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 and are equal to and the mass is . 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.