Various ways of generalizing classical canonical dynamics are considered. It is found that there exist systems of c-number Hamiltonian dynamics possessing neither canonically conjugate dynamical variables nor Lagrangian formulation. The possibility of a non-c-number classical dynamics is then considered and realized. Elementary examples of both types of dynamics are examined, and emerge as classical analogues of a Fermi oscillator. This is remarkable in view of the commonly held belief that a Fermi oscillator does not have a classical analogue. Not all possible classical analogues possess a Lagrangian formulation: those which do are of particular interest, since they provide the means of setting up a kind of Feynman principle.