Rigorous treatment of the averaging process for co-orbital motions in the planetary problem

TL;DR

This paper develops a rigorous Hamiltonian formalism for co-orbital planetary motions, providing bounds and theorems that describe their behavior over large timescales.

Contribution

It introduces a complex domain approach and estimates transformations to rigorously analyze co-orbital resonance in planetary systems.

Findings

Bounded the difference between averaged and integrable Hamiltonians.

Proved theorems on co-orbital motion stability over finite times.

Constructed a complex domain for Hamiltonian analysis.

Abstract

We develop a rigorous analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. By constructing a complex domain of holomorphy for the planetary Hamilto-nian, we estimate the size of the transformation that maps this Hamil-tonian to its first order averaged over one of the fast angles. After having derived an integrable approximation of the averaged problem, we bound the distance between this integrable approximation and the averaged Hamiltonian. This finally allows to prove rigorous theorems on the behavior of co-orbital motions over a finite but large timescale.