Dynamics in the Schwarzschild Isosceles Three Body Problem

Abstract

We emphasize the distinct features of the Schwarzschild model when compared to its Newtonian counterpart. We prove that, in contrast to the Newtonian case, on any level of energy the measure of the set on initial conditions leading to triple collision is positive. Further, whereas in the Newtonian problem triple collision is asymptotically reached only for zero angular momentum, in the Schwarzschild problem the triple collision is possible for non-zero total angular momenta (e.g., when two of the mass points spin infinitely many times around the centre of mass). This phenomenon is known in celestial mechanics as the black-hole effect and it is understood as an analogue in the classical context of the behaviour near a Schwarzschild black hole. Also, while in the Newtonian problem all triple collision orbits are necessarily homothetic, in the Schwarzschild problem this is not necessarily true. In fact, in the Schwarzschild problem there exist triple collision orbits which are neither homothetic, nor homographic.

On the Manev spatial isosceles three-body problem

“…Using similar arguments as in [Arredondo & al. 2014], one may prove that motions ejecting/ending from/to the equilibria P ± are homographic, i.e., they maintain a self-similar shape of the triangle formed by the three bodies.…”

Rational Parameterizations Approach for Solving Equations in Some Dynamical Systems Problems

“…Distances among bodies and, therefore, square roots appear often in equations modeling many problems in Celestial Mechanics. In this section we show two applications of rational parameterization and polynomial tools in this field: a first one concerning critical points of a certain potential (see [6]) and a second one dealing with central configurations in a system of four charged particles (see [7,21]).…”

“…Relative equilibria in the Schwarzschild isosceles three body problem. The so-called Schwarzschild isosceles three body problem with masses m 1 = m 2 = M > 0 and m 3 = 1 is introduce in detail in [6]. In particular, in cylindrical coordinates, it has the following reduced Hamiltonian…”

“…In [6] it is studied, joint to other problems, the number of solutions of (9) on z = 0 for positive values of A, A 1 , B, and B 1 in terms of C. In this case, there is only one bifurcation value, C 2 in the next result, that distinguish the cases with 0, 1 or 2 relative equilibria. The following result extends the study of the number of solutions of (9), also in terms of C, for arbitrary non-zero values of A, A 1 , B, and B 1 .…”

“…Here, square roots appear due to the Euclidean distances among the bodies. Two examples are presented: one concerning critical points of a certain potential (see [6]) and a second one dealing with central configurations (see [7,21]). And the third type, in Section 5, where we seek for estimates for the number of limit cycles of planar non-autonomous differential equations by studying the zeros of some Melnikov functions.…”