A new proof of Poincar\'e's result on the restricted three-body problem

TL;DR

This paper presents a simplified proof of Poincaré's classical result on the nonintegrability of the restricted three-body problem, enhancing understanding of its complex dynamical behavior.

Contribution

The paper introduces a new, simpler proof of Poincaré's nonintegrability result using an approach developed for nearly integrable systems.

Findings

Confirmed nonexistence of real-analytic first integrals in the problem

Provided a clearer, more accessible proof of classical nonintegrability results

Applied a novel method to both planar and spatial cases

Abstract

The problem of nonintegrability of the circular restricted three-body problem is very classical and important in dynamical systems. In the first volume of his masterpieces, Henri Poincar\'e showed the nonexistence of a real-analytic first integral which is functionally independent of the Hamiltonian and real-analytic in a small parameter representing the mass ratio as well as in the state variables, in both the planar and spatial cases. However, his proof was very complicated and unclear. In this paper, we give a new and simple proof of a very similar result for both the planar and spatial cases, using an approach which the author developed recently for nearly integrable systems.