International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 72978, 17 pages

On Eulerian equilibria in K-order approximation of the gyrostat in the three-body problem

J. A. Vera and A. Vigueras

Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena 30203, Murcia, Spain

Received 9 June 2006; Revised 21 September 2006; Accepted 9 November 2006

Copyright © 2006 J. A. Vera and A. Vigueras. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the noncanonical 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 of Euler type in analogy with classic results on the topic, are considered. Necessary and sufficient conditions for their existence in an approximate dynamics of order k are obtained and we give explicit expressions of these equilibria, useful for the later study of the stability of the same ones. A complete study of the number of Eulerian equilibria is made in approximate dynamics of orders zero and one. We obtain the main result of this work, the number of Eulerian equilibria in an approximate dynamics of order k for k1 is independent of the order of truncation of the potential if the gyrostat S0 is close to the sphere. The instability of Eulerian equilibria is proven for any approximate dynamics if the gyrostat is close to the sphere. In this way, we generalize the classical results on equilibria of the three-body problem and many of those obtained by other authors using more classic techniques for the case of rigid bodies.