Cassini states and Lunar free librations from a reduced 3-DoF Hamiltonian model

The spin–orbit coupled motion of a rigid body on a Keplerian orbit is a classical problem in celestial mechanics, historically motivated by the rotational dynamics of the Moon. A Hamiltonian formulation is developed in this work to describe the motion of a slow-rotating rigid body subject to a gravity-gradient potential. Modified Serret–Andoyer variables and Delaunay elements, combined with canonical transformations and averaging over the fast orbital mean anomaly, reduce the original four-degree-of-freedom system to a three-degree-of-freedom system governing the slow resonant dynamics. In this formulation, the fourth canonical pair consists of the orbital mean anomaly and its conjugate action, while the nodal and apsidal rates are prescribed as external orbital precessions. Fourier–Bessel expansions of the orbital motion identify the dominant spin–orbit commensurabilities and drive the reduction. Relative equilibria corresponding to generalized Cassini states are computed, and closed-form algebraic conditions for the ecliptic obliquity are derived across the five dominant commensurabilities. Applied to the Earth–Moon system, linearization about the 1 : 1 resonant equilibrium recovers longitudinal, latitudinal, and wobble libration periods of 2.895, 24.076, and 74.686 years, with relative errors of $$0.13%$$ 0.13 % , $$0.33%$$ 0.33 % , and $$0.08%$$ 0.08 % , respectively, relative to the values reported by Rambaux and Williams (2011).