On the Manev spatial isosceles three-body problem

Abstract

We study the isosceles three-body problem with Manev interaction. Using a McGehee-type technique, we blow up the triple collision singularity into an invariant manifold, called the collision manifold, pasted into the phase space for all energy levels. We find that orbits tending to/ejecting from total collision are present for a large set of angular momenta. We also discover that as the angular momentum is increased, the collision manifold changes its topology.

Total Collision in a Four-Body Problem with Jacobi Potential

“…We also find that the topology of the collision manifold C varies with the absolute value of the total angular momentum c. This is similar to the spatial Manev three-body problem [10]. For unit masses in the horizontal plane and arbitrary masses α in the vertical axis, we find that: ❼ For |c| < 1/2, the collision manifold C is homeomorphic to a 2-sphere with 4 points removed, as in the Newtonian case [4].…”

“…Related to the rhomboidal problem is the isosceles three body problem, initially studied by Devaney [5] for the gravitational potential, and then by Diacu [6] for Manev's gravitational law (−1/x − γ/x 2 ), where x is the distance between two bodies and γ > 0. Recently, the planar isosceles problem was considered in the spatial case for the Manev [10] and the Schwarzschild (−1/x − γ/x 3 ), γ > 0 potentials [2] . The present paper investigates the dynamics of the spatial rhomboidal four-body problem with particular emphasis on a binary attractive potential interaction of the form −1/x 2 .…”

“…Along such solutions the particle spins infinitely many times before reaching a collision, creating a black-hole effect, a terminology classical celestial mechanics researchers adopted as counterpart of the equivalent of the relativistic black hole effect. Second, in a recent study [10] it was observed that in the isosceles three body problem with Manev interaction (and so dominated near collision by a −1/x 2 term), the collision manifold changes its topology as the angular momentum increases, having profound implication on the dynamics in the proximity of total collision. Last, the presence of a third integral of motion allowed us to give a qualitative description of the global flow.…”