Nonintegrability of the restricted three-body problem

The problem of nonintegrability of the circular restricted three-body problem is very classical and important in dynamical systems. The problem was partially solved by Poincare in the nineteenth century: He showed that there exists no first integral which depends analytically on the mass of the second body 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.