Asymptotic Hamiltonian reduction for the dynamics of a particle on a surface

TL;DR

This paper develops an asymptotic Hamiltonian reduction method to analyze the motion of a particle on a surface close to a sphere, providing detailed descriptions of its dynamics through a simplified Hamiltonian system.

Contribution

It introduces a novel Hamiltonian reduction technique for perturbed spherical surfaces, offering a new way to understand particle dynamics on nearly spherical surfaces.

Findings

Derivation of a subsidiary Hamiltonian system resembling a top with a 4th order Hamiltonian.

Asymptotic description of particle motion in terms of graphs on the sphere.

Qualitative analysis of precessing great circles as an approximation of particle trajectories.

Abstract

We consider the motion of a particle on a surface which is a small perturbation of the standard sphere. One may qualitatively describe the motion by means of a precessing great circle of the sphere. The observation is employed to derive a subsidiary Hamiltonian system that has the form of equations for the top with a 4-th order Hamiltonian, and provides the detailed asymptotic description of the particle's motion in terms of graphs on the standard sphere.