Nonintegrability of the restricted three-body problem

開催日時
2021/11/26 金 15:00 - 17:00
講演者
矢ヶ崎 一幸
講演者所属
京都大学
概要

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 first integral which depends analytically on the mass ratio of the second body to the first one 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. In this talk, 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 explained 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. I also review my recent related work on the Duffing oscillator, for which Ueda and Holmes's work from 1960's to 80's are well-known, and a new proof of Poincare's original result on the restricted three-body problem.