Projective Transformations for Regularized Central-Force Dynamics: Hamiltonian Formulation

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 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. Governing equations are numerically validated for the classic two-body problem incorporating the J2 gravitational perturbation.

💡 Research Summary

The paper presents a novel Hamiltonian‑based regularization and linearization technique for central‑force dynamics, extending the classic Burdet‑Ferrándiz (BF) projective transformation into a full canonical (symplectic) map that includes both configuration and momentum variables. By introducing a family of point transformations parameterized by two exponents (n) and (m) (with the authors ultimately selecting (n=m=-1)), the physical position vector (\mathbf r) is expressed as (\mathbf r = \mathbf q / u), where (\mathbf q) is a unit direction vector and (u) is the reciprocal radial distance. This mapping raises the dimensionality of the phase space (from 6 to 8 variables) but preserves the canonical structure when supplemented with an appropriate generating function.

A second key ingredient is the re‑parameterization of the evolution variable. While traditional regularizations use (dt = r^2 ds) or (dt = r^2/\ell, d\tau), the authors adopt (dt = u^{-2} ds), which together with the chosen projective map converts the original nonlinear Hamiltonian into a completely linear one: \