Restricted rhomboidal five-body problem

Abstract

The restricted rhomboidal five-body problem (RRFBP) is a problem in which four positive masses, called the primaries, move two by two in circular motions such that their configuration is always a rhombus, the fifth mass being small and not influencing the motion of the four primaries. In our model, we assume that the fifth mass is in the same plane of the primaries and that the masses of the primaries are m1 = m2 = m and and the radius associated with the circular motion of m1 and m2 is and the one for the masses of m3 and m4 is 1. Similar to the circular restricted three-body problem, we obtain the first integral of motion. The Hamiltonian function which governs the motion of the fifth mass is obtained and has two degrees of freedom depending periodically on time. We use a synodical system of coordinates to eliminate the time dependence. With the help of the Hamiltonian structure, we characterize the regions of possible motion. We show the existence of equilibrium solutions along the coordinate axis as well as off them. We verify that the number of equilibria depends on λ and there can be 11, 13 or 15 equilibrium solutions all unstable. We prove the existence of periodic solutions with short as well as long period. Also we prove the existence of transversal ejection–collision orbits (binary collisions) for certain large values of the Jacobi constant, for an uncountable number of invariant punctured tori in the corresponding energy surface.