Talk at Arnold's seminar on Sep 11, 2001

Elena Kudryavtseva
(Moscow State University)

Periodic solutions of the $N$-body problem
and applications to planet systems with satellites

The periodic solutions of the $N$-body problem were investigated in
partial cases by G.W.Hill (Sun and a planet with a satellite),
H.Poincar\'e (Sun and two planets) and later by G.A.Krassinskii
(Sun and planets), V.N.Tkhai (Sun and a planet with satellites).
We consider the {\it planar $N$-body problem}
for the Sun and the arbitrary number of planets and the arbitrary number of
satellites, $N\ge 3$. Here the mass of the Sun is supposed to be $1$, the
masse
have orders $O(\mu\nu)$ where $\mu$ and $\nu$ are `small parameters''.
We investigate the periodic solutions of
this problem applying technics of the perturbation theory and the average
method on a submanifold.

The solution of the $N$-body problem is called {\it relative periodic}
or {\it periodic} if
there exists a pair of real numbers $(T,\alpha)$, called {\it relative
period}, such that $T>0$, $-\pi < \alpha \le \pi$ and,
for any $t\in \RR$, the configuration of the mass points at time $t+T$
is obtained by rotation of the configuration at time $t$
by the angle $\alpha$ around the barycenter.
The solution is called {\it symmetric} if there exists time
at which all the mass points
are on the same line (i.e.\ one can watch all the planets and
the satellites of the Solar system on `parade'') and their velocities
are orthogonal to this line.

As the `unperturbed problem'' one considers the collection of $N-1$
Kepler problems for independent motions of planets around the Sun
and satellites around planets.
As the `generating periodic solutions'' one considers the `circular''
solutions of the Kepler problems with the common relative period
$(T,\alpha)$.
That is, planets uniformly rotate around the Sun along different circular
orbits with constant angle velocities $\omega_i$, and satellites
uniformly rotate around their planets along different circular
orbits with constant angle velocities $\omega_{ij}$.

We prove: The $N$-body problem under consideration has
$(T,\alpha)$-periodic solutions close to the generating solutions,
provided that the nondegeneracy condition
$|\alpha| > {\omega_i^2 \over |\omega_{ij}|} T$ holds, that the
fractions $|{\omega_i \over \omega_{ij}}|$ of the `months'' to the
`years'' are sufficiently small, and that the parameters $\mu$ and
$\nu$ are small enough. Moreover, exactely $2^{N-3}$ of these
solutions are symmetric, and hence, every half-period $T\over 2$ one
can watch a parade for them.

In the case $\alpha=0$ there does not exist, in general, such a solution.

For the case that all planets rotate in the same (positive) direction,
sufficient conditions are given for the stability of some periodic
solutions in linear approximation.