Section: New Results
Computation of necessary integrability conditions for parametrized Hamiltonian systems
Let be a homogeneous function whose coefficients depend rationally on parameters . InΒ [10] we designed an algorithm to compute polynomial necessary conditions on the parameters such that the dynamical system associated to the potential is integrable. These conditions originate from those of the classical Morales-Ramis-SimΓ³ integrability criterion. The implementation of the algorithm allows to treat applications that were out of reach before, for instance concerning the non-integrability of polynomial potentials up to degree 9. Another striking application is the first complete proof of the non-integrability of the collinear three-body problem.