First Integrals of Extended Hamiltonians in n +1 Dimensions Generated by Powers of an Operator

[摘要] We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians H obtained as one-dimensional extensions of natural (geodesic) n -dimensional Hamiltonians L . The Liouville integrability of L implies the (minimal) superintegrability of H . We prove that, as a consequence of natural integrability conditions, it is necessary for the construction that the curvature of the metric tensor associated with L is constant. As examples, the procedure is applied to one-dimensional L , including and improving earlier results, and to two and three-dimensional L , providing new superintegrable systems.

[关键词] superintegrable Hamiltonian systems;polynomial first integrals;constant curvature;Hessian tensor