A first integral to the partially averaged Newtonian potential of the three-body problem

TL;DR

This paper introduces a new first integral for the partially averaged Newtonian potential in the three-body problem, revealing its role in cancellations, resonances, and symmetries, with implications for understanding the system's dynamics.

Contribution

It establishes a novel first integral for the partial average of the Newtonian potential, elucidating its impact on resonances and symmetries in the three-body problem.

Findings

Identifies a non-trivial first integral for the partial average.

Shows the integral's role in cancellations and the Herman resonance.

Discusses applications and open problems related to the integral.

Abstract

We consider the partial average i.e., the Lagrange average with respect to {\it just one} of the two mean anomalies, of the Newtonian part of the perturbing function in the three--body problem Hamiltonian. We prove that such a partial average exhibits a non--trivial first integral. We show that this integral is fully responsible of certain cancellations in the averaged Newtonian potential, including a property noticed by Harrington in the 60s. We also highlight its joint r\^ole (together with certain symmetries) in the appearance of the so called "Herman resonance". Finally, we discuss an application and an open problem.