Ejection-Collision orbits in the symmetric collinear four–body problem

Abstract

In this paper, we consider the collinear symmetric four-body problem, where four masses m 3 = α, m 1 = 1, m 2 = 1, and m 4 = α, α > 0, are aligned in this order and move symmetrically about their center of mass. We introduce regularized variables to deal with binary collisions as well as McGehee coordinates to study the quadruple collision manifold for a negative value of the energy. The paper is mainly focused on orbits that eject from (or collide to) quadruple collision. The problem has two hyperbolic equilibrium points, located in the quadruple collision manifold. We use high order parametrizations of their stable/unstable manifolds to devise a numerical procedure to compute ejectioncollision orbits, for any value of α. Some results from the explorations done for α = 1 are presented. Furthermore, we prove the existence of ejection-direct escape orbits, which perform a unique type of binary collisions.

Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

“…where f is as defined in Equation (27). Let φ α (x, t) denote the flow generated by the the vector field of Equation (42). For c ∈ R consider the fixed energy level set M .…”

“…These works depend on results from geometry/topology, the calculus of variations, and the KAM theory. For parameter and energy regimes where analytical results are unavailable, numerical studies illuminate the dynamics of the collision set [41,42,43,44,45,46,47,48]. Our work goes towards providing a framework for computer assisted proofs for collision orbits, for the parameter regimes where the perturbative methods cannot be applied.…”

Ejection–Collision Orbits in Two Degrees of Freedom Problems in Celestial Mechanics

“…One direction to tackle this problem has been to consider few body problems, as the collinear three body problem [15,7] or the isosceles three body problem [6]. Some others have some symmetries involved, as the symmetric collinear four body problem [10,1,4], the trapezoidal four body problem [9,3,2], and the rhomboidal four body problem [5] and [12]. Some of them depend on parameters associated to the masses.…”

“…These problems have been given much attention, and they share several common properties that we generalize and use to obtain information to prove analytically the existence of ECOs. In a prior paper by the authors of this article [4], a family of ECO was obtained from a numerical point of view in the symmetric collinear four body problem.…”

On the spatial collinear restricted four-body problem with non-spherical primaries

“…Palacios et al (2019) have illustrated the evolution of families of symmetric periodic orbits as the function of mass parameter and evolution of spiral points, which show the connection between heteroclinic orbits and equilibrium points. Alvarez-Ramirez et. al.…”