Abstract
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 primariesm2andm3are equal toμand the massm1is1−2μ. 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.
Highlights
Dynamical systems with few bodies three have been extensively studied in the past, and various models have been proposed for research aiming to approximate the behavior of real celestial systems
We study the motion of a mass point of negligible mass under the Newtonian gravitational attraction of three mass points of masses m1, m2, and m3 called primaries moving in circular periodic orbits around their center of mass fixed at the origin of the coordinate system
The description of the number of equilibrium points is given in Section 5, and in the symmetrical case i.e., μ 1/3, we describe the kind of stability of each equilibrium
Summary
Dynamical systems with few bodies three have been extensively studied in the past, and various models have been proposed for research aiming to approximate the behavior of real celestial systems. At any instant of time, the primaries form an equilateral equilibrium configuration of the three-body problem which is a particular solution of the three-body problem given by Lagrange see 4 or 3. Two of these primaries have equal masses and are located symmetrically with respect to the third primary. The equilateral restricted four-body problem shortly, ERFBP consists in describing the motion of the infinitesimal mass, m4, under the gravitational attraction of the three primaries m1, m2, and m3. The authors studied the motion of a fourth body of small mass which moves under the combined attractions of these three massive bodies This model is called bicircular four-body problem. The restricted four-body problem with radiation pressure was considered in , while the photogravitational restricted four body problem was considered in
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