Mathematics – Dynamical Systems
Scientific paper
2012-01-29
Mathematics
Dynamical Systems
40 pages, 1 figure, 1 table, English translation of the Russian version v3
Scientific paper
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 slow and fast variables close to the systems of semi-direct product type.
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