Large Lagrangian Actions on Elliptical Solutions of 2-Body and 3-Body Problems with Fixed Energies

TL;DR

This paper investigates the Lagrangian actions of elliptical solutions in Newtonian 2-body and 3-body problems with fixed energies, revealing a relationship between period and energy using variational principles.

Contribution

It applies the constrained variational principle to compute actions and explores the period-energy relationship for Lagrangian elliptical solutions.

Findings

Computed Lagrangian actions for elliptical solutions.

Discovered a relationship between period and energy.

Extended variational methods to multi-body problems.

Abstract

Based on the works of Gordon ([4]) and Zhang-Zhou([8])) on the variational minimizing properties for Keplerian orbits and Lagrangian solutions of Newtonian 2-body and 3-body problems, we use the constrained variational principle of Ambrosetti-Coti Zelati ([1]) to compute the Lagrangian actions on Keplerian and Lagrangian elliptical solutions with fixed energies, we also find an interesting relationship between period and energy for Lagrangian elliptical solutions with Newtonian potentials.