Higher order first integrals in classical mechanics

We present a practical algorithm for computing first integrals of motion which are polynomial in the momenta for natural Hamiltonian systems defined in a flat pseudo-Riemannian space of arbitrary dimension and signature. We then apply our algorithm to explore the integrability of two physical systems. First, we study the Holt potential in two dimensions and derive analogous potentials which admit an additional first integral quartic in the momenta. Second, we analyze a class of cylindrically symmetric potentials in three-dimensional Euclidean space and recover known families of second-order maximally superintegrable systems. (C) 2007 American Institute of Physics.