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