On the periodic orbits of the circular double Sitnikov problem

• Published in: Comptes rendus. Mathématique Vol. 347; no. 5; pp. 333 - 336
• Main Authors: Jiménez Pérez, Hugo, Lacomba, Ernesto A.
• Format: Journal Article
• Language: English
• Published: Paris Elsevier SAS 01.03.2009; Elsevier
• Subjects: Combinatorics; Combinatorics. Ordered structures; Designs and configurations; Exact sciences and technology; General mathematics; General, history and biography; Global analysis, analysis on manifolds; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Configuration; Collision; Ensemble; Periodic orbit; Mathematics; Body; Regularization; Surface energy
• ISSN: 1631-073X; 1778-3569
• DOI: 10.1016/j.crma.2009.01.011

We introduce a restricted four body problem in a 2 + 2 configuration extending the classical Sitnikov problem to the Double Sitnikov problem. The secondary bodies are moving on the same perpendicular line to the plane where the primaries evolve, so almost every solution is a collision orbit. We extend the solutions beyond collisions with a symplectic regularization and study the set of energy surfaces that contain periodic orbits. To cite this article: H. Jiménez Pérez, E.A. Lacomba, C. R. Acad. Sci. Paris, Ser. I 347 (2009). On présente un problème à quatre corps en configuration 2 + 2 qui étend le problème de Sitnikov au problème double de Sitnikov. Dans cette configuration presque toute solution du problème est une orbite avec collisions. On étend les solutions qui présentent des collisions en utilisant une regularisation symplectique et on étudie l'ensemble des surfaces d'énergie qui contiennent des orbites périodiques. Pour citer cet article : H. Jiménez Pérez, E.A. Lacomba, C. R. Acad. Sci. Paris, Ser. I 347 (2009).