Planar rotations for a gyrostat in the three-body problem

Abstract

We consider the non-canonical Hamiltonian dynamics of a gyrostat in the three body problem. By means of geometric-mechanics methods we study the approximate Poisson dynamics that arises when we develop the potential in series of Legendre and truncate this in an arbitrary order k. Working in the reduced problem,the existence and number of equilibria, that we denominate planar rotation type in analogy with classic results on the topic, is considered. Necessary and sufficient conditions for their existence in a approximate dynamics of order k is obtained and we give explicit expressions of this equilibria, useful for the later study of the stability of the same ones. A complete study of the planar rotation type equilibria is made in approximate dynamics or order zero and one.