We show that the Hamiltonian systems on Sternberg-Wein- stein phase spaces which yield Wong’s equations of motion for a classical particle in a gravitational and a Yang-Mills field, naturally arise as the first order approximation of generic Hamiltonian systems on Poisson manifolds at a critical La- grangian submanifold. We further define a second order ap- proximated system involving scalar fields which first appeared in Einstein-Mayer theory. Reduction and symplectic realiza- tion of this system are interpreted in terms of dimensional reduction and Kaluza-Klein theory.