1201.6356 (Elena A. Kudryavtseva)

Periodic solutions of planetary systems with satellites and the
averaging method in systems with fast and slow variables    [PDF]

Elena A. Kudryavtseva

We study the partial case of the planar $N+1$ body problem, $N\ge2$, of the
type of planetary system with satellites. We assume that one of the bodies (the
Sun) is much heavier than the other bodies ("planets" and "satellites"),
moreover the planets are much heavier than the satellites, and the "years" are
much longer than the "months". We prove that, under a nondegeneracy condition
which in general holds, there exist at least $2^{N-2}$ smooth 2-parameter
families of symmetric periodic solutions in a rotating coordinate system such
that the distances between each planet and its satellites are much shorter than
the distances between the Sun and the planets. We describe generating symmetric
periodic solutions and prove that the nondegeneracy condition is necessary. We
give sufficient conditions for some periodic solutions to be orbitally stable
in linear approximation. Via the averaging method, the results are extended to
a class of Hamiltonian systems with fast and slow variables close to the
systems of semi-direct product type.

View original: http://arxiv.org/abs/1201.6356