solution. In order to define the Galois

group G associated to [6], we consider a differential

field K of functions over C (i.e., a field of functions

equipped with a derivation). The field of constants

of K is C; it is the subfield of K whose elements have

a derivative equal to zero. We denote by K , , ...

the differential field extension obtained from K by

the adjunct ion of the functions , , ....If(’, )isa

fundamental system of solutions of eqn [6], then

L = K ’,

is the smallest differential field exten-

sion which contains all the solutions of [6]. The

field of constants of L is the same as the one of K,

that is, C. By definition, L is a Picard–V essiot

extension of K.

The differential Galois group of L is defined as

the group of the automorphisms  of L (that map a

solution of [6] onto a solution) leaving the field of

constants fixed. Given a fundamental system of

solutions (’, ), we can associate to each automorph-

ism  the matrix M such that ((’), ( )) = (’, ).M.

By definition, the set of these matrices M is the

Galois group G of eqn [6]. It is a linear algebraic

group (because, the matrices M being symplectic,

their coefficients verify polynomial equations) and a

subgroup of the linear group of matrices GL(C).

We note that, for a given linear system, the mono-

dromy group is contained in the Galois group and

both are subgroups of the symplectic group Sp(C).

In the Galois group G of eqn [6], we consider G

the connected component of the identity. The integr-

ability of the initial Hamiltonian system is connected

to the integrability of the variational equation [6] and,

through it, to the properties of its Galois group:

Theorem of Morales and Ramis (Morales-Ruiz

1999). If an analytic Hamiltonian system is com-

pletely integrable, then the Galois group associated

to the variational equation along a particular

solution  is such that its connected component of

identity G

is Abelian.

Thus, if a Hamiltonian system is such that G

is not

Abelian, there cannot exist a complete set of first

integrals in involution in a neighborhood of the

particular solution  and the system is not integrable.

In the concrete applications of this theory, an

algorithm of Kovacic allows us to determine the

Galois group explicitly. By this method, several

Hamiltonian systems were proved to be nonintegrable:

for instance, systems of points on a line with a

potential in 1=r

,studiedbyJulliard-Tosel (1998),

but also ancient proofs of nonintegrability of homo-

geneous potentials, which were improved by Yoshida

and Umeno, thanks to the theorem of Morales–Ramis.

See also: Billiards in Bounded Convex Domains;

Infinite-Dimensional Hamiltonian Systems; Integrable

Systems: Overview; Peakons; Separatrix Splitting.

Further Reading

Abraham R and Marsden J (1967) In: Benjamin WA (ed.)

Foundations of Mechanics. Reading, MA: Benjamin.

Adler M and van Moerbeke P (1989) Inventiones Mathematicae

97(1): 3–51.

Arnol’d VI (1976) In: Mir (ed.) Me´thodes mathe´matiques de la

Me´canique classique. Moscow: Editions Mir.

Arnol’d VI (1985) Dynamical Systems III, Encyclopedia of

Mathematical Sciences. Berlin: Springer.

Birkhoff D (1927) Dynamical Systems. American Mathematical

Society Colloquium Publication, 9.

He´non M and Heiles C (1964) Astronomical Journal 69: 73–79.

Julliard-Tosel E (1998) Comptes-Rendus de l’Academie des

Sciences, Paris t. 327: (Se´rie I) 387–389.

Kowalevski S (1889) Acta Mathematica 12: 177–232.

Kozlov VV (1983) Russian Mathematical Surveys 38-1: 1–76.

Kozlov VV (1989) Sovietic Science Review, Mathematical

Physics. 8: 1–81.

Lagrange JL (1810) Me´canique analytique. Œuvres comple´tes,

Gauthier-Villars.

Morales-Ruiz JJ (1999) Differential Galois Theory and Non-

Integrability. Basel: Birkhau¨ ser-Verlag.

Poincare´ H (1890) Acta Mathematica 13: 1–270.

Poincare´ H (1892) Les Me´thodes Nouvelles de la Me´canique

Ce´leste. tome I, A. Blanchard.

Whittaker ET (1904) A Treatise on the Analytical Dynamics.

Cambridge: Cambridge University Press.

Ziglin SL (1983) Functional Analysis and Applications 16:

181–189; 17: 6–17.

630 Hamiltonian Systems: Obstructions to Integrability

Hamiltonian Systems: Stability and Instability Theory

P Bernard, Universite

de Paris Dauphine, Paris,

France

The solar system has long appeared to astronomers

and mathematicians as a model of stability. On the

other hand, statistical mechanics relies on the

assumption that large assemblies of particles form

highly unstable systems (at the microscopic scale).

Yet all these physical situations are described, at

least to a certain degree of approximation, by

Hamiltonian systems.

One may hope that Hamiltonian systems can be

classified in two different categories, stable and

unstable ones. However, the situation is much more

complicated and both stable and unstable behaviors

cohabit in typical systems. Even our examples are

not perfect paradigms of stability and instability.

Indeed, it is now clear from numerical as well as

theoretical points of views that some instability is

present over long timescales in the solar systems, so

that for example future collisions between planets

cannot be completely ruled out in view of our

present understanding. On the other hand, unex-

pected patterns of stability have been discovered in

systems involving a large number of particles.

Understanding the impact of stable and unstable

effects in Hamiltonian systems has been considered

ever since Poincare´ as one of the most important

questions in dynamical syst ems. In this article, we

will discuss model Hamiltonian systems of the form



ðq; pÞ¼hðpÞþG



ðq; p Þ

where (q, p) 2T

U, with U a bounded open

subset of R

. Recall that the equations of motion are

qðtÞ¼@

hðpÞþ@



ðq; pÞ½1

pðtÞ¼@



ðq; p Þ½2

The textbook by Arnol’d (1964) is a good general

introduction on Hamiltonian systems. We will always

denote by !(p) the frequency map @

h(p), which plays

a crucial role. Here, as is obvious in [2], the action

variables p are preserved under the evolution in the

unperturbed case  = 0. We will try to explain what is

known on the evolution of these action variables for

the perturbed system. As we will see, in many

situations, these variables are extremely stable. For

example, KAM theorem implies that, for a positive

measure of initial conditions (q

, p

) the trajectory

(q(t ), p(t)) satisfies kp(t)  p(0)kC  for all times.

Examples show that some initial conditions may lead

to unstable trajectories, that is, trajectories such that

kp(t)  p(0)k1=C for some t (depending on )and

some fixed constant C independent of .However,this

is, as we will see, possible only for very large time t

(meaning that t as a function of  hastogotoinfinity

very quickly when  !0). The main questions here

are to understand in what situation instability is or is

not possible, and what kind of evolutions can have the

action variable p. Another important question is to

estimate the speed (as a function of the parameter )of

the evolutions of p.

A Convention

We assume, unless otherwise stated, that the

Hamiltonians are real analytic. The norm jHj of

the Hamiltonian H is the uniform norm of its

holomorphic extension to a certain complex strip.

We do not specify the width of this strip.

Whenever we consider a family H



, F



... of

Hamiltonians, we mean that the norm jH



j is

bounded when  ! 0.

Averaging and Exponential Stability

The first observation concerning the action variables

is that they should evolve at a speed of the order of

. However, averaging effects occur. More precisely,

in the equation

p(t) = @



(q(t ), p(t)), the variable

q(t) is moving fast compared to p(t). If the evolution

of q(t) nicely fills the torus T

, it is tempting to

think that the averaged equation



pðtÞ¼



pðtÞÞ

should approximate accurately the actual behavior

of p(t ), where



ðpÞ:¼

Hðq; pÞdq

We have



V  0, which leads us to think that the

evolution should consist mainly of oscillations of

small amplitude with no large evolution. This

reasoning is limited by the presence of resonances.

Frequencies

A frequency ! 2R

is said to be resonan t if there

exists k 2Z



(= Z

 {0}) such that hk, !i= 0. The

resonance module of !,

Zð!Þ¼fk 2Z

=hk;!i¼0g

is a subgroup Z

; we denote by R(!) the vector

space generated by Z(!)inR

. The order of

resonance r( !) is the dimension of R(!). The main

examples of resonances of order r are the

Hamiltonian Systems: Stability and Instability Theory 631

frequencies ! = (!

, 0), where !

dr

is nonreso-

nant. This example is universal. Indeed, if ! is a

resonant frequency, then there exists a matrix

A 2Gl

(Z) such that A! = (!

, 0), where !

dr

is not resonant. The matrix A can be seen as a

diffeomorphism of T

, which transports the constant

vector field ! to the constant vector field A! = (!

, 0).

It is useful to distinguish, among nonresonant

frequencies, some which are sufficiently nonresonant.

Afrequency! 2R

is called Diophantine if there exist

real constants >0and  d such that

jh!; kij  kkk

1

for each k 2Z



. Finally, a frequency is called

resonant Diophantine if there exists a matrix

A 2Gl

(Z) such that ! = A(!

, 0), where !

a Diophantine frequency.

Symplectic Diffeomorphisms and Normal Forms

An efficient mathematical method to take averaging

effects into account is the use of normal forms.

Normal form theory consists in finding new coordi-

nates in which the fast angles have been eliminated

from the equations up to a small remainder. This is

done exploiting the existence of a large group of

diffeomorphisms preserving the Hamiltonian struc-

ture of equations, called symplectic diffeomorphisms

or canonical transformations. We refer the reader to

standard textbooks for these notions, for example to

Arnol’d (1964). An important point is that a

symplectic diffeomorphism  sends the trajectories

of the Hamiltonian H   to the trajectories of the

Hamiltonian H. A Hamiltonian N(q, p)issaidtobein

R-normal form, where R is a linear subspace of R

,if

N 2R for each (q, p). Let us give an illustrative

result, taken from Lochak et al. (2003). Note that this

result is not sufficient to obtain uniform stability

estimates, as in Nekhoroshev theorem below. More

precise normal form results are given in Nekhoroshev

(1977) and Po¨ schel (1993).

Normal Form Theorem

Let !

= !(p

) be a given Diophantine or resonant-

Diophantine frequency. Let us denote B

) the

open ball of radius r in R

centered at p

. There

exists a constant a which depends only on !, and

constants 

> 0 and C > 0 such that the following

holds: for each <

, there exists an analytic

symplectic embedding 



: T

B

r()

! T

U,

which is -close to identity and such that



 



ðq; pÞ¼hðpÞþN



ðq; pÞþð ÞF



ðq; pÞ

where N is in R(!

)-normal form, r()

ﬃﬃ



, and

() e

C

a

This means that the motions with resonant initial

conditions are confined, up to small oscillations, in

the associated affine plane p(0) þ R(!(p(0)) until

they live in the domain of the normal form, or until

time 

1

().

Geometry of Resonances

In view of the normal form theorem, we are led to

consider the curves P():R ! R

which satisfy

Pð

ÞPðÞ2Rð!ðPðÞÞÞ

for each  and 

. Indeed, it appears that these curves

are the ones the action variables can follow on

timescales not involving the remainders of the

normal forms. Note that here the parameter  is

not the physical time. Assuming that P() is such a

curve, we can define the affine space

R :¼ Pð0Þþ\

 2R

Rð!ðPðÞÞÞ

We then have P() 2R for each . In addition, each

point P(),  2R, is a critical point of the restriction

of the unpertur bed Hamiltonian h to the affine

space R. It follows that the curve P() has to be

constant if the unperturbed Hamiltonian satisfies the

following hypothesis.

Nekhoroshev Steepness

We say that the unperturbed Hamiltonian h is steep

if, for each affine subspace  in R

, the restriction

j

has only isolated critical points.

This formulation, due to Niederman, is much

simpler than the equivalent one first given by

Nekhoroshev. It turns out that this condition,

which was made natural by our heuristic explana-

tion, implies stability over exponential timescales for

all initial conditions (see Nekhoroshev (1977)). We

first need another condition.

Kolmogorov Nondegeneracy

We say that the unperturbed Hamiltonian h is

nondegenerate in the sense of Kolmogorov if it has

nondegenerate Hessian at each point, or equivalently

if the frequency map p 7! !(p) is an immersion.

Nekhoroshev Stability Theorem

Assume that the unperturbed Hamiltonian does not

have critical points (!(p) does not vanish), satisfies

Nekhoroshev steepness and Kolmogorov nondegene-

racy conditions. Then there exists constants a > 0

and b > 0, which depend only on h, and constants



> 0 and C > 0 such that the following holds: for

<

, each trajectory (q



(t), p



(t)) satisfies the

estimate

632 Hamiltonian Systems: Stability and Instability Theory



ðtÞp



ð0Þk  C

for all t such that jtje

C

a

Herman’s Example

In order to illustrate the necessity of the condition of

steepness, let us consider the Hamiltonian



ðq

; q

; p

Þ¼p

þ Vðq

with V : T ! R. The associated equations are

¼ 0;

¼V

;

¼ p

;

¼ p

The trajectories whose initial conditions are sub-

jected to p

(0) = 0 and V

(0)) 6¼ 0 satisfy

ðtÞ¼p

ð0ÞtV

ðq

ð0ÞÞ

ðtÞ¼0; q

ðtÞ¼q

ð0Þ

We see an evolution at speed  of the action variable

contradicting the conclusion of Nekhoroshev

theorem. In this example, we have R(!(p(t))) =

R {0}, and h

jR {0}

 0, so that the curve

PðÞ¼ð; 0Þ

is indeed a curve of critical points of h

jR {0}

Genericity of Steepness

The condition of steepness is frequently satisfied. In

order to be more precise, we mention that, for N 2N

large enough (how large depends on the dimension d),

steepness is a generic condition in the finite-dimen-

sional space of polynomials of degree less than N.

Note in contrast that a quadratic Hamiltonian is steep

if and only if it is positive definite. Finally, it is

important to mention that convex Hamiltonians h

with positive-definite Hessian are steep. More gener-

ally, quasiconvex Hamiltonians are steep. A function

h : U ! R is said to be quasiconvex if, at each point,

the restriction of its Hessian to the kernel of its

differential is positive definite.

The Quasiconvex Case

It is interesting to be more precise about the values

of a and b in Nekhoroshev theorem. We shall do so

in the quasiconvex case, which is the most stable

case, and where much more is known. If h is

quasiconvex, one can take

a ¼ b ¼

as was proved by Lochak (1992). It is a question of

active present research whether these exponents are

optimal. It now appears that this is almost so, and

that the optimal exponent a should not be larger

than 1=2(d  3). That this exponent deteriorates as

the dimension increases is of course very natural in

the perspective of statistical mechanics. As a matter

of fact, not only the exponent a but also the

threshold 

of validity of Nekhoroshev theorem

deteriorates with the dimension, as was noticed in

Bourgain and Kaloshin.

Another important fact was proved in Lochak

(1992): in these expressions, the important value of

d is not the total number of degrees of freedom, but

the number of active degrees of freedom. More

precisely, resonant initial conditions are more stable

than generic ones. If r is the order of resonance of a

given initial condition, then the number d  r of fast

angles can be substituted to the total number of

degrees of freedom for the computation of the

stability exponent. This phenomenon may account

for the surprising stability obtained numerically by

Fermi, Pasta, and Ulam.

Permanent Stability

Many initial conditions satisfy more than exponen-

tial stability: they are permanently stable.

Kolmogorov Theorem

Assume that h satisfies Kolmogorov nondegeneracy

condition (‘‘Kolmogorov nondegeneracy’’). Then for

each open subset V  R

such that



V  U, there

exists 

> 0 such that, for each <

, there exists



a smooth symplectic embedding 



: T

V !

U, which is -close to the identity,



a co mpact subset F



of V, whose relative measure

in V is converging to 1 as  ! 0,

such that the Hamiltoni an system H



 



preserves

the torus T

{p} for each p 2F



The union



¼ 



ðT

F



of all the invariant tori has positive measure. Its

complement is usually an open dense subset of

U. All the orbits starting in this invariant set

obviously undergo oscillations of amplitude of the

order of  for all times. It is worth mentioning that

some energy surfaces may not intersect the invariant

set F



. This is illustrated in example, i.e., ‘‘Herman’s

example,’’ where the surface of zero energy does not

contain invariant tori. The following condition

guarantees the existence of invariant tori on each

energy surface.

Hamiltonian Systems: Stability and Instability Theory 633

Arnol’d Nondegeneracy

The Hamiltoni an h is said to be nondegenerate in

the sense of Arnol’d if it does not have critical points

and if the map

p 7!

!ðpÞ

k!ðpÞk

is a local diffeomorphism between each level set of h

and S

d1

. This is equivalent to say that the function

(, p) 2R U 7! h(p) has nondegenerate Hessian

at each point of the form (1, p).

Arnol’d Theorem

If h satisfies Arnol’d nondegeneracy con dition, then

the relative meas ure of the set F



of invariant tori is

converging to 1 in each energy surface.

This theorem prevents ergodicity of the perturbed

systems for the canonical invariant measure on its

energy surface. This may be considered as a very

disappointing result for statistical mechanics, whose

mathematical foundation has often been considered

to be the Boltzmann hypothesis of ergodicity.

However, statistical mechanics is first of all a

question of letting d go to infinity, and ergodicity

might not be such a crucial hypothesis (see

Khinchin).

When d = 2, the Arnol’d theorem has particularly

strong consequences. Indeed, in this case, the

invariant tori cut the energy surfaces in small

connected components. The motion is then confined

in these connected components. As a consequence,

we obtain permanent stability for all initial

conditions.

In higher dimensions however, the complement of



in each energy shell is usually a dense, connected

open set. There may exist orbits wandering in this

large connected set, although the speed of evolution

of these orbits is limited by Nekhoroshev theory.

Understanding the dynamics in this open set is a

very important and difficult question. It is the

subject of the next section.

Relaxed Assumption

For many applications, such as celestial mechanics,

the nondegeneracy conditions of Arnol’d or Kolmo-

gorov are not satisfied, or difficult to check.

However, the existence of invariant tori has been

proved under much milder assumptions. As a rule,

invariant tori exist in the perturbed systems if the

frequency map p 7! !(p) stably contains Diophan-

tine vectors in its image.

The Mechanism of Arnol’d

Understanding instability is the subject of intense

present research. General methods of construction

of interesting orbits as well as clever classes of

examples are being developed. These methods are

exploring the limits of stability theory. Here we shall

only describe the fundamental ideas of Arnol’d (see

Arnol’d 1964), where most of the present activity

finds its roots. Although these ideas have some

ambition of universality, they are best presented,

like in Arnol’d (1964), on an example. We consider

the quasiconvex Hamiltonian

Hðq

; q

; p

¼ðp

þ p

Þ=2  p

þ  cos 2q

þ ðcos 2q

Þðcos 2q

þ cos 2q

As we have seen, this system is typical of the kind of

Hamiltonians one gets after reduction to resonant

normal form. However, it is illuminating to consider

 not as a function of  but as an independent

parameter. This is an idea of Poincare´ then follow ed

by Arnol’d. We shall expose the main steps of the

proof of the following result.

Theorem

Let us fix numbers 0 < A < B. For each >0, there

exists a number 

() such that, when 0 <<

(),

there exists a trajectory

ðq

ðtÞ; q

ðtÞ; p

ðtÞÞ

and a time T > 0(which depends on  and ) such

that

ð0ÞA; p

ðTÞB

The Truncated System

Let us begin with some remarks about the truncated

Hamiltonian obtained when  = 0:

ðq; pÞ¼H

ðq

; q

; p

ÞþH

ðq

; p

¼ p

=2  p

þ p

=2 þ  cos 2q

This system is the uncoupled product of H

and of

the pendulum described by H

. The variable p

constant along motion; hence, the theorem can not

hold for  = 0.

Recall that the point q

= 0, p

= 0 is a hyperbolic

fixed point of the pendulum H

, p

) = p

=2 þ

 cos 2q

. The stable and unstable manifolds of this

integrable system coincide; they form the energy

level H

= . As a consequence, in the product

system of Hamiltonian H

= H

þ H

, there exists,

634 Hamiltonian Systems: Stability and Instability Theory

in the zero energy level, a one-parameter family T

of invariant tori of dimension 2:

¼fp

¼ !; p

¼ !

=2 þ ; q

¼ 0; p

¼ 0g

 T

R

Each of these tori is hyperbolic in the sense that it has a

stable manifold of dimension 3 and an unstable

manifold of dimension 3, which are nothing but the

liftings of the stable and unstable manifolds of the

hyperbolic fixed point of H

.Noticethatthese

manifolds do not intersect transversally along T

When  6¼ 0, the perturbation is chosen in such a

way that the tori T

are left invariant by the

Hamiltonian flow.

Splitting

For 0 <<

(), the invariant tori T

still have

stable and unstable manifolds of dimension 3. These

stable and unstable manifolds intersect transversally

in the energy surface, along an orbit which is

homoclinic to the torus.

The first point is that the tori remain hyperbolic,

and that the stable and unstable manifolds are

deformed, but not destroyed by the additional

term. This results from the observation that the

manifold M formed by the union of the invariant

tori is normally hyperbolic in its energy surface.

Note that this step does not require exponential

smallness of .

It is then a very general result that the stable and

unstable manifolds have nonempty intersection. It is

a global property, which can be established by

variational methods, and which still does not rest on

exponential smallness of .

The key point, where exponential smallness is

required, is transversality. Since transversality is a

generic phenomenon, one may think that this step is

not so crucial. And indeed, it is very likely that the

statement remains true for most values of  2]0, ]

(and not only for   

()). However, there are two

important issues here. First, trans versality is difficult

to estab lish on explicit examples. Second, it is useful

for many further discussion s to obtain some quanti-

tative estimates.

Indeed, we can associat e to the intersection

between the stable and unstable manifolds a

quantity, the splitting, which in a sense measures

transversality. Discussions on such a definition are

available in Lochak et al. (2003). Using methods of

Poincare´ and Melnikov, Arnol’d showed that this

splitting can be estimated, for sufficiently small ,by

  e

C=

ﬃﬃ



þ Oð

Þ½3

This implies non-nullity of the splitting, hence

transversality, for small .

Transition Chain

We have established the existence, when >0is

small enough, of a family T

of hyperbolic invariant

tori such that the stable manifold W

and the

unstable manifold W



intersect transversally along a

homoclinic orbit (but not along T

!) for each !.

A stabi lity argument shows that the stable mani-

fold W

of the torus T

intersects transversally the

stable manifold W



of the torus T

when ! is close

enough to !

. How close directly depends on the

size of the splitting. We obtain heteroclinic orbits

between tori close to each other.

Given two values ! and !

, we can find a sequence

,1 i  N, such that !

= !, !

= !

, and W



intersects transversa lly W

iþ1

for all i. The associated

family T

of tori is called a transition chain.

The left step consists in proving that some orbits

shadow the transition chain. Arnol’d solved this step

by a very simple topological argument which,

however, does not provide any estimate on the

time T. He proves the existence of an orbit joining

any neighborhood of T

to any neighborhood of T

This ends the proof of the mai n theore m, since we

can chose ! and !

such that !<A < B <!

The dynamics associat ed to hyperbolic tori and

transition chains have later been studied more

carefully. It particul ar, a -lemma can be proved in

this context, which allows us to conclude that, in a

transition chain, the unstable manifold W



of the

first torus intersects transversally the stable manifold

of the last torus W

. These detailed studies also

allow us to relate the speed of diffusion to the

splitting of the invariant manifolds.

Diffusion Speed

It is interesting to estimate the speed of evolution of

the variable p

, or in other words the time T in the

statement. It follows from Nekhoroshev theory that

this time T has to be exponentially large as a

function of . In fact, it is possible to prove, either by

recent developments on the ideas of Arnol’d exposed

above, or more easily by variational methods, (Bessi

1996) that

T 

ﬃﬃ



 log 

for   

(). This time is of course highly related to

the estimate [3] of the splitting. In addition, Ugo Bessi

proved that one can take 

() = e

C=

ﬃﬃ



. Plugging this

value of  in the estimate of T,wegettheestimate

T  e

ﬃﬃ



as a function of the only parameter .

Hamiltonian Systems: Stability and Instability Theory 635