What is the hamiltonian formalism?

In this paper the basic concepts of the classical Hamiltonian formalism are translated into algebraic language. We treat the Hamiltonian formalism as a constituent part of the general theory of linear differential operators on commutative rings with identity. We take particular care in motivating the concepts we introduce. As an illustration of the theory presented here, we examine the Hamiltonian formalism in Lie algebras. We conclude by presenting a version of the “orbit method” in the theory of representations of Lie groups, which is a natural corollary of our view of the Hamiltonian formalism.