Symplectic Regularization of Binary Collisions in the Circular N+2 Sitnikov Problem

TL;DR

This paper introduces a symplectic regularization method for binary collisions in the circular N+2 Sitnikov problem, connecting classical transformations with symplectic geometry to handle singularities.

Contribution

It develops a linear symplectic transformation framework to regularize rectilinear binary collisions within the N-body problem, including the Sitnikov configuration.

Findings

Regularization of binary collisions via symplectic transformations

Connection between Euler transformation and symplectic structure

Application to the circular N+2 Sitnikov problem

Abstract

We present a brief overview of the regularizing transformations of the Kepler problem and we relate the Euler transformation with the symplectic structure of the phase space of the N-body problem. We show that any particular solution of the N-body problem where two bodies have rectilinear dynamics can be regularized by a linear symplectic transformation and the inclusion of the Euler transformation into the group of symplectic local diffeomorphisms over the phase space. As an application we regularize a particular configuration of the circular N+2 Sitnikov problem.