Eulerian equilibria of a triaxial gyrostat in the three body problem: rotational Poisson dynamics in Eulerian equilibria

We consider the noncanonical Hamiltonian dynamics of a triaxial gyrostat in the three body problem. By means of geometric-mechanics methods, we will study the approximated dynamics that arises when we develop the potential in series of Legendre and truncate the series to the second harmonics. Working in the reduced problem, we will study the existence of equilibria that we will denominate of Euler in analogy with classic results on the topic. 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. The instability of Eulerian equilibria is proven in this approximate dynamics if the gyrostat is close to the sphere. The rotational Poisson dynamics of the gyrostat placed at an Eulerian equilibrium and the study of the nonlinear stability of some equilibria is considered. The analysis is done in vectorial form avoiding the use of canonical variables and the tedious expressions associated with them.