Introduction

The account we present of the foundations of Hamiltonian mechanics and its mathematical language — the Hamiltonian formalism — represents the contents of a course oflectures given by A. M. Vinogradov in the Faculty of Applied Mathematics at the Moscow Institute of Electronic Machine Construction in the Faculty of Mechanics and Mathematics at the Moscow State University, and at the eighth Voronezh winter school. What these lectures aimed at was a logical account of mechanics, for example, the question when basic facts can be derived from basic concepts by “rules of grammar”. Since the language of Hamiltonian mechanics is the calculus of differential forms and vector fields on smooth manifolds, the basic formulae of this calculus play the role of such grammatical rules. A pleasant consequence of this approach is the possibility of avoiding the manner of arguing by calculations which is usual in analytical mechanics.

Lack of space prevents us from considering more special questions (non-holonomic systems, the theory of impact and refraction, friction, optimal control). So we only remark that they too admit a very simple invariant treatment starting out from the point of view we have taken. We have also not described the natural connection of mechanics with the general theory of differential equations, which is a problem of paramount importance. The algebraic—logical side of this connection is indicated in [16]; for the geometrical basis see [10] and [13].