Eulerian Equilibria of a Gyrostat in Newtonian Interaction with Two Rigid Bodies

In this paper the noncanonical Hamiltonian dynamics of a gyrostat in the three body problem will be examined. By means of geometric-mechanics methods we will study the approximate dynamics that arises when we develop the potential in Legendre series and truncate the series to the second harmonics. Some relative equilibria, called Eulerian, of the dynamics of a gyrostat in Newtonian interaction with two rigid bodies will be studied. Taking advantage of the results obtained in previous papers, working on the reduced problem, we will study the bifurcations of these relative equilibria. The instability of Eulerian relative equilibria if the gyrostat is close to a sphere is proven. The rotational Poisson dynamics of the gyrostat placed in an Eulerian equilibrium and the study of the nonlinear stability of some equilibria are considered. The analysis is done in vectorial form avoiding the use of canonical variables and the tedious expressions associated with these variables. In this way, the classic results on equilibria of the three body problem, many of them obtained by other authors who had made used of more classic techniques, are generalized.