Secular dynamics for curved two-body problems

TL;DR

This paper investigates the motion of two masses on curved surfaces under an inverse-square law, revealing how curvature influences their dynamics through averaged perturbation analysis.

Contribution

It introduces action-angle coordinates for curved two-body problems and analyzes the effects of curvature on their dynamics.

Findings

Curvature affects the stability and trajectories of the two-body system.

Averaged equations reveal curvature-induced dynamical effects.

Perturbation methods extend classical two-body analysis to curved spaces.

Abstract

Consider the dynamics of two point masses on a surface of constant curvature subject to an attractive force analogue of Newton's inverse square law. When the distance between the bodies is sufficiently small, the reduced equations of motion may be seen as a perturbation of an integrable system. We define suitable action-angle coordinates to average these perturbing terms and observe dynamical effects of the curvature on the motion of the two-bodies.