March  2013, 33(3): 1157-1175. doi: 10.3934/dcds.2013.33.1157

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

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona

2. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona

Received  May 2011 Revised  November 2011 Published  October 2012

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.
Citation: Regina Martínez, Carles Simó. On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1157-1175. doi: 10.3934/dcds.2013.33.1157
References:
[1]

C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, Users' guide to PARI/GP,, (freely available from \url{http://pari.math.u-bordeaux.fr/})., ().   Google Scholar

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F. Diacu and E. Pérez-Chavela, Homographic solutions of the curved 3-body problem,, Journal of Differential Equations, 250 (2011), 340.  doi: 10.1016/j.jde.2010.08.011.  Google Scholar

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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.  doi: 10.1016/0022-0396(89)90161-7.  Google Scholar

[4]

T. Kapela and C. Simó, Rigorous KAM results around arbitrary periodic orbits for Hamiltonian systems,, Preprint, ().   Google Scholar

[5]

R. Martínez, A. Samà and C. Simó, "Stability of Homographic Solutions of the Planar Three-Body Problem with Homogeneous Potentials,", Proceedings EQUADIFF (2003), (2003).   Google Scholar

[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.  doi: 10.1016/j.jde.2006.01.014.  Google Scholar

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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.  doi: 10.1016/j.jde.2005.09.012.  Google Scholar

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C. Siegel and J. Moser, "Lectures on Celestial Mechanics,", Springer, (1971).   Google Scholar

[9]

C. Simó, On the analytical and numerical approximation of invariant manifolds,, Modern methods in celestial mechanics, (1990), 285.   Google Scholar

[10]

E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,", Cambridge Univ. Press, (1970).   Google Scholar

show all references

References:
[1]

C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, Users' guide to PARI/GP,, (freely available from \url{http://pari.math.u-bordeaux.fr/})., ().   Google Scholar

[2]

F. Diacu and E. Pérez-Chavela, Homographic solutions of the curved 3-body problem,, Journal of Differential Equations, 250 (2011), 340.  doi: 10.1016/j.jde.2010.08.011.  Google Scholar

[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.  doi: 10.1016/0022-0396(89)90161-7.  Google Scholar

[4]

T. Kapela and C. Simó, Rigorous KAM results around arbitrary periodic orbits for Hamiltonian systems,, Preprint, ().   Google Scholar

[5]

R. Martínez, A. Samà and C. Simó, "Stability of Homographic Solutions of the Planar Three-Body Problem with Homogeneous Potentials,", Proceedings EQUADIFF (2003), (2003).   Google Scholar

[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.  doi: 10.1016/j.jde.2006.01.014.  Google Scholar

[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.  doi: 10.1016/j.jde.2005.09.012.  Google Scholar

[8]

C. Siegel and J. Moser, "Lectures on Celestial Mechanics,", Springer, (1971).   Google Scholar

[9]

C. Simó, On the analytical and numerical approximation of invariant manifolds,, Modern methods in celestial mechanics, (1990), 285.   Google Scholar

[10]

E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,", Cambridge Univ. Press, (1970).   Google Scholar

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