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

Martha Alvarez-Ramírez; Joaquín Delgado

doi:10.3934/dcds.2003.9.1149

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2003, Volume 9, Issue 5: 1149-1173. Doi: 10.3934/dcds.2003.9.1149

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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.
2.
Departamento de Matemáticas, Universidad Autónoma Metropolitana – Iztapalapa, 09340 Iztapalapa, México, D. F.

Received: April 30, 2002

Revised: November 30, 2002

Published: August 2003

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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.

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Mathematics Subject Classification: 70F15, 70F16, 70F05, 70F07.

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