Multiple Periodic Orbits Connecting a Collinear Configuration and a Double Isosceles Configuration in the Planar Equal-Mass Four-Body Problem

By applying our variational method, we show that there exist 24 local action minimizers connecting two prescribed configurations: a collinear configuration and a double isosceles configuration in {H^{1}([0,1],\chi)} in the planar equal-mass four-body problem. Among the 24 local action minimizers, we prove that the one with the smallest action has no collision singularity and it can be extended to a periodic or quasi-periodic orbit. Furthermore, if all the 24 local action minimizers are free of collision, we show that they can generate sixteen different periodic orbits.

Global Regularization of a Restricted Four-Body Problem

In the n-body problem, a collision singularity occurs when the position of two or more bodies coincide. By understanding the dynamics of collision motion in the regularized setting, a better understanding of the dynamics of near-collision motion is achieved. In this paper, we show that any double collision of the planar equilateral restricted four-body problem can be regularized by using a Birkhoff-type transformation. This transformation has the important property to provide a simultaneous regularization of three singularities due to binary collision. We present some ejection–collision orbits after the regularization of the restricted four-body problem (RFBP) with equal masses, which were obtained by numerical integration.

Collision dynamics in Hénon-Heiles’ two-body problem

We tackle the two-body problem associated to H?non-Heiles? potential in the special case of the collision singularity. Using McGehee-type transformations of the second kind, we blow up the singularity and replace it by the collision manifold Mc pasted on the phase spece. We fully describe the flow on Mc. This flow is similar to analogous flows met in post-Newtonian two-body problems.

The zonal satellite problem - I: Near-collision flow

The force field described by a potential function of the form U = ?n k=1 ak/rk (r = distance between particles, ak = real parameters) models various concrete situations belonging to astronomy, physics, mechanics, astrodynamics, etc. The two-body problem is being tackled in such a field. The motion equations and the first integrals of energy and angular momentum are established. The McGehee-type coordinates are used to blow up the collision singularity and to paste the resulting manifold on the phase space. The flow on the collision manifold is depicted. Then, using the rotational symmetry of the problem and the angular momentum integral, the local flow near collision is described and interpreted in terms of physical motion.