# On the n-body problem on surfaces of revolution

## TL;DR

This paper investigates the n-body problem on surfaces of revolution, revealing new dynamical properties, symmetry results, and bifurcation phenomena, extending classical mechanics to curved geometries.

## Contribution

It introduces a geometric framework for the n-body problem on surfaces of revolution, proving new results about invariant manifolds, symmetry, and bifurcations not previously known.

## Key findings

- Saari's conjecture fails on certain surfaces of revolution.
- Homographic motions with equal masses form an invariant manifold.
- Regular n-gon relative equilibria undergo pitchfork bifurcations.

## Abstract

We explore the $n$-body problem, $n\geq 3,$ on a surface of revolution with a general interaction depending on the pairwise geodesic distance. Using the geometric methods of classical mechanics we determine a large set of properties. In particular, we show that Saari's conjecture fails on surfaces of revolution admitting a geodesic circle. We define homographic motions and, using the discrete symmetries, prove that when the masses are equal, they form an invariant manifold. On this manifold the dynamics are reducible to a one-degree of freedom system. We also find that for attractive interactions, regular $n$-gon shaped relative equilibria with trajectories located on geodesic circles typically experience a pitchfork bifurcation. Some applications are included.