Comptes Rendus
Mathematical Problems in Mechanics
On the periodic orbits of the circular double Sitnikov problem
Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 333-336.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.01.011
Hugo Jiménez Pérez 1; Ernesto A. Lacomba 1

1 Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, C.P. 09340, México D.F.
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     author = {Hugo Jim\'enez P\'erez and Ernesto A. Lacomba},
     title = {On the periodic orbits of the circular double {Sitnikov} problem},
     journal = {Comptes Rendus. Math\'ematique},
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Hugo Jiménez Pérez; Ernesto A. Lacomba. On the periodic orbits of the circular double Sitnikov problem. Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 333-336. doi : 10.1016/j.crma.2009.01.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.011/

[1] M. Corbera; J. Llibre On symmetric periodic orbits of the elliptic sitnikov problem via the analytic continuation method, Contemporary Mathematics, vol. 292, American Mathematical Society, 2002, pp. 91-127

[2] K. Sitnikov Existence of oscillating motions for the three-body problem, Dokl. Akad. Nauk URSS, Volume 133 (1960) no. 2, pp. 303-306

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