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Projective Transformations for Regularized Central-Force Dynamics: Hamiltonian Formulation

Joseph T. A. Peterson, Manoranjan Majji, John L. Junkins

TL;DR

The paper develops a Hamiltonian framework that extends Burdet–Ferrándiz projective transformations into canonical, symplectic coordinates with an augmented (redundant) configuration space. By selecting the n=m=−1 case, it yields a preferred projective coordinate set tied to the LVLH frame, decoupling angular and radial dynamics and enabling closed-form, linear solutions for Kepler- and Manev-type central forces using time reparameterizations t → s or t → τ (true anomaly). The approach generalizes to arbitrary central forces and perturbations in an extended phase space, includes systematic derivations from Hamilton's principle, and provides a concrete J2-perturbed two-body validation, highlighting the method’s accuracy and numerical stability. Compared to KS, the projective canonical transform offers similar linearization/regularization benefits with different geometric intuition (LVLH-based), broader Manev applicability, and a more direct decoupling of motion components. Overall, the work broadens the toolbox for regularized central-force dynamics by delivering a family of canonical projective transformations with explicit, closed-form solutions and practical computational benefits.

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.

Projective Transformations for Regularized Central-Force Dynamics: Hamiltonian Formulation

TL;DR

The paper develops a Hamiltonian framework that extends Burdet–Ferrándiz projective transformations into canonical, symplectic coordinates with an augmented (redundant) configuration space. By selecting the n=m=−1 case, it yields a preferred projective coordinate set tied to the LVLH frame, decoupling angular and radial dynamics and enabling closed-form, linear solutions for Kepler- and Manev-type central forces using time reparameterizations t → s or t → τ (true anomaly). The approach generalizes to arbitrary central forces and perturbations in an extended phase space, includes systematic derivations from Hamilton's principle, and provides a concrete J2-perturbed two-body validation, highlighting the method’s accuracy and numerical stability. Compared to KS, the projective canonical transform offers similar linearization/regularization benefits with different geometric intuition (LVLH-based), broader Manev applicability, and a more direct decoupling of motion components. Overall, the work broadens the toolbox for regularized central-force dynamics by delivering a family of canonical projective transformations with explicit, closed-form solutions and practical computational benefits.

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.

Paper Structure

This paper contains 73 sections, 206 equations, 2 figures.

Table of Contents

  1. Projective Point Transformations for Linearized Newtonian Kepler Dynamics.
  2. Projective Canonical Transformations for Linearized Hamiltonian Kepler Dynamics.
  3. The Present Work.
  4. Comparisons to the KS Transformation.
  5. OUTLINE & SUMMARY OF THIS WORK
  6. Summary: Canonical Transformations from Minimal to Redundant Coordinates.
  7. Summary: A Family of Canonical Transformations for Projective Coordinates
  8. Original Cartesian Coordinate Formulation.
  9. A Family of Canonical Transformations.
  10. The Inverse Transformation.
  11. Summary: A Preferred Canonical Projective Coordinate Set
  12. Recovering Cartesian Coordinate Numerical Solutions.
  13. Summary: "Time" Reparameterization, Linearization, Closed-Form Solutions
  14. Rotational Motion.
  15. Radial Motion.
  16. ...and 58 more sections

Figures (2)

  • Figure 1: Verifying accuracy of the canonical equations of motion for $({{\boldsymbol{q}}},u,{{\boldsymbol{p}}},p_u)$, showing errors in the classic orbit elements (COEs), as compared to propagating the modified equinoctial elements (MEEs). ($J_2$ gravitational term included.)
  • Figure 3: Verification that the STM for $({{\boldsymbol{q}}},u,{{\boldsymbol{p}}},p_u)$, with $t$ as the evolution parameter, is a symplectic matrix. ($J_2$ gravitational term included.)