Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 181360, 23 pages
Research Article

Dynamical Aspects of an Equilateral Restricted Four-Body Problem

1Departamento de Matemáticas, UAM-Iztapalapa, A.P. 55-534, 09340 Iztapalapa, México, Mexico
2Departamento de Matemáticas, Facultad de Ciencias, Universidad del Bío Bío, Casilla 5-C, Concepción, VIII-Región 4081112, Chile

Received 26 July 2009; Revised 10 November 2009; Accepted 9 December 2009

Academic Editor: Tadashi Yokoyama

Copyright © 2009 Martha Álvarez-Ramírez and Claudio Vidal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The spatial equilateral restricted four-body problem (ERFBP) is a four body problem where a mass point of negligible mass is moving under the Newtonian gravitational attraction of three positive masses (called the primaries) which move on circular periodic orbits around their center of mass fixed at the origin of the coordinate system such that their configuration is always an equilateral triangle. Since fourth mass is small, it does not affect the motion of the three primaries. In our model we assume that the two masses of the primaries m2 and m3 are equal to μ and the mass m1 is 12μ. The Hamiltonian function that governs the motion of the fourth mass is derived and it has three degrees of freedom depending periodically on time. Using a synodical system, we fixed the primaries in order to eliminate the time dependence. Similarly to the circular restricted three-body problem, we obtain a first integral of motion. With the help of the Hamiltonian structure, we characterize the region of the possible motions and the surface of fixed level in the spatial as well as in the planar case. Among other things, we verify that the number of equilibrium solutions depends upon the masses, also we show the existence of periodic solutions by different methods in the planar case.