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

[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

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

[1]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[2]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

[3]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[4]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[5]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[6]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[7]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005

[8]

Yongming Luo, Athanasios Stylianou. On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3455-3477. doi: 10.3934/dcdsb.2020239

[9]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[10]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[11]

Akio Matsumot, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021069

[12]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

[13]

Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021068

[14]

Yongjian Liu, Qiujian Huang, Zhouchao Wei. Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3357-3380. doi: 10.3934/dcdsb.2020235

[15]

Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329

[16]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200

[17]

Guodong Wang, Bijun Zuo. Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021078

[18]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[19]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[20]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (17)

Other articles
by authors

[Back to Top]