Regularization of the Symmteric Four-Body Problem by Elliptic Functions

Consider two pairs of equal masses moving under Newtonian forces in a planar configuration such that central symmetry with respect to the origin holds at all times $t$. This system, referred to as the Caledonian Four-Body Problem, has been extensively studied B.A. Steves, A.E. Roy, and many others. Binary collisions can occur as single collisions in each of the symmetric pairs. Also, two kinds of binary simultaneous binary collisions can occur. Regularization according to Levi-Civita is possible in every case (for the simultaneous collisions simply as a consequence of the symmetry). A single coordinate transformation involving Jacobian elliptic functions is able to regularize every binary collision.

Presentation (19 slides) caledvancouver.pdf

An even simpler constellation, referred to as the rhomboidal symmetric four-body problem, is obtained when the two symmetric binaries move on perpendicular axes. Owing to the simplicity of the equations of motion this problem allows regularization by pure Levi-Civita transformations. It is also well suited to study homothetic solutions and central configurations, as well as the four-body collision and escape manifolds. Furthermore, resonance phenomena between the two interacting rectilinear binaries play an important role.