Nonintegrability of the restricted three-body problem

TL;DR

This paper proves the nonintegrability of the circular restricted three-body problem in both planar and spatial cases for any nonzero mass of the second body, using advanced techniques related to perturbations and integrability theories.

Contribution

It introduces a new technique based on generalized Morales-Ramis theories to establish nonintegrability near resonant periodic orbits in perturbed systems.

Findings

Proves nonintegrability for all nonzero second body masses.

Develops a novel method for analyzing non-Hamiltonian perturbations.

Extends nonintegrability results to spatial three-body problems.

Abstract

The problem of nonintegrability of the circular restricted three-body problem is very classical and important in the theory of dynamical systems. It was partially solved by Poincare in the nineteenth century: He showed that there exists no real-analytic first integral which depends analytically on the mass ratio of the second body to the total and is functionally independent of the Hamiltonian. When the mass of the second body becomes zero, the restricted three-body problem reduces to the two-body Kepler problem. We prove the nonintegrability of the restricted three-body problem both in the planar and spatial cases for any nonzero mass of the second body. Our basic tool of the proofs is a technique developed here for determining whether perturbations of integrable systems which may be non-Hamiltonian are not meromorphically integrable near resonant periodic orbits such that the first integrals and commutative vector fields also depend meromorphically on the perturbation parameter. The technique is based on generalized versions due to Ayoul and Zung of the Morales-Ramis and Morales-Ramis-Simo theories.