Covariant Hamiltonian dynamics

We discuss the covariant formulation of the dynamics of particles with Abelian and non-Abelian gauge charges in external fields. Using this formulation we develop an algorithm for the construction of constants of motion, which makes use of a generalization of the concept of Killing vectors and tensors in differential geometry. We apply the formalism to the motion of classical charges in Abelian and non-Abelian monopole fields.