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Moser Regularization of a Stochastically Perturbed Kepler Problem

Archishman Saha

TL;DR

This work analyzes a Kepler problem perturbed by angular-momentum–oriented stochastic noise and shows that key invariants, specifically $||\mathbf{J}||$ and $||\mathbf{A}||$, remain constant in magnitude while the distance $||\mathbf{q}||$ and speed follow deterministic trajectories. By applying a Moser-type regularization, the stochastic dynamics are mapped to a stochastic geodesic flow on the 3-sphere, with the noise structure preserved by the map, enabling a geometric treatment of collisions. The study demonstrates that collision solutions correspond to great circles on $S^3$ through the north pole and that the regularization extends to general negative energy levels via scaling, maintaining a symplectic framework. This provides a robust bridge between stochastic Hamiltonian perturbations and classical geometric regularization, offering a deterministic backbone for certain stochastic Keplerian phenomena.

Abstract

We consider a stochastic Kepler problem perturbed by a Hamiltonian noise affecting the angular momentum vector. We show that the angular momentum and the Laplace-Runge-Lenz vectors are conserved in magnitude and as a consequence, the distance and speed of the particle follow deterministic dynamics. Further, in a procedure similar to Moser's regularization, we transform the stochastic Kepler problem to obtain its dynamics as a stochastic geodesic flow on a 3-sphere.

Moser Regularization of a Stochastically Perturbed Kepler Problem

TL;DR

This work analyzes a Kepler problem perturbed by angular-momentum–oriented stochastic noise and shows that key invariants, specifically and , remain constant in magnitude while the distance and speed follow deterministic trajectories. By applying a Moser-type regularization, the stochastic dynamics are mapped to a stochastic geodesic flow on the 3-sphere, with the noise structure preserved by the map, enabling a geometric treatment of collisions. The study demonstrates that collision solutions correspond to great circles on through the north pole and that the regularization extends to general negative energy levels via scaling, maintaining a symplectic framework. This provides a robust bridge between stochastic Hamiltonian perturbations and classical geometric regularization, offering a deterministic backbone for certain stochastic Keplerian phenomena.

Abstract

We consider a stochastic Kepler problem perturbed by a Hamiltonian noise affecting the angular momentum vector. We show that the angular momentum and the Laplace-Runge-Lenz vectors are conserved in magnitude and as a consequence, the distance and speed of the particle follow deterministic dynamics. Further, in a procedure similar to Moser's regularization, we transform the stochastic Kepler problem to obtain its dynamics as a stochastic geodesic flow on a 3-sphere.

Paper Structure

This paper contains 5 sections, 3 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Stochastic Perturbations of Hamiltonian Systems
  3. Kepler Problem with Perturbed Angular Momentum
  4. Hamilton’s velocity vector circle theorem
  5. Collisions and the Moser Regularization

Key Result

Proposition 1

Assume the setup of equation pertham. Suppose $\Gamma_t$ solves the SDE pertham with initial condition $\Gamma_0$. Then, for every $f\in C^{\infty}(M)$, where the integral on the right is the Stratonovich integral and where the integral on the right is the Itô integral.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Theorem 1