On the existence of generalised syzygies in the planar three-body problem

Alexei Tsygvintsev

doi:10.5802/crmath.409

Dynamical systems

On the existence of generalised syzygies in the planar three-body problem

Alexei Tsygvintsev ¹

¹ ENS de Lyon, UMPA, 46 allée d’Italie, 69364 Lyon Cedex 07, FRANCE

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 331-335.

Abstract
Résumé

We consider the Newtonian planar three-body problem. One defines a generalised syzygy as a configuration where the three bodies or their velocities become collinear. Assuming that the motion is bounded and collision-free, we provide a simple sufficient condition for the existence of such configurations. Our proof is elementary and uses only basic tools from the Sturm–Liouville theory.

Nous considérons le problème plan des trois corps. On définit une syzygie généralisée comme une configuration où les trois corps ou leurs vitesses deviennent colinéaires. En supposant que le mouvement est borné et sans collision, nous fournissons une condition suffisante pour l’existence de telles configurations. Nos principaux outils sont élémentaires et basés sur la théorie de Sturm–Liouville.

Received: 2022-06-22
Revised: 2022-07-27
Accepted: 2022-07-30
Published online: 2023-01-12

DOI: 10.5802/crmath.409

Classification: 92B05

Author's affiliations:

Alexei Tsygvintsev ¹

¹ ENS de Lyon, UMPA, 46 allée d’Italie, 69364 Lyon Cedex 07, FRANCE

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {On the existence of generalised syzygies in the planar three-body problem},
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Alexei Tsygvintsev. On the existence of generalised syzygies in the planar three-body problem. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 331-335. doi : 10.5802/crmath.409. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.409/

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