Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Renato Calleja, Carlos García-Azpeitia, Olivier Hénot, Jean-Philippe Lessard, Jason D. Mireles James

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.

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

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 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.

Paper Structure

This paper contains 15 sections, 14 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some historical remarks
  3. Reformulation of the problem as a functional equation
  4. Symmetry reduction
  5. Desingularization
  6. Figure eight
  7. Lagrange triangle
  8. Polynomial embedding
  9. Zero-finding problem
  10. Existence of a branch of solutions of the functional equation
  11. Choice of norms
  12. Verification of the contraction hypotheses
  13. Proof of Marchal's Conjecture
  14. Future work
  15. Acknowledgments

Key Result

Theorem \oldthetheorem

In the three body problem with equal masses, the figure eight choreography of Moore, Chenciner, and Montgomery is contained in the $P_{12}$ family of Marchal.

Figures (1)

  • Figure 1: Branch of relative choreographies connecting the Lagrange triangle to the figure eight. A dense set of the relative choreographies in the family corresponds to choreographies in inertial frame.

Theorems & Definitions (30)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 20 more