Projective Transformations for Regularized Central Force Dynamics: Hamiltonian Formulation

Abstract: This work introduces a Hamiltonian approach to regularization and linearization of central force particle dynamics through a new canonical extension of the so-called "projective decomposition". The regularization scheme is formulated within the framework of classic analytical Hamiltonian dynamics as a redundant-dimensional canonical/symplectic coordinate transformation, combined with an evolution parameter transformation, on extended phase space. By considering a generalized version of the standard projective decomposition, we obtain a family of such canonical transformations which differ at the momentum level. From this family of transformations, a preferred canonical coordinate set is chosen that possesses a simple and intuitive connection to the particle's local reference frame. Using this transformation, closed-form solutions are readily obtained for inverse square and inverse cubic radial forces (or any superposition thereof) on any finite-dimensional Euclidean space. From these solutions, a new set of orbit elements for Kepler-Coulomb dynamics is derived, along with their variational equations for arbitrary perturbations (singularity-free in all cases besides rectilinear motion). Governing equations are numerically validated for the classic two-body problem, incorporating the J_2 gravitational perturbation.