Linear and Regular Kepler-Manev Dynamics via Projective Transformations: A Geometric Perspective

TL;DR

This paper introduces a geometric approach using projective transformations to regularize and linearize Kepler and Manev dynamics, providing explicit solutions and handling perturbations.

Contribution

It develops a novel geometric framework for transforming and linearizing nonconservative Hamiltonian systems like Kepler and Manev dynamics.

Findings

Achieves full linearization of Kepler and Manev systems in any finite dimension.

Provides explicit solutions for unperturbed Kepler and Manev cases.

Includes a method to handle arbitrary perturbations within the geometric framework.

Abstract

This work presents a geometric formulation for transforming nonconservative mechanical Hamiltonian systems and introduces a new method for regularizing and linearizing central force dynamics -- in particular, Kepler and Manev dynamics -- through a projective transformation. The transformation is formulated as a configuration space diffeomorphism (rather than a submersion) that is lifted to a cotangent bundle (phase space) symplectomorphism and used to pullback the original mechanical Hamiltonian system, Riemannian kinetic energy metric, and other key geometric objects. Full linearization of both Kepler and Manev dynamics (in any finite dimension) is achieved by a subsequent conformal scaling of the projectively-transformed Hamiltonian vector field. Two such conformal scalings are given, both achieving linearization. Arbitrary conservative and nonconservative perturbations are included, with closed-form solutions readily obtained in the unperturbed Kepler or Manev cases.