| 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 Poincar\`{e} 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 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.
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-Sim\`{o} theories. |
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