12. Higher variational equation techniques for the integrability of homogeneous potentials

Abstract

We present several methods using higher variational equations to study the integrability of Hamiltonian systems from the algebraic and computational point of view. Through the Morales Ramis Simo theorem, strong integrability conditions can be computed for Hamiltonian systems, allowing us to prove nonintegrability even for potentials with parameters. This theorem can, in particular, be applied to potentials, even transcendental ones, by properly defining them on complex Riemann surfaces. In the even more particular case of homogeneous potentials, a complete computation of integrability conditions of variational equation near straight line orbits is possible at arbitrary order, allowing us to prove the nonintegrability of certain n-body problems which were inaccessible due to the complicated central configuration equation.