Integrability of Hamiltonian systems with homogeneous potentials of degree zero

Guy Casale (1) , Guillaume T. Duval , Andrzej J. Maciejewski , Maria Przybylska

1 IRMAR - Institut de Recherche Mathématique de Rennes

We derive necessary conditions for integrability in the Liouville sense of classical Hamiltonian systems with homogeneous potentials of degree zero. We obtain these conditions through an analysis of the differential Galois group of variational equations along a particular solution generated by a non-zero solution View the MathML source of nonlinear equation gradV(d)=d. We prove that when the system is integrable the Hessian matrix V″(d) has only integer eigenvalues and is diagonalizable.

Guy Casale, Guillaume T. Duval, Andrzej J. Maciejewski, Maria Przybylska. Integrability of Hamiltonian systems with homogeneous potentials of degree zero. Physics Letters A, 2010, 374 (3), pp.448-452. ⟨10.1016/j.physleta.2009.11.018⟩. ⟨hal-00459098⟩