Projective Transformations for Regularized Central-Force Dynamics: Hamiltonian Formulation

TL;DR

This paper develops a Hamiltonian-based regularization method for central-force dynamics using projective transformations, enabling simplified solutions and numerical validation for perturbed two-body problems.

Contribution

It introduces a novel canonical extension of projective decomposition within Hamiltonian dynamics for regularizing and linearizing central-force systems.

Findings

Closed-form solutions for inverse-square and inverse-cubic forces.

Numerical validation with two-body problem including J2 perturbation.

A new coordinate transformation linked to the particle's local frame.

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 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.