Basic Concepts and Theorems

Abstract

Let us consider a vector field on a smooth manifold. Let x ¹, ..., x ⁿ be local coordinates, then we can write down the vector field in the form

$$\frac{{d{x^i}}}{{dt}} = {\xi ^i}\left( {{x^1}, \ldots ,{x^n}} \right),\quad i = n, \ldots ,n,$$

((1.1))

, where ξ ⁱ (x ¹, ..., x ⁿ) are smooth functions being the components of the field. Thus, each vector field is interpreted as a system of ordinary differential equations on a manifold. And conversely, each system of ordinary differential equations describes the vector field on the corresponding manifold. In classical mechanics a motion of a system can be described with the help of ordinary differential equations. Among mechanical systems there exists the important class of systems which are described by Hamiltonian equations. These systems are realized on symplectic manifolds.