Closed Form Approximations For The Three Body Problem

TL;DR

This paper develops closed-form analytical approximations for the classical three body problem with spherically symmetric masses, providing explicit time-dependent position solutions where exact solutions are impossible due to inherent instabilities.

Contribution

It introduces a novel method to approximate solutions of the three body problem using Lambert's wave function, replacing the need for exact solutions.

Findings

Derived explicit time-dependent position vectors.

Validated the approximation method for symmetric mass distributions.

Provided a new analytical framework for a historically unsolvable problem.

Abstract

In this paper, an approach is developed to solve the three body problem involving masses which posses spherical symmetry. The problem dates back to the times of Poincare, and is undoubtedly one of the oldest of unsolved problems of classical mechanics. The Poincares Dictum comprehensively proves that the problem is truly insolvable as a result of the nature of the instabilities involved. We therefore refute the idea of finding exact solutions. Instead, we develop closed form analytical approximations in place of exact solutions. We will solve the problem for the case when all the masses involved have spherically symmetric mass distributions. The method of solution would include the use of a single mass to replicate the effect of two individual masses on each body. The derivation of solutions will involve the use of the Lamberts wave function and the solution will comprise of the position vectors expressed as explicit time functions.