Potentials allowing integration of the perturbed two-body problem in regular coordinates

TL;DR

This paper identifies specific potentials enabling the separation of variables in a perturbed two-body problem using regular coordinates, providing explicit solutions and numerical insights into bounded and unbounded motions.

Contribution

It introduces new potentials with arbitrary smooth functions that allow variable separation in the perturbed two-body problem using $L$-transformations.

Findings

Potentials enabling separation of variables are characterized.

Explicit solutions are obtained using elliptic functions.

Numerical experiments illustrate bounded and unbounded motions.

Abstract

The problem of separation of variables in some coordinate systems obtained with the use of $L$-transformations is studied. Potentials are shown that allow separation of regular variables in a perturbed two-body problem. The potential contains two arbitrary smooth functions. An example of a potential is considered allowing explicit solution of the problem in terms of elliptic functions. The cases of bounded and unbounded motion are shown. The results of numerical experiments are given.