On the existence of regular tetrahedral non-homothetic homographic solution

TL;DR

This paper extends the analysis of symmetrical solutions in the three-body problem to four masses forming a tetrahedron, proving that only homothetic solutions exist under these constraints.

Contribution

It introduces a new geometrical proof demonstrating the exclusivity of homothetic solutions for four masses in a tetrahedral configuration, expanding prior three-body results.

Findings

Only homothetic solutions are possible for four masses on a tetrahedron

Extension of three-body solutions to four-body tetrahedral configurations

Geometrical proof aligning with Wintner's classical results

Abstract

It is well known that the three-body problem has few analytical solutions in certain symmetrical constraints; the Lagrangian triangular solution is one of them. This triangular solution has been revisited by R.Broucke and H.Lass in 1971, concerning three relative position vectors pointing from one mass to another. This paper proposes a significant advance to the method, extended to four arbitrary masses on the vertices of a tetrahedron. The research provides a geometrical proof that under such constraint, only homothetic solution is possible which agrees with the conclusion brought by article 371 of Wintner (1941).