ABSTRACTS
| Alain Albouy (Paris) |
Self-Similar Motions. The N-body Problem and
the N-Vortex Problem
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The relative equilibria in the problem
of N point vortices (or Helmholtz' vortices) and
in the Newtonian problem of N point masses are
given by closely related algebraic systems. We will
try to compare the theory of self-similar
motion (or homographic motion) in both cases.
We will focus on the question of the description
of the sets of the possible configurations and of
the possible motions, using the tools developped in
A. Albouy, A. Chenciner, Le probleme des N corps et les
distances mutuelles, Inventiones Mathematicae 131 (1998)
pp. 151--184
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| Mitchell Berger (UCL) |
Hamiltonian Dynamics Generated by Vassiliev Invariants
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We employ higher order winding numbers to generate Hamiltonian motion of
particles in two dimensions. The ordinary winding number counts how many
times two particles rotate about each other. Higher order winding numbers
measure braiding motions of three or more particles. These winding numbers
relate to various invariants known in topology and knot theory and can be
derived from Vassiliev-Kontsevich integrals.
For just two particles, the Hamiltonian gives the familiar motion of two
point-vortices. However, for three or more particles, the Hamiltonian
generates more complicated intertwining patterns. We examine the dynamics
for the cases of 3 or 4 particles, and show these are completely
integrable. The Hamiltonian provides an elegant method for generating
simple geometrical examples of complicated braids and links.
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| Leo Butler (Northwestern) |
The Fundamental Group of a Manifold with an Integrable
Geodesic Flow
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Let us say that a flow is integrable if an open, dense subset
L of its phase space M is fibred by invariant tori, and that it is
tamely integrable if the complement of L is a tamely embedded
polyhedron. We show that if a C^2 geodesic flow \phi_t : S\Sigma \to
S\Sigma is tamely integrable then \pi_1(\Sigma) contains a polycyclic
subgroup of finite index.
As a corollary, we obtain a topological obstruction to the integrability
of the Euler-Lagrange flow of a positive-definite Lagrangian system.
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| Kuo-Chang Chen (Northwestern) |
Constructing Periodic Solutions for the Planar 4N-Body
Problem by Variational Methods
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Following the idea of constructing the celebrated
Figure-8 orbit [Chenciner and Montgomery, Ann. of Math., 2000],
we use variational methods to construct a class of periodic
solutions for the planar 4N-body problem with equal masses. These
orbits share the feature that the configuration changes periodically
from a regular 4N-gon to two homothetic regular 2N-gons. The
existence for the case N=1 can be rigorously proved. For the case
N >1, we will show some numerical evidences for their existence.
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| Alain Chenciner (Paris) |
From Lagrange to the Eight: Marchal's P12 Family
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Christian Marchal discovered a very natural way to connect the Eight
solution of the equal mass three-body problem to the Lagrange equilateral
relative equilibrium solution, through spatial choreographies in a
rotating frame. The proof rests on a remarkable theorem on action minimizing
paths with fixed ends but leaves open the problem of unicity and hence the
continuity of the family with respect to the parameter.
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| Jacques Fejoz (Paris) |
Diffusion in the Five-Body Problem
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I will give a model of Arnold diffusion in the planar five-body
problem. In this purpose I build some normal forms called the secular
systems. I hope the global dynamics of these secular systems will show
some drift behavior, along the lines of geometric mechanism due to
R. Moeckel.
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| Angel Jorba (Barcelona) |
Dynamics Near the Lagrangian Points of the Earth-Moon System
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In this work we consider the motion of an infinitessimal particle near
the equilateral points of the real Earth-Moon system. We use, as real
system, the one provided by the JPL ephemeris: the ephemeris give the
positions of the main bodies of the solar system (Earth, Moon, Sun and
planets) so it is not difficult to write the vectorfield for the
motion of a small particle under the attraction of those bodies.
Numerical integrations of this vectorfield show that trajectories with
initial conditions in a vicinity of the equilateral points escape
after a short time.
On the other hand, it is known that the Restricted Three Body Problem
is not a good model for this problem, since it predicts a quite large
region of practical stability. For this reason, we will discuss some
intermediate models that try to account for the effect of the Sun and
the eccentricity of the Moon. As we will see, they are more similar to
the real system in the sense that the vicinity of the equilateral
points is also unstable. However, these models have some families of
lower dimensional tori (2-D and 3-D), some of them elliptic and some
of them hyperbolic. The ellipic ones give rise to a region of
effective stablity at some distance of the triangular points in the
above mentioned models. It is remarkable that these regions seem to
persist in the real system, at least for time spans of 1000 years.
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| Wang Sang Koon (Caltech) |
Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design
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The invariant manifold structures of the collinear libration points for
the spatial
restricted three-body problem provide the framework for understanding
complex dynamical phenomena from a geometric point of view. In particular, the
stable and unstable invariant manifold ``tubes'' associated to libration
point orbits are the phase space structures that
provide a conduit for orbits between
primary bodies for separate three-body systems. These invariant manifold
tubes can be used to construct new spacecraft trajectories, such as a
''Petit Grand Tour'' of the moons of Jupiter. Previous work focused on the
planar circular restricted three-body problem. The current work extends
the results to the spatial case.
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| Vadim Kuznetsov (Leeds) |
Solution of Inverse Problem for Integrable Lattices
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We develop a new technique that allows to solve
the inverse problem for integrable lattices in the most explicit
form, namely to describe a map from the linearizing (separation)
variables to the initial (local) variables of an integrable lattice.
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| Eduardo Leandro (Paris) |
Finiteness of Some Symmetrical Classes of Central Configurations
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We address the Wintner-Smale conjecture on the finiteness of the
number of classes of central configurations of the N-body problem with
fixed masses. We prove finiteness for some symmetrical classes
corresponding to d-dimensional configurations of (d+2)-bodies, where d>=2.
Our proof is based on the concept of rational parametrization, which we
introduced in our research, and is inspired in the classical Be'zout's
theorem from Algebraic Geometry. Possible applications to more general problems
are discussed.
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| Ken Meyer (Cincinatti) |
Are Hamiltonian Flows Geodesic Flows?
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When a Hamiltonian system has a ''Kinetic + Potential'' structure,
the resulting flow is locally a geodesic flow. But there may be
singularities of
the geodesic structure, so the local structure does not always imply
that the
flow is globally a geodesic flow. In order for a flow to be a geodesic
flow,
the underlying manifold must have the structure of a unit tangent
bundle. We
develop homological conditions for a manifold to have such a structure.
We apply these criteria to several classical examples: a particle in
a potential well, double spherical pendulum, the Kovalevskaya top, and the
N-body problem. We show that the flow of the reduced planar N-body
problem and the reduced spatial 3-body are never geodesic flows except when
the angular momentum is zero and the energy is positive.
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| Shane Ross (Caltech) |
Invariant Manifolds and Transport in the Three-Body Problem
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The dynamics of comets and other solar system objects which have a
three-body energy close to that of the collinear Lagrange points are known
to exhibit a complicated array of behaviors such as rapid transition
between the interior and exterior Hill's regions, temporary capture, and
collision. The invariant manifold structures of the collinear Lagrange
points for the restricted three-body problem, which exist for a range of
energies, provide the framework for understanding these complex dynamical
phenomena from a geometric point of view. In particular, the stable and
unstable invariant manifold ''tubes'' associated to Lagrange point orbits
are the phase space structures that provide a conduit for orbits
travelling to and from the smaller primary body (e.g., Jupiter), and
between primary bodies for separate three-body systems (e.g., Saturn and
Jupiter).
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| Luca Sbano (Warwick) |
Exchange Orbits with Negative Energy in the Three-Body Problem
with Short-Range Newtonian Potential
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We study the problem of exchange orbits in a three-body system
in R^3 with short range Newtonian potential. In the three body problem,
'exchange orbits' are trajectories in which
at large times the system is decomposed in a binary system and the third
body at infinite relative distance from the center of mass of the
binaries. The binary system is composed of different bodies at t=-\infty
and t=+\infty. The system has generic masses and we find an orbit which
has total negative energy and the
bodies exchange takes place possibly with collisions.
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| Zhang Shiqing (Chongqing) |
Non-Planar and Non-Collision Periodic Solutions for Newtonian n > 3 Body
Problems in 3-dimensional Space.
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Using the variational methods and the theory of central configurations,we
prove the existence of a non-planar and non-collision periodic solution for
Newtonian n > 3 body problems with any masses in 3-dimensional space.
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| Ian Stewart (Warwick) |
Coupled Cell Systems and Multibody Problems
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The study of symmetry in nonlinear dynamics has led to
the concept of a 'coupled cell system', a dynamical system
equipped with distinguished projections onto variables
that represent component 'cells' of the system.
An N-body system can be viewed as a coupled cell system,
with the attractive extra ingredient of Hamiltonian dynamics.
The talk will review interesting phenomena known to occur
in coupled cell systems, including phase-locking and
'multirhythms', or symmetry-induced resonances. Possible
connections with N-body dynamics will be discussed. For
example, `choreographies' are discrete travelling waves
of a kind typical in coupled cell systems. (However,
this observation does not of itself make it any easier to
study choreographies!)
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| Susanna Terracini (Milan) |
Topological Aspects in the Variational Approach to the Search
for Periodic Solutions
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The topological properties of the free loop space
play a key role in the study to the periodic N-body problem,
both in the 2 and 3 dimensional case. We shall discuss some
recent results that may enlight the link between topology and
periodic solutions.
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| Tadashi Tokieda (Montreal) |
Point Vortices on Surfaces
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| Alexei Tsygvintsev (Loughborough) |
Complex Monodromy, Differential Galois Theory and the Three-Body Problem.
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In this talk we present some results about the monodromy (Galois)
group of normal variational equations of the three-body problem around the
Lagrangian orbits.
[1] A. Tsygvintsev, The meromorphic non-integrability of the three-body
problem, Journal fur die Reine und Angewandte Mathematik de Gruyter, N 537, 2001
[2] A. Tsygvintsev,Sur l'absence d'une integrale premiere supplementaire
meromorphe dans le probleme plan des trois corps C.R. Acad. Sci. Paris, t. 333,
Serie I, p. 125-128, 2001
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| Andre Vanderbauwhede (Gent) |
Continuation and Bifurcation of Periodic Orbits in the 3-Body Problem
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In Hamiltonian systems (with or without additional symmetries) the
typical conditions
needed for the (numerical) continuation of periodic orbits by
pseudo-arclength techniques
are never satisfied, due to the existence of one or more first
integrals. To remedy this
problem we generalize the concept of normal periodic orbits
introduced a few years ago
by Sepulchre and MacKay, and we show how pseudo-arclength continuation can be
implemented at such normal periodic orbits by unfolding the
Hamiltonian vectorfield with
some appropriate dissipative terms. In the second part of the talk we
describe the results
which came out of the application of this technique to the restricted
3-body problem and
to the continuation of the figure eight solution of the full 3-body
problem found
recently by Chenciner and Montgomery.
This is joint work with Jorge Galan (Sevilla), Eusebius Doedel
(Montreal) and Randy Paffenroth (Pasadena).
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| Andrea Venturelli (Milan) |
A Family of Z_4 x Z_{2q} - Symmetric Periodic
Solutions in the Spatial Four Body Problem with Equal Masses
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We consider four equal masses in R^3 with a newtonian gravitational interaction
and we impose a given Z_4-symmetry on the
configuration in such a way that the system has only three degree of freedom
with a cyclic angular coordinate. We impose
a given Z_{2q}-symmetry on the space of periodic loop with fixed period in
the configuration space and we study the minima of the
action functional. We then prove that minima are collision free and are not
homographic solutions in certains homotopy classes.
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| Laurent Wiesenfeld (Grenoble) |
Transport in N>2 Degree of Freedom Hamiltonian Systems :
Fractal Domains and Transition States
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Transport in some simple scattering n=3 degree of freedom system will be
described in two limits. The short time limit is dominated by
transition states which separate two regions of phase
space usually called ''reactants'' and ''products''. The transition
state of a Henon-Heiles with 3 degrees of freedom will be described,
in linearized and non linearized setting.
In the long-time limit, we shall describe the domains that
correspond to the various exit channels (transition states) as
transported backwards in time towards an entrance channel. The domains
have a Wada Lake (fractal) structure, which will be shown.
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