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

Pages: 1149 - 1173, Volume 9, Issue 5, September 2003

doi:10.3934/dcds.2003.9.1149       Abstract        Full Text (384.5K)       Related Articles

Martha Alvarez-Ramírez - Departamento de Matemáticas, Universidad Autónoma Metropolitana – Iztapalapa, A. P. 55-534, 09340 Iztapalapa, México, D. F., Mexico (email)
Joaquín Delgado - Departamento de Matemáticas, Universidad Autónoma Metropolitana – Iztapalapa, 09340 Iztapalapa, México, D. F., Mexico (email)

Abstract: 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:  3–body problem, blow up, restricted isosceles problem, symbolic dynamics.
Mathematics Subject Classification:  70F15, 70F16, 70F05, 70F07.

Received: May 2002;      Revised: December 2002;      Published: June 2003.