Arnold's seminar
September 11, 2001

E.A.Kudryavtseva (Moscow State University)

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

We analyse the following partial case of the planar $N$--body problem,
$N\ge3$, which describes motions of the `planet system with satellits''
type. The mass of one body (Sun) is much greater than the masses of other
bodie
(planets and satellites), and the mass of any planet is much greater than
the masses of its satellites. Besides, distances between each planet and its
satellites are much smaller than the distance between the Sun and this
planet, and the angular velocity of planet rotation about the Sun is much
smaller than angular velocities of satellite rotations about the planet. We
prove that in the case of some natural relation between small parameters
of the problem, under some nondegeneracy condition, there exist at least
$2^{N-3}$ families of periodic solutions of the problem under
consideration. We describe `generating'' symmetric periodic solutions and
prove that the nondegeneracy condition is necessary. We give sufficient
conditions for the stability in linear approximation of some of periodic
solutions.