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Nonintegrability of the restricted three-body problem

Kazuyuki Yagasaki

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.

Nonintegrability of the restricted three-body problem

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.

Paper Structure

This paper contains 6 sections, 5 theorems, 129 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Determination of nonintegrability for \ref{['eqn:aasys']}
  3. Planar Case
  4. Spatial Case
  5. Differential Galois Theory
  6. Monodromy matrices

Key Result

Theorem 1.1

The circular restricted three-body problem eqn:pp$($resp. eqn:sp$)$ does not have a complete set of first integrals in involution that are functionally independent almost everywhere and meromorphic in $(x,y,p_x,p_y,u_1,u_2)$$($resp. in $(x,y,z,p_x,p_y,p_z,u_1,u_2))$ except on $\Sigma(\mathscr{S}_2)$ for any $\mu\in(0,1)$, as Hamiltonian systems on $\mathscr{S}_2$$($resp. on $\mathscr{S}_3)$.

Figures (6)

  • Figure 1: Configuration of the circular restricted three-body problem in the rotational frame.
  • Figure 2: Assumption (A2).
  • Figure 3: Riemann surface $\Gamma$. The monodromy matrix $M_\gamma$ is computed along the loop $\gamma$.
  • Figure 4: Dependence of $K_1$ on $e$.
  • Figure 5: Closed path $\gamma_\theta$.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Theorem 1.1
  • Definition 1.2: Bogoyavlenskij
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • proof : Proof of Theorem $\ref{['thm:tool']}$
  • Lemma 2.5
  • proof
  • Remark 2.6
  • ...and 1 more