From the Lagrange Triangle to the Figure Eight Choreography: Proof of Marchal's Conjecture

TL;DR

This paper proves that the symmetric continuation of Lagrange's equilateral triangle solution in the three-body problem includes the figure eight choreography, confirming a longstanding conjecture of Marchal.

Contribution

It establishes a rigorous link between Lagrange's solution and the figure eight choreography, settling Marchal's conjecture from 1999.

Findings

The $P_{12}$ family contains the figure eight choreography.

The result confirms the conjecture of Marchal from 1999.

It advances understanding of symmetric solutions in the three-body problem.

Abstract

For the three body problem with equal masses, we prove that the most symmetric continuation class of Lagrange's equilateral triangle solution, also referred to as the $P_{12}$ family of Marchal, contains the remarkable figure eight choreography discovered by Moore in 1993, and proven to exist by Chenciner and Montgomery in 2000. This settles a conjecture of Marchal which dates back to the 1999 conference on Celestial Mechanics in Evanston Illinois, celebrating Donald Saari's 60th birthday.