A new proof of Poincaré’s result on the restricted three-body problem

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 Poincaré showed the nonexistence of a 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 the planar case. 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 that the author developed recently for nearly integrable systems. The recent result of Yagasaki [Ergod. Theory Dyn. Syst. 44, 3012–3049 (2024)] on this problem removes the dependence of the first integral on the small parameter, but it is limited to a small region in the phase space. We will see that the progress in the theory of nonintegrability of dynamical systems since the seminal work of Ziglin [Funct. Anal. Appl. 16, 181–189 (1982)] can now simplify the proof of Poincaré’s result drastically.