Hamiltonian structure and relative equilibria of an axisymmetric rigid body in a second degree and order gravity field: cylindrical and generalized hyperbolic equilibria

The full dynamics of an axisymmetric rigid body in a uniformly rotating second degree and order gravity field are investigated, where orbit and attitude motions of the body are coupled through the gravity. Compared with the classical orbital dynamics with the body considered as a point mass, the full dynamics is a higher-precision model in close proximity of the central body where the gravitational orbit–attitude coupling is significant, such as a spacecraft about a small asteroid or an irregular-shaped natural satellite about a planet. The full dynamics are modeled by using the non-canonical Hamiltonian structure, in terms of variables expressed in the frame fixed with the central body. A Poisson reduction is carried out by means of the axial symmetry of the body, and a reduced system with lower dimension, as well as its non-canonical Hamiltonian structure and equations of motion, is obtained through the reduction process. With the second-order potential, three types of relative equilibria are found to be possible: cylindrical equilibria, generalized hyperbolic equilibria, and conic equilibria, which are counterparts to cylindrical equilibria, hyperbolic equilibria, and conic equilibria of an axisymmetric rigid body in a spherical gravity field, respectively. The geometrical properties and existence of the cylindrical equilibria and generalized hyperbolic equilibria are investigated in detail. It has been found that compared with the classical results in a spherical gravity field, the relative equilibria in this study are more complicated and diverse. The most significant difference is that the non-spherical gravity field enables the existence of non-Lagrangian hyperbolic equilibria, called generalized hyperbolic, which cannot exist in a spherical gravity.