The meromorphic non-integrability of the three-body problem (Q2755159)

The paper is devoted to the non-integrability of planar three-body problem. First, the author gives a short review of classical and modern works showing that classical integrals or their combinations are only independent algebraic integrals. Then the author applies known methods to Lagrangian parabolic orbits. The integrability of three-body problem is studied in a sufficiently small complex neighbourhood. The author describes a reduction of plane three-body problem from Hamiltonian system with 6 degrees of freedom to a Hamiltonian system with 3 degrees of freedom. The definition of meromorphic first integral is given, and normal variational equations of the system along integral curves are obtained. The author proves the nonexistence of the additional meromorphic integral in a set of theorems. The final result is formulated as follows: ``the plane three-body problem is meromorphically non-integrable near the Lagrangian parabolic solution''. Finally, the author gives a dynamical interpretation of this result in terms of the theory of splitting and transverse asymptotic manifolds.