Evolution of the regions of the 3$D$ particle motion in the regular polygon problem of ($N+1$) bodies with a quasi-homogeneous potential

Abstract

The regular polygon problem of ($N+1$) bodies deals with the dynamics of a small body, natural or artificial, in the force field of $N$ big bodies, the $\nu=N-1$ of which have equal masses and form an imaginary regular $\nu$-gon, while the $N$th body with a different mass is located at the center of mass of the system. In this work, instead of considering Newtonian potentials and forces, we assume that the big bodies create quasi-homogeneous potentials, in the sense that we insert to the inverse square Newtonian law of gravitation an inverse cube corrective term, aiming to approximate various phenomena due to their shape or to the radiation emitting from the primaries. Based on this new consideration, we apply a general methodology in order to investigate by means of the zero-velocity surfaces, the regions where 3$D$ motions of the small body are allowed, their evolutions and parametric variations, their topological bifurcations, as well as the existing trapping domains of the particle. Here we note that this process is definitely a fundamental step of great importance in the study of many dynamical systems characterized by a Jacobian-type integral of motion in the long way of searching for solutions of any kind.