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.

A curious fact that has come to light in the course of the last ten or fifteen years is the appearance of the Hamiltonian formalism of classical mechanics as an essential component of a number of mathematical theories, far removed from mechanics as such. Examples of this kind are the orbit method in the theory of representations of Lie groups (see [10] and [11]), the theory of local solubility of linear differential operators (see the survey [8]), the theory of a canonical operator (see [13] and [14]), and others.