Benettin, Henrard, Kuksin. Hamiltonian dynamics.. theory and applications (LNM1861, Springer, 2005)(ISBN 3540240640)(188s)_PD_.pdf

Lecture Notes in Mathematics

1861

Editors:

J.--M. Morel, Cachan

F. Takens, Groningen

B. Teissier, Paris

Subseries:

Fondazione C.I.M.E., Firenze

Adviser: Pietro Zecca

Giancarlo Benettin

Jacques Henrard

Sergei Kuksin

Hamiltonian Dynamics

Theory and Applications

Lecturesgivenatthe

C.I.M.E.-E.M.S. Summer School

held in Cetraro, Italy,

July

1999

Editor: Antonio Giorgilli

123

Editors and Authors

Giancarlo Benettin

Dipartimento di Matematica Pura e Applicata

Universit`adiPadova

ViaG.Belzoni7

35131

Padova, Italy

e-mail: benettin@math.unipd.it

Antonio Giorgilli

Dipartimento di Matematica e Applicazioni

Universita degli Studi di Milano Bicocca

Via Bicocca degli Arcimboldi

20126

Milano, Italy

e-mail: antonio@matapp.unimib.it

Jacques Henrard

Departement de Mathematiques

FUNDP 8

Rempart de la Vierge

5000

Namur, Belgium

e-mail: Jacques.Henrard@fundp.ac.be

Sergei Kuksin

Department of Mathematics

Heriot-Watt University

Edinburgh

EH14 4AS

,UnitedKingdom

and

Steklov Institute of Mathematics

8GubkinaSt.

111966

Moscow, Russia

e-mail: kuksin@ma.hw.ac.uk

LibraryofCongressControlNumber:

2004116724

Mathematics Subject Classification (2000):

70H07, 70H14, 37K55, 35Q53, 70H11, 70E17

ISSN

0075-8434

ISBN

3-540-24064-0

Springer Berlin Heidelberg New York

DOI:

10.1007

b104338

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Preface

“ Nous sommes donc conduit a nous proposer le probleme suivant:

Etudier les equations canoniques

dx i

∂F

∂x i

en supposant que la function F peut se developper suivant les

puissances d’un parametre tres petit µ de la maniere suivante:

∂F

∂y i

dy i

−

F = F 0 + µF 1 + µ 2 F 2 + ... ,

en supposant de plus que F 0 ne depend que des x et est independent

des y ;etque F 1 ,F 2 ,... sont des fonctions periodiques de periode

2 π par rapport aux y .”

This is all of the contents of

13 in the ﬁrst volume of the celebrated treatise

Les methodes nouvelles de la mecanique celeste of Poincare, published in 1892.

In more usual notations and words, the problem is to investigate the dy-

namics of a canonical system of diﬀerential equations with Hamiltonian

H ( p, q, ε )= H 0 ( p )+ εH 1 ( p, q )+ ε 2 H 2 ( p, q )+ ... ,

(1)

R n

where p

≡

( p 1 ,...,p n )

∈G⊂

are action variables in the open set

T n are angle variables, and ε is a small parameter.

The lectures by Giancarlo Benettin, Jacques Henrard and Sergej Kuksin

published in the present book address some of the many questions that are

hidden behind the simple sentence above.

≡

( q 1 ,...,q n )

∈

1. A Classical Problem

It is well known that the investigations of Poincare were motivated by a clas-

sical problem: the stability of the Solar System. The three volumes of the

Preface

Methodes Nouvelles had been preceded by the memoir Sur le probleme des

trois corps et les equations de la dynamique; memoire couronn´eduprixde

S. M. le Roi Oscar II le 21 janvier 1889 .

It may be interesting to recall the subject of the investigation, as stated

in the announcement of the competition for King Oscar’s prize:

“ A system being given of a number whatever of particles attracting

one another mutually according to Newton’s law, it is proposed,

on the assumption that there never takes place an impact of two

particles to expand the coordinates of each particle in a series pro-

ceeding according to some known functions of time and converging

uniformly for any space of time. ”

In the announcement it is also mentioned that the question was suggested

by a claim made by Lejeune–Dirichlet in a letter to a friend that he had

been able to demonstrate the stability of the solar system by integrating the

diﬀerential equations of Mechanics. However, Dirichlet died shortly after, and

no reference to his method was actually found in his notes.

As a matter of fact, in his memoir and in the Methodes Nouvelles Poincare

seems to end up with diﬀerent conclusions. Just to mention a few results of his

work, let me recall the theorem on generic non–existence of ﬁrst integrals, the

recurrence theorem, the divergence of classical perturbation series as a typical

fact, the discovery of asymptotic solutions and the existence of homoclinic

points.

Needless to say, the work of Poincare represents the starting point of most

of the research on dynamical systems in the XX–th century. It has also been

said that the memoir on the problem of three bodies is “the ﬁrst textbook

in the qualitative theory of dynamical systems”, perhaps forgetting that the

qualitative study of dynamics had been undertaken by Poincareina Memoire

sur les courbes deﬁnies par une equation diﬀerentielle , published in 1882.

2. KAM Theory

Let me recall a few known facts about the system (1). For ε = 0 the Hamilto-

nian possesses n ﬁrst integrals p 1 ,...,p n that are independent, and the orbits

lie on invariant tori carrying periodic or quasi–periodic motions with frequen-

cies ω 1 ( p ) ,...,ω n ( p ), where ω j ( p )= ∂H 0

∂p j

. This is the unperturbed dynamics.

For ε

= 0 this plain behaviour is destroyed, and the problem is to understand

how the dynamics actually changes.

The classical methods of perturbation theory, as started by Lagrange and

Laplace, may be resumed by saying that one tries to prove that for ε

the system (1) is still integrable. However, this program encountered major

diculties due to the appearance in the expansions of the so called secular

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