Equivalence of Rigid Motions and Relative Equilibria in the N-Body Problem on the Two-Sphere

TL;DR

This paper proves that in the N-body problem on a sphere, any rigid motion corresponds to a relative equilibrium, extending classical results and providing insights into dynamics on curved surfaces.

Contribution

It establishes the equivalence between rigid motions and relative equilibria for N-body systems on the sphere, extending to systems in three-dimensional space.

Findings

Rigid motions on S2 are necessarily relative equilibria.

Results extend to N-body systems in R3.

Provides a broader understanding of dynamics on curved surfaces.

Abstract

We investigate the relationship between rigid motions and relative equilibria in the N-body problem on the two-dimensional sphere, S2. We prove that any rigid motion of the N-body system on S2 must be a relative equilibrium. Our approach extends the classical study of rigid body dynamics by Euler and utilizes a rotating frame attached to the particles to derive the corresponding equations of motion. We further show that our results can be extended to the N-body gravitational system in R3. The results are oriented to a broader understanding of the dynamics of N-body systems on curved surfaces.