Deterministic and Random Perturbations of the Kepler Problem

TL;DR

This paper studies how deterministic relativistic corrections and stochastic potentials affect the dynamics of the Kepler problem, combining analytical and simulation methods to understand orbit perturbations.

Contribution

It provides a comprehensive analysis of both deterministic and stochastic perturbations in the Kepler problem using classical mechanics and Monte Carlo simulations.

Findings

Relativistic correction causes orbit precession.

Stochastic perturbations influence mean exit times.

Analytical and simulation results align well.

Abstract

We investigate perturbations in the Kepler problem. We offer an overview of the dynamical system using Newtonian, Lagrangian and Hamiltonian Mechanics to build a foundation for analyzing perturbations. We consider the effects of a deterministic perturbation in the form of a first order relativistic correction which change bounded orbits from standard to precessing ellipses. We also consider the effects of stochastic perturbations with certain potentials and evaluate the analytical results of mean exit times using Monte Carlo simulations.