Effective Hamiltonian for the D’Alembert Planetary Model near a Spin/Orbit Resonance

Abstract

The D’Alembert model for the spin/orbit problem in celestial mechanics is considered. Using a Hamiltonian formalism, it is shown that in a small neighborhood of a p : q spin/orbit resonance with (p, q) different from (1, 1) and (2, 1) the ‘effective’ D’Alembert Hamiltonian is a completely integrable system with phase space foliated by maximal invariant curves; instead, in a small neighborhood of a p : q spin/orbit resonance with (p, q) equal to (1, 1) or (2, 1) the ‘effective’ D’Alembert Hamiltonian has a phase portrait similar to that of the standard pendulum (elliptic and hyperbolic equilibria, separatrices, invariant curves of different homotopy). A fast averaging with respect to the ‘mean anomaly’ is also performed (by means of Nekhoroshev techniques) showing that, up to exponentially small terms, the resonant D’Alembert Hamiltonian is described by a twodegrees-of-freedom, properly degenerate Hamiltonian having the lowest order terms corresponding to the ‘effective’ Hamiltonian mentioned above.

Supported by MURST-Italy Variational Methods and Nonlinear Differential Equations.