A translation of L. Euler's "On the rectilinear motion of three bodies mutually attracting each other"

TL;DR

This paper provides an annotated translation of Euler's work on a special three-body problem where three bodies attract each other via inverse-square law, leading to a specific aligned motion solution.

Contribution

It offers a detailed translation and analysis of Euler's exact solution for three collinear bodies under inverse-square attraction, a historically significant problem in celestial mechanics.

Findings

Euler's solution describes three bodies moving in a line with mutual inverse-square attraction.

The solution models a hypothetical Sun-Earth-Moon alignment scenario.

A quintic function parameter governs the distances among the bodies.

Abstract

This is an annotated translation from Latin of E327 'De motu rectilineo trium corporum se mutuo attrahentium'. In this publication, Euler considers three bodies lying on a straight line, which are attracted to each other by central forces inversely proportional to the square of their separation distance (inverse-square law). Although not explicitly mentioned by Euler, this is an exact solution of three bodies that move around the common center of mass and always line up. The solution given by Euler could represent a hypothetical situation of Sun, Earth and Moon in perpetual alignment in syzygy, for which the parameter that controls the distances among the planets was found to be given by a quintic function.