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Section: New Results

Computation of necessary integrability conditions for parametrized Hamiltonian systems

Let $V({𝐪}_1,{𝐪}_2)$ be a homogeneous function whose coefficients depend rationally on parameters ${𝐚}_1,...,{𝐚}_n$. In [10] we designed an algorithm to compute polynomial necessary conditions on the parameters $({𝐚}_1,...,{𝐚}_n)$ such that the dynamical system associated to the potential $V$ is integrable. These conditions originate from those of the classical Morales-Ramis-Simó 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.