Eulerian equilibria of a triaxial gyrostat in the three-body problem: rotational Poisson dynamics in Eulerian equilibria. (English) Zbl 1200.70007
Summary: We consider the noncanonical Hamiltonian dynamics of a triaxial gyrostat in the three-body problem. By means of geometric-mechanics methods, we study the approximated dynamics that arises when we expand the potential in series of Legendre functions and then truncate the series to the second harmonics. Working in the reduced problem, we study the existence of equilibria that we denominate Eulerian in analogy with classical results on the topic. In this way, we generalize the classical results on equilibria of the three-body problem and many results obtained by other authors using more classical techniques for rigid bodies. The instability of Eulerian equilibria is proven in this approximate dynamics, if the gyrostat is close to the sphere. We examine the rotational Poisson dynamics of the gyrostat placed at an Eulerian equilibrium, and study the nonlinear stability of some equilibria. The analysis is done in vectorial form, avoiding the use of canonical variables and tedious expressions associated with them.
MSC:
| 70F07 | Three-body problems |
| 70E05 | Motion of the gyroscope |
| 70E50 | Stability problems in rigid body dynamics |
| 70H05 | Hamilton’s equations |
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