Table Of ContentTHE DYNAMICAL BEHAVIOUR OF OUR PLANETARY SYSTEM
THE DYNAMICAL BEHAVIOUR OF OUR
PLANETARY SYSTEM
Proceedings of the Fourth
Alexander von Humboldt
Colloquium on Celestial Mechanics
Edited by
R. DVORAK
Institut für Astronomie, Universitätssternwarte, Vienna, Austria
and
J. HENRARD
Departement de Mathematique, FNDP, Namur, Belgium
Partly reprinted from Celestial Mechanics and Dynamical Astronomy
Volume 65, Nos. 1-2,1996/7
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C.LP. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6320-3 ISBN 978-94-011-5510-6 (eBook)
DOI 10.1007/978-94-011-5510-6
Printed on acid-free paper
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©1997 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1997
Softcover reprint of the hardcover 1st Edition 1997
No part of the material protected by this copyright notice may be reproduced or
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TABLE OF CONTENTS
Proceedings of the
Fourth Alexander von Humboldt Colloquium on Celestial Mechanics
THE DYNAMICAL BEHAVIOUR OF OUR PLANETARY SYSTEM
Preface
INVITED PAPERS
Theoretical Dynamics
A. NEISHTADT I Scattering by Resonances 1
S. ICHTIAROGLOU I Non-Integrability in Hamiltonian Mechanics 21
G. CONTOPOULOS and E. GROUSOUZAKOU /
Regular and Irregular Periodic Orbits 33
C. SIOPIS, H.E. KANDRUP, G. CONTOPOULOS and R. DVORAK I
Universal Properties of Escape in Dynamical Systems 57
Numerical Methods of Analysis
M. SIDLICHOVSKY / Mapping and Dynamical Systems 69
K. WODNAR / A New Fast Fourier Method for Evaluating Fourier
Spectra at Arbitrary Frequencies 85
Planets and Satellites
P. BRETAGNON I On the Use of Analytical Theories
in the Calculation of Precession-Nutation 95
Asteroids
A. MORBIDELLI and M. GUZZO / The Nekhoroshev Theorem
and the Asteroid Belt Dynamical System 107
M. SIDLICHOVSKY and D. NESVORNY I Frequency Modified
Fourier Transform and its Application to Asteroids 137
B. ERDI / The Trojan Problem 149
Ch. FROESCHLE / Dynamical Transport to Planet-Crossing Orbits 165
M. MOONS I Review of the Dynamics in the Kirkwood Gaps 175
Miscellaneous
V. SZEBEHELY / Open Problems on the Eve of the Next Millennium 205
H. EICHHORN I Limit of Comprehension, Reference Frames
Reference Frames and General Relativity 213
Theoretical Dynamics
A. LEMAITRE / Stable Chaos: A First Model 223
H. VARVOGLIS, Ch. VOZIKIS and B. BARBANIS /
Transport in Perturbed Integrable Hamiltonian Systems
and the Fractality of Phase Space 233
V.W. KUDIELKA / Equilibria Bifurcations of Satellite Orbits 243
Numerical Methods of Analysis
E. LEGA and C.FROESCHLE / Fast Lyapunov Indicators.
Comparison with Other Chaos Indicators.
Application to Two and Four Dimensional Maps 257
J. HADJIDEMETRIOU and A. LEMAITRE / : Asteroid Motion
near the 2: 1 Resonance: A Symplectic Mapping Approach 277
J. WALDVOGEL / Symplectic Integrators for Hill's Lunar Problem 291
VI-SUI SUN and JI-LIN ZHOU / Rotation Number
and Global Stability of Symplectic Integrator 307
Planets and Satellites
C. BEAU GE / On a Global Expansion of the Planar Disturbing Function 321
x. MOISSON / Links betwen TIme Scales
Using Barycentric Relativistic Ephemerides 333
S. MIKKOLA and K. INNANEN / Orbital Stability
of Planetary Quasi-Satellites 345
P.J. MESSAGE / Alternative Derivations of Perturbations
with Use of a Mixed Reference-Plane System:
Application to the Motion of the Orbit Plane of Hyperion 357
C. MARCHAL / Astrodynamic Study of the Earth Rotation 365
Asteroids
S. FERRAZ-MELLO and T.A. MICHTCHENKO /
Orbital Evolution of Asteroids in the Hecuba Gap 377
E. LOHINGER/ A Study of the Local Lyapunov Numbers
for Orbits in the Outer Solar System 385
Miscellaneous
W. SCHAFFENBERGER and A. HANSLMEIER / On the Influence
of Supernova Shockfronts on the Stability of the Solar System 393
C. MARCHAL / Henri Poincare:
A Decisive Contribution to Special Relativity 403
J. KALLRATH, R. DVORAK and J. SCHLODER /
Periodic Orbits in the Sitnikov Problem 415
Participants in the Fourth Alexander von Humboldt Colloquium on Celestial Mechanics
PREFACE
It is now a well established tradition that every four years, at the end of winter,
a group of "celestial mechanicians" from all over the world gather at the "Alpen
gasthof Peter Rosegger" in the Styrian Alps (Ramsau, Austria).
This time the colloquium was held from March 17 to March 23, 1996 and
was devoted to the Dynamical Behaviour of our Planetary System. The papers
covered a large range of questions of current interest: theoretical questions (reso-
nances, universal properties, non integrability, transport, ... ) and questions about
numerical tools ( symplectic maps, indicators of chaos, ... ) were particularly well
represented; the never ending problem of the sculpting of the asteroid belt was also
qui te popular.
You will find in the following pages a pot-pourri of what we listen to; you will
miss of course the diversity of accents with which the tunes were delivered: from
China, from Japan, from Brazil, from the United-States of America and from all
over Europe, East and West.
Let us not forget that the comet 199682 (Hyakutake) came to visit us; many
an evening was spent on the deck of the Alpengasthof contemplating this celestial
visitor who liked to play hide-and-seek behind the spruce trees.
To be able to organize this Fourth Alexander von Humbolt Colloquium, we
have to thank primarily the generosity of the Austrian Ministry of Science. Also the
Steiermarkischen Wissenschafts-und Landesfonds, the Osterreichische Forschungs
gemeinschaft and especially the Bank -Austria made it possible to invite partici
pants from abroad. Latter supported our meeting also in form of "Tagungsunterla
gen"; coffee during the whole meeting time was donated by Hornig-Kaffee Graz.
Many thanks are due to the Chairwoman A. Lemal'tre and Chairmen: G. Con
topoulos, R. Dvorak, H. Eichhorn, C. Froeschle, S. Ferraz-Mello, P.K. Seidelmann,
V. Szebehely, Yi-Sui Sun. They acted as Editors of the papers presented during
"their" session, arranging for competent and fast refereeing so that the papers could
be reviewed and, when necessary, corrected.
Our model hosts Barbara and Fritz Walcher contributed a lot to the success of
the meeting, as they did their best providing pleasant and friendly surroundings.
We all are looking forward to the next millennium when we plan to meet again at
the AlpengasthofRosegger.
R. Dvorak J. Henrard
SCATTERING BY RESONANCES
ANATOLY NEISHTADT
Space Research Institute,
Pro!soyuznaya 84132 , Moscow 117810, Russia
Abstract. Small perturbations imposed on an integrable system produce an evolution of the motion.
In the course of this evolution the system can pass through a state 6f resonance. Quasi-random
phenomena of capture into the resonance and scattering on the resonance are discussed in this
paper for the case of a two-frequency system. Major topics are calculation of probability of capture,
description of the motion of the captured phase points, calculation of a probabilistic distribution of
the scattering amplitude.
Key words: Resonance - adiabatic invariant - capture into resonance - scattering
1. Introduction
Consider a system which differs from an integrable one by a small perturbation. The
motion in such a system can be approximately represented as a slowly evolving un
perturbed one. For the description of this evolution the classical averaging method
prescribes to average the differential equations of the perturbed motion over the
phases of the unperturbed oscillations. In the course of the evolution the frequen
cies of motion are changing slowly and at some time they become resonant, i.e.
linearly rationally dependent. Due to the influence of resonances the actual motion
can be considerably different from the one predicted by the averaging method.
The two basic phenomena that are.associated with the effect of a single resonance
are capture into the resonance and scattering by the resonance.
Capture into a resonance can be described as follows. First the system evolves
as it is predicted by the averaging method. At a certain time the system approaches
a resonance. After that the system evolves in such a way that the resonant relation
between frequencies is approximately kept. As a result, after a time '" 1 / c the
state of the system is completely different from the one predicted by the aver
aging method. Here c is a small parameter characterizing the magnitude of the
perturbation. The set of initial conditions for trajectories with capture has a small
measure O( Vi) (we consider capture at time t E (0,1/ c). Initial conditions for
trajectories with capture and trajectories without capture are mixed if c is small.
Thus, it is reasonable to consider the capture as a random event and to calculate a
probability of this event. Capture into a resonance was first discussed in [(10; 17)]
in connection with problems of celestial mechanics.
Scattering by a resonance can be described as follows. Far from the resonance
the system oscillates with an amplitude'" c near the state predicted by the averaging
method. During a passage through a narrow neighborhood of the resonant state,
Vi.
the parameters undergo a jump'" The amplitude of this jump is very sensitive
to the variation of initial conditions if c is small. Thus, it is reasonable to consider
this jump as a random scattering by the resonance. Scattering by a resonance was
Celestial Mechanics and Dynamical Astronomy 65: 1-20, 1997.
©1997 Kluwer Academic Publishers.
2
ANATOLY NEISHfADT
first discussed in [(8)] for a Hamiltonian system depending on a slowly varying
parameter and in [(4)] in a general context.
The word "scattering" in the title of the paper is used in a broad sense which
includes the changes in the system state due both to capture into resonance and
scattering by a resonance in the restricted sense that was described above.
Major topics of this review paper are:
-calculation of probability of a capture into resonance,
-description of motion after capture into resonance,
- calculation of the statistical distribution of the amplitude of scattering by a
resonance.
For Hamiltonian systems with slowly varying parameters these topics are con
sidered in [(25; 26)].
2. Perturbed Systems with Rotating Phases
The differential equations of motion of a system which differs by a small pertur
bation from an integrable one can often be written in the following standard fonn
(see, e.g., [(1; 2)]):
i=€I(l,<p,€), <p=w(l)+€g(l,<p,€). (1)
Here € is a small parameter characterizing the amplitude of the perturbation. If
€ = 0, the system is unperturbed. Variables lED C Rn are called slow variables.
Variables <p modd 211" E Tm = Rm /(211" Zr are called fast variables or phases, w
I,
is a vector of (unperturbed) frequencies, the functions g are 211"-periodic in the
components of <p. System (1) is called a system with rotating phases.
The averaging method gives a recipe for the approximate description of the slow
variables 1 evolution in a time interval of order 1/ €. According to this method, one
should replace the system (1) with the averaged system
1
. 1
J = €F( J), F( J) = (211" )m JTm I( J, <p, 0) d<p. (2)
We denote by J( t) the solution of the averaged system (2) with initial condition
10 E Do c D at t = 0 and by (l(t), <pet)) the solution of system with rotating
= =
phases (1) with initial condition (lo, <po) E Do x Tm. So, 1(0) J(O) 10. We
suppose that the solution J (t) is defined and kept at a positive distance from the
boundary of the domain D for 0 ::; t ::; 1/€ .
If the system possesses only one frequency (m = 1) and this frequency w does
notvanish,thenthe following estimate is valid: 11(t)-J(t)1 = O(€), 0::; t::; 1/€
(see, e.g. [(1; 7)]).
In general multi-frequency case (m ;::: 2, ~~ -=I- 0) a point I is called a resonant
one, if (k,w(I)) = 0 for some integer vector k E zm; here (.,.) is the standard
scalar product in Rm. The set of resonant points for a given k is a surface in the
space of the slow variables, which is called a resonant surface. When a phase point
SCA'ITERING BYRESONANCES 3
of (1) approaches a resonant surface, it can be captured into the resonance and
undergo a scattering as it is described in the introduction.
= =
Henceforth we will consider two-frequency systems: m 2, <P (<pI, !.p2), W =
(WI'Wz). We will assume thatw, Ile=o,gle=o E C4,and I,g E C2.
In the two-frequency case, resonances are represented, in the frequency plane,
by straight lines with rational slopes passing through the origin of the coordinates.
Such a simple structure of the resonant set simplifies the problem essentially. This
first has been used in [(9)] for a Hamiltonian system depending on a slowly varying
parameter and in [(4)] for general system in the form (1). Under some additional
assumptions (see (3), (5)below) it is possible to investigate the effect of each
resonance separately and combine the results afterwards.
A two-frequency system is considered in [(4; 2)] under the assumption that the
rate at which the frequency ratio changes along the trajectories of the system (1) is
different from zero:
I(Wz OoWJl - WI ooWJz )I(I, <p, 0)1 > ci- I , (3)
CI = const > O. This condition prevents capture into resonances. It is shown that
IJ(t) - J(t)1 = O( ve), 0::; t ::; lie:. (4)
Henceforth system (1) is considered under the assumption that the rate at which
the frequency ratio changes along the trajectories of the averaged system (2) is
different from zero [(4)]:
1£1 > (5)
£
Under this assumption and some additional "general position" assumptions it is
shown [(19)] that
IJ(t) - J(t)1 = O(veloge:), 0::; t ::; lie:, (6)
for all initial conditions except of a set of a measure O(.,fi) (see, also, [(15)]).
Henceforth we will use assumption (5) also with some additional "general
position" assumptions that are formulated in Subsection 3.3.
3. Reduction Near a Resonant Surface
Near the resonant surface of a given resonance system, (1) can be transformed into
a standard "perturbed pendulum-like system" form. This transformation has been
used in a number of works (see [(5; 8; 13; 16; 17; 18; 19)]).
Let k = (kl' k2) be a vector with integer co-prime components. The value
Ik I = Ik tl + Ik 21 is called the order of the k- resonance. We assume that Ik I 1.
rv
Description:It is now a well established tradition that every four years, at the end of winter, a group of "celestial mechanicians" from all over the world gather at the "Alpen gasthof Peter Rosegger" in the Styrian Alps (Ramsau, Austria). This time the colloquium was held from March 17 to March 23, 1996 and
