Bounded, periodic relative motion using canonical epicyclic orbital elements

In this paper, we use a Hamiltonian approach to derive the equations of motion for an object relative to a circular or slightly elliptical reference orbit. By solving the Hamilton-Jacobi equation we develop constants of the relative motion called epicyclic elements. A perturbation Hamiltonian is formulated in order to derive variational equations for the " constants" via Hamilton's equations. We use this formalism to derive bounded, periodic orbits in the presence of various perturbations. In particular, we show a simple no-drift condition that guarantees bounded orbits in the presence of J ₂ forces. We also derive the relative motion deviations and boundedness conditions due to eccentricity of the reference orbit and higher-order terms in the gravitational potential.