Article Contents
Article Contents

# On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$

• The problem of three bodies with equal masses in $\mathbb{S}^2$ is known to have Lagrangian homographic orbits. We study the linear stability and also a "practical'' (or effective) stability of these orbits on the unit sphere.
Mathematics Subject Classification: Primary: 70F07, 70H12; Secondary: 34D08, 37J25.

 Citation:

•  [1] C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, Users' guide to PARI/GP, (freely available from http://pari.math.u-bordeaux.fr/). [2] F. Diacu and E. Pérez-Chavela, Homographic solutions of the curved 3-body problem, Journal of Differential Equations, 250 (2011), 340-366.doi: 10.1016/j.jde.2010.08.011. [3] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem, Journal of Differential Equations, 77 (1989), 167-198.doi: 10.1016/0022-0396(89)90161-7. [4] T. Kapela and C. Simó, Rigorous KAM results around arbitrary periodic orbits for Hamiltonian systems, Preprint, arXiv:1105.3235. [5] R. Martínez, A. Samà and C. Simó, "Stability of Homographic Solutions of the Planar Three-Body Problem with Homogeneous Potentials," Proceedings EQUADIFF (2003), 1005–1010. World Scientific, 2005. [6] R. Martínez, A. Samà and C. Simó, Stability diagram for 4D linear periodic systems with applications to homographic solutions, Journal of Differential Equations, 226 (2006), 619-651.doi: 10.1016/j.jde.2006.01.014. [7] R. Martínez, A. Samà and C. Simó, Analysis of the stability of a family of singular-limit linear periodic systems in $R^4.$ applications, Journal of Differential Equations, 226 (2006), 652-686.doi: 10.1016/j.jde.2005.09.012. [8] C. Siegel and J. Moser, "Lectures on Celestial Mechanics," Springer, 1971. [9] C. Simó, On the analytical and numerical approximation of invariant manifolds, Modern methods in celestial mechanics, (eds, D. Benest and C. Froeschlé), 285-330, Editions Frontières, Paris, 1990. (Also available at {http://www.maia.ub.es/dsg/2004/}). [10] E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies," Cambridge Univ. Press, fourth edition, reprinted, 1970.