Table Of ContentHamiltonian Mechanics
Integrability and Chaotic Behavior
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Series B: Physics
Hamiltonian Mechanics
Integrability and Chaotic Behavior
Edited by
John Seimenis
University of the Aegean
Samos, Greece
and N. Copernicus University
Torun, Poland
Springer Science+Business Media, LLC
Proceedings of a NATO Advanced Research Workshop on
Hamiltonian Mechanics,
held June 28-July 2, 1993, in Torun, Poland
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Library of Congress Cataloging in Publication Information
Hamiltonian mechanics : integrabi 1 ity and chaotic behavior / edited by
John Seimenis.
p. cm. — (MATO ASI series. Series B, Physics ; v. 331)
xPublished in cooperation with NATO Scientific Affairs Division."
"Proceedings of a NATO Advanced Research Workshop on Hamiltonian
Mechanics, held June 28-Ju1 y 2, 1993, In Torun, Poland"—T.p. verso.
Includes bibliographical references and index.
ISBN 978-1-4899-0966-4
1. Mathematical physics—Congresses. 2. Hamiltonian systems-
-Congresses. 3. Chaotic behavior in systems—Congresses.
4. Astrophysics—Mathematics—Congresses. I. Seimenis. John.
II. North Atlantic Treaty Organization. Scientific Affairs
Division. III. NATO Advanced Research Workshop on Hamiltonian
Mechanics (1993 : Torun, Poland) IV. Series.
QC19..H36 1994
530. 1 ' 55352~dc20 94-40008
CIP
ISBN 978-1-4899-0966-4 ISBN 978-1-4899-0964-0 (eBook)
DOI 10.1007/978-1-4899-0964-0
© Springer Science+Business Media New York 1994
Originally published by Plenum Press, New York in 1994
Softcover reprint of the hardcover 1st edition 1994
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SPECIAL PROGRAM ON CHAOS, ORDER, AND PATTERNS
This book contains the proceedings of a NATO Advanced Research Workshop held
within the program of activities of the NATO Special Program on Chaos, Order, and
Patterns.
Volume 208 - MEASURES OF COMPLEXITY AND CHAOS
edited by Neal B. Abraham, Alfonso M. Albano,
Anthony Passamante, and Paul E. Rapp
Volume 225 - NONLINEAR EVOLUTION OF SPATIO-TEMPORAL STRUCTURES
IN DISSIPATIVE CONTINUOUS SYSTEMS
edited by F. H. Busse and L. Kramer
Volume 235 - DISORDER AND FRACTURE
edited by J. C. Charmet, S. Roux, and E. Guyon
Volume 236 - MICROSCOPIC SIMULATIONS OF COMPLEX FLOWS
edited by Michel Mareschal
Volume 240 - GLOBAL CLIMATE AND ECOSYSTEM CHANGE
edited by Gordon J. MacDonald and Luigi Sertorio
Volume 243 - DAVYDOV'S SOLITON REVISITED: Self-Trapping of Vibrational
Energy in Protein
edited by Peter L. Christiansen and Alwyn C. Scott
Volume 244 - NONLINEAR WAVE PROCESSES IN EXCITABLE MEDIA
edited by Arun V. Holden, Mario Markus, and Hans G. Othmer
Volume 245 - DIFFERENTIAL GEOMETRIC METHODS IN THEORETICAL
PHYSICS: Physics and Geometry
edited by Ling-Lie Chau and Werner Nahm
Volume 256 - INFORMATION DYNAMICS
edited by Harald Atmanspacher and Herbert Scheingraber
Volume 260 - SELF-ORGANIZATION, EMERGING PROPERTIES, AND
LEARNING
edited by Agnessa Babloyantz
Volume 263 - BIOLOGICALLY INSPIRED PHYSICS
edited by L. Peliti
Volume 264 - MICROSCOPIC ASPECTS OF NONLINEARITY IN CONDENSED
MADER
edited by A. R. Bishop, V. L. Pokrovsky, and V. Tognetti
Volume 268 - THE GLOBAL GEOMETRY OF TURBULENCE: Impact of Nonlinear
Dynamics
edited by Javier Jimenez
SPECIAL PROGRAM ON CHAOS, ORDER, AND PATTERNS
Volume 270 - COMPLEXITY, CHAOS, AND BIOLOGICAL EVOLUTION
edited by Erik Mosekilde and Lis Mosekilde
Volume 272 - PREDICTABILITY, STABILITY, AND CHAOS IN N-BODY
DYNAMICAL SYSTEMS
edited by Archie E. Roy
Volume 276 - GROWTH AND FORM: Nonlinear Aspects
edited by M. Ben Amar, P. Pelce, and P. Tabeling
Volume 278 - PAINLEVE TRANSCENDENTS: Their Asymptotics and Physical
Applications
edited by Decio Levi and Pavel Winternitz
Volume 280 - CHAOS, ORDER, AND PATTERNS
edited by Roberto Artuso, Predrag Cvitanovic, and Giulio Casati
Volume 284 - ASYMPTOTICS BEYOND ALL ORDERS
edited by Harvey Segur, Saleh Tanveer, and Herbert Levine
Volume 291 - PROTEIN TRANSFER IN HYDROGEN-BONDED SYSTEMS
edited by T. Bountis
Volume 292 - MICROSCOPIC SIMULATIONS OF COMPLEX HYDRODYNAMIC
PHENOMENA
edited by Michel Mareschal and Brad Lee Holian
Volume 298 - CHAOTIC DYNAMICS: Theory and Practice
edited by T. Bountis
Volume 312 - FUTURE DIRECTIONS OF NONLINEAR DYNAMICS IN PHYSICAL
AND BIOLOGICAL SYSTEMS
edited by P. L. Christiansen, J. C. Eilbeck, and R. D. Parmentier
Volume 320 - SINGULAR LIMITS OF DISPERSIVE WAVES
edited by N. M. Ercolani, I. R. Gabitov, C. D. Levermore, and
D. Serre
Volume 323 - SOFT ORDER IN PHYSICAL SYSTEMS
edited by Y. Rabin and R. Bruinsma
Volume 329 - NONLINEAR COHERENT STRUCTURES IN PHYSICS
AND BIOLOGY
edited by K. H. Spatschek and F. G. Mertens
Volume 331 - HAMILTONIAN MECHANICS: Integrability and Chaotic Behavior
edited by John Seimenis
PREFACE
This volume contains invited papers and contributions delivered at the International
Conference on Hamiltonian Mechanics: Integrability and Chaotic Behaviour, held in
Tornn, Poland during the summer of 1993. The conference was supported by the NATO
Scientific and Environmental Affairs Division as an Advanced Research Workshop. In
fact, it was the first scientific conference in all Eastern Europe supported by NATO.
The meeting was expected to establish contacts between East and West experts as well as
to study the current state of the art in the area of Hamiltonian Mechanics and its
applications. I am sure that the informal atmosphere of the city of Torun, the birthplace
of Nicolaus Copernicus, stimulated many valuable scientific exchanges.
The first idea for this cnference was carried out by Prof Andrzej J. Maciejewski and
myself, more than two years ago, during his visit in Greece. It was planned for about
forty well-known scientists from East and West. At that time participation of a scientist
from Eastern Europe in an Organising Committee of a NATO Conference was not
allowed. But always there is the first time. Our plans for such a "small" conference, as a
first attempt in the new European situation -the Europe without borders - quickly passed
away. The names of our invited speakers, authorities in their field, were a magnet for
many colleagues from all over the world. More than one hundred and forty colleagues
from twenty-five countries and all continents, expressed their interest to participate in our
conference. Among them, distinguished scientists from the United States and the former
Soviet Union. We were very glad to have the majority of them among us, in Tonm.
The conference was funded by a major grant from NATO. Additional support was
provided by the N. Copernicus University. I am indebted to Professor Andrzej
Jamiolkowski, Rector of the N. Copernicus University and Chairman of the Local
Organising Committee of our conference, for his continued help and support during the
organisational period. I am sure that without his help, we would not have been able to
finish this meeting succesfully.
I would like to express my thanks to my colleague and friend Andrzej J. Maciejewski for
his efforts during the time he was a member of the Scientific Organising Committee.
Many thanks are due to Professors Carles Simo and Tassos Bountis for their advice as
members of the Scientific Organising Committee. I would like to also thank, the
members of the Local Organising Committee, Krzysztof Gozdziewski and Ewa
Derkowska as well as the members of the staff of the Faculty of Physics and Astronomy
of the N. Copernicus University for their help, during the period of the conference.
All papers of the volume are refereed. I am indebted to the referees - especially to the
external experts -for their kind assistance and their efforts to be fast, in order to give the
volume for publication, in proper time.
In the end, I would like to apologise to my son Kimon for the time I spent far from him,
in order to finish successfully my work as Director of the NATO Advanced Research
Workshop and editor of the proceedings. The part of the work that was carried out by
me during the last two years, is dedicated to him.
John Seimenis, Editor
vii
CONTENTS
Invited papers
Non-integrability criterion of Hamiltonian systems based on Ziglin's Theorem
and its· relation to the singular point analysis ...................................................... ..
H. Yoshida
Averaging under fast quasiperiodic forcing . ..... ... .... ...... .............. .... ......... .... ...... 13
C. Simo
Natural boundaries of normalizing transformations ........................................... 35
L. Billi, M. Malavasi and O. Turchetti
Singular perturbation in Hamiltonian Mechanics .............. .................. .............. 43
A.D. Bruno
The structure of Chaos ......................................................................................... 51
O. Contopouios
From Integrability to Chaos: Examples of interrelations between physics
and dynamics for minor bodies in the solar system .... .............. .......... ............... 73
C. Froeschle, Ph. Bendjoya and A. Cellino
Successive elimination of harmonics: A way to explore the resonant
structure of a Hamiltonian system ...................... .............. ........ .......... ................ 89
A. MorbideIli
Periodic solutions of nonlinear Schrodinger equations and the Nash-Moser
method ............................................................................................................... 103
W. Craig and C. Eugene Wayne
On the tendency toward ergodicity with increasing number of degrees
of freedom in Hamiltonian systems .. ....... ....... .......... .......... ............ ............... 123
L. Hurd, C. Grebogi and E.Ott
Gibbsian check of the validity of Gibbsian calculation through
dynamical observables ........ .... .............. .......... ....................... ................... ......... 131
D.F. Escande, H. Kantz, R. Livi and S. Ruffo
Adiabatic invariants and time scales for energy sharing in models
of classical gases ................................................................................................ 139
O. Benettin
Numerical integration of Hamiltonian systems in the presence of
additional integrals: Application of the observer method ................................. 153
A.J. Maciejewski and J-M Strelcyn
viii
Symmetries and topology of dynamical systems with two degrees
of freedom .......................................................................................................... 167
V. Kozlov
Contributed papers
Variational criteria for nonintegrability and chaos in Hamiltonian
systems .............................................................................................................. 173
S. Bolotin
Exponentially small splitting in Hamiltonian systems ........ ...... .......... ...... ....... 181
A. Delshams and T.M. Seara
Integrable and chaotic behaviour in the Paul trap and the hydrogen atom
in a generalized van der Waals potential .......................................................... 189
D. Farrelly, I.E. Howard and T. Uzer
Recent applications of Hamiltonian dynamics to accelerator physics ............. 197
E. Todesco
Singularity analysis of 2D complexified Hamiltonian systems ....................... 205
S. Abenda and A. Bazzani
Perturbation theory for systems without global action-angle
coordinates .................................................. ..................................................... 213
F. Fasso
A non-integrability test for perturbed Hamiltonian systems of two
degrees of freedom ... ....................................................................................... 221
E. Meletlidou and S. Ichtiaroglou
Librational invariant surfaces in the spin-orbit problem ................................ 229
A. Celletti
Normalization of resonant Hamiltonians ......................................................... 237
D. Farrelly, I. Hwripherys and T. Uzer
Effective stability for periodically perturbed Hamiltonian systems ." ............. 245
A. Jorba and C. Sima
Bihamiltonian systems and Lax representation ............................................... 253
I.I. Morales and R. Ramirez
An efficient method for computing periodic orbits of conservative
dynamical systems ............................................................................................. 261
M.N. Vrahatis and T.C. Bountis
The dynamics of trace maps ............................................................................. 275
I.A.G. Roberts and M. Baake
Scars in groups of eigenfunctions .................................................................... 287
F. Borondo, G.G. de Polavieja and R.M. Benito
A model of Poincare and rigorous proof of the second element
of thermodynamics from mechanics ............................................................... 295
L.D. Pustyl'nikov
Nekhoroshev and KAM theorems revisited via a unified approach ............... 299
A. Delshams and P. Gutierrez
ix
