A Counterexample to a Generalized Saari's Conjecture with a Continuum of Central Configurations

TL;DR

This paper presents a counterexample in the three-body problem with harmonic potential, challenging a generalized version of Saari's conjecture and offering insights for future research on the classical conjecture.

Contribution

It provides a continuum of central configurations for three bodies and a counterexample to a generalized Saari's conjecture interpretation.

Findings

Existence of a continuum of central configurations for n=3

Counterexample to a generalized Saari's conjecture

Implications for classical Saari's conjecture for n≥4

Abstract

In this paper we show that in the $n$-body problem with harmonic potential one can find a continuum of central configurations for $n=3$. Moreover we show a counterexample to an interpretation of Jerry Marsden Generalized Saari's conjecture. This will help to refine our understanding and formulation of the Generalized Saari's conjecture, and in turn it might provide insight in how to solve the classical Saari's conjecture for $n\geq 4$.