Dynamical Aspects of an Equilateral Restricted Four-Body Problem

The spatial equilateral restricted four-body problem (ERFBP) is a fourbody problem where a mass point of negligible mass is moving under theNewtonian gravitational attraction of three positive masses (called the primaries) which move on circular periodic orbits around their center of massfixed at the origin of the coordinate system such that their configuration isalways an equilateral triangle. Since fourth mass is small, it does not affectthe motion of the three primaries. In our model we assumethat the two masses of the primaries m2 and m3 are equal to μ and the mass m1 is 1−2μ. 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