Oscillatory motions in the rectangular four body problem

Pages: 557 - 587, Volume 1, Issue 4, December 2008

doi:10.3934/dcdss.2008.1.557       Abstract        Full Text (1582.0K)       Related Articles

Ernesto A. Lacomba - Departamento de Matemáticas, Universidad Autónoma Metropolitana–Iztapalapa, Av. San Rafael Atlixco 186, C.P. 09340 México, D.F., Mexico (email)
Mario Medina - Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, México DF, CP 09340, Mexico (email)

Abstract: In this paper we describe a symbolic dynamics for the rectangular four body problem by applying blow ups at total collisions and at infinity, studying the homoclinic or heteroclinic orbits obtained as intersection of corresponding two dimensional invariant submanifolds in a 3 dimensional energy level plus a convenient Poincaré map. With this tool we show the existence of a very rich dynamics and obtain the Main Theorem of this article. It gives the transition matrix for the symbolic dynamics of the images of conveniently chosen rectangles in the Poincaré section of the flow.

Keywords:  Four body problem, invariant manifolds, symbolic dynamics, Poincaré section.
Mathematics Subject Classification:  Primary: 70F07; Secondary: 70F15.

Received: March 2008;      Revised: August 2008;      Published: September 2008.