American Institute of Mathematical Sciences

Advanced
September  2003, 9(5): 1149-1173. doi: 10.3934/dcds.2003.9.1149

Blow up of the isosceles 3--body problem with an infinitesimal mass

Martha Alvarez-Ramírez 1, and  Joaquín Delgado 2,

1. 

Departamento de Matemáticas, Universidad Autónoma Metropolitana – Iztapalapa, A. P. 55-534, 09340 Iztapalapa, México, D. F., Mexico
2. 

Departamento de Matemáticas, Universidad Autónoma Metropolitana – Iztapalapa, 09340 Iztapalapa, México, D. F., Mexico

Received  May 2002 Revised  December 2002 Published  June 2003

We consider the isosceles $3$--body problem with the third particle having a small mass which eventually tend to zero. Classical McGehee's blow up is not defined because the matrix of masses becomes degenerate. Following Elbialy [3] we perform the blow up using the Euclidean norm in the planar $3$--body problem. We then complete the phase portrait of the flow in the collision manifold giving the behavior of some branches of saddle points missing in [3]. The homothetic orbits within the fixed energy level then provide the necessary recurrence in order to build a symbolic dynamics. This is done following ideas of S. Kaplan [6] for the collinear $3$--body problem. Here the difficulty is the larger number of critical points.
Keywords: symbolic dynamics., blow up, restricted isosceles problem, 3–body problem.
Mathematics Subject Classification: 70F15, 70F16, 70F05, 70F0.
Citation: Martha Alvarez-Ramírez, Joaquín Delgado. Blow up of the isosceles 3--body problem with an infinitesimal mass. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1149-1173. doi: 10.3934/dcds.2003.9.1149
[1]

Samuel R. Kaplan, Ernesto A. Lacomba, Jaume Llibre. Symbolic dynamics of the elliptic rectilinear restricted 3--body problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 541-555. doi: 10.3934/dcdss.2008.1.541
[2]

Gianni Arioli. Branches of periodic orbits for the planar restricted 3-body problem. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 745-755. doi: 10.3934/dcds.2004.11.745
[3]

Elbaz I. Abouelmagd, Juan Luis García Guirao, Jaume Llibre. Periodic orbits for the perturbed planar circular restricted 3–body problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1007-1020. doi: 10.3934/dcdsb.2019003
[4]

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
[5]

Giovanni F. Gronchi, Chiara Tardioli. The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1323-1344. doi: 10.3934/dcdsb.2013.18.1323
[6]

Daniel Offin, Hildeberto Cabral. Hyperbolicity for symmetric periodic orbits in the isosceles three body problem. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 379-392. doi: 10.3934/dcdss.2009.2.379
[7]

Marcel Guardia, Tere M. Seara, Pau Martín, Lara Sabbagh. Oscillatory orbits in the restricted elliptic planar three body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 229-256. doi: 10.3934/dcds.2017009
[8]

Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229
[9]

Eduardo S. G. Leandro. On the Dziobek configurations of the restricted $(N+1)$-body problem with equal masses. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 589-595. doi: 10.3934/dcdss.2008.1.589
[10]

Nicola Soave, Susanna Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3245-3301. doi: 10.3934/dcds.2012.32.3245
[11]

Tiancheng Ouyang, Duokui Yan. Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3989-4018. doi: 10.3934/dcds.2017169
[12]

Marina Chugunova, Chiu-Yen Kao, Sarun Seepun. On the Benilov-Vynnycky blow-up problem. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1443-1460. doi: 10.3934/dcdsb.2015.20.1443
[13]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[14]

Niraj Pathak, V. O. Thomas, Elbaz I. Abouelmagd. The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 849-875. doi: 10.3934/dcdss.2019057
[15]

Hadia H. Selim, Juan L. G. Guirao, Elbaz I. Abouelmagd. Libration points in the restricted three-body problem: Euler angles, existence and stability. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 703-710. doi: 10.3934/dcdss.2019044
[16]

Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074
[17]

Davide L. Ferrario, Alessandro Portaluri. Dynamics of the the dihedral four-body problem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 925-974. doi: 10.3934/dcdss.2013.6.925
[18]

Nicola Soave, Susanna Terracini. Addendum to: Symbolic dynamics for the $N$-centre problem at negative energies. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3825-3829. doi: 10.3934/dcds.2013.33.3825
[19]

Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009
[20]

Alain Albouy, Holger R. Dullin. Relative equilibria of the 3-body problem in $ \mathbb{R}^4 $. Journal of Geometric Mechanics, 2020, 12 (3) : 323-341. doi: 10.3934/jgm.2020012

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences