Visual Symplectic Topology and Visual Hamiltonian Mechanics

Abstract

We shall consider a vector field υ on a smooth manifold. Relative to local coordinates x1,..., x n this field can be written as dx i /dt = υ i (x1,..., x n ) where 1 ≤ i ≤ n and υ i (x1,..., x n ) are smooth functions which are components of the field υ. Thus, each vector field is interpreted as a system of ordinary differential equations on a manifold. Inversely, each system or ordinary differential equations can be represented as the vector field on a corresponding manifold. Many physical laws are written in the form of differential equations, and therefore the study of topological properties of vector fields provides a good deal of qualitative information on the behaviour of certain physical systems. Among the variety of mechanical and physical systems there exists an important class described using so-called Hamiltonian equations. These systems can be realized on even-dimensional manifolds only.