Hamilton-Jacobi modelling of stellar dynamics

One of the physical settings emerging in the galaxy and stellar dynamics is motion of a single star and a stellar cluster about a galaxy center. The potential availability of analytical treatment of this problem stems from the smallness of mass of the star and cluster relative to the galactic mass, giving rise to Hill's restricted three-body problem in the galaxy-cluster-star context. Based on this observation, this paper presents a Hamiltonian approach to modelling stellar motion by the derivation of canonical coordinates for the dynamics of a star relative to a star cluster. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton-Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic orbital elements. The effect of an arbitrary cluster potential is incorporated into the analysis by a variation of parameters procedure. A numerical optimization technique is developed based on the new orbital elements, and quasiperiodic stellar orbits are found.