The few-body problem of particles with only gravitational interactions

TL;DR

This paper explores the complex dynamics of N gravitationally interacting particles, highlighting the challenges in solving their equations of motion and analyzing specific cases like three and four particles on a line.

Contribution

It provides insights into the qualitative behavior of the N-body gravitational problem, including special cases and phenomena like finite-time escape.

Findings

Three masses on a line exhibit specific motion patterns.

Four masses on a line can all escape to infinity in finite time.

General solutions for N > 2 are not obtainable in closed form.

Abstract

In this article, I discuss the motion of $N$ point masses in non-relativistic mechanics, when the interaction between them is purely the Newtonian gravitational interaction, with $N$ greater than or equal to 2. The dynamical equations of motion cannot be solved in closed form, for general initial conditions, for any $N$ greater than 2. However, the qualitative behavior of the solutions can be understood from general considerations. I discuss in particular motion the three masses on a line, and the counter-intuitive case of four masses on a line that leads to all particles escaping to infinity in a finite time.