Computing necessary integrability conditions for planar parametrized homogeneous potentials

Abstract

Let V ∈ Q(i)(a₁,..., a_n)(q₁, q₂) be a rationally parametrized planar homogeneous potential of homogeneity degree k ≠ −2, 0, 2. We design an algorithm that computes polynomial necessary conditions on the parameters (a₁,..., a_n) such that the dynamical system associated to the potential V is integrable. These conditions originate from those of the Morales-Ramis-Simó integrability criterion near all Darboux points. 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.