ON THE GEOMETRY OF HAMILTONIAN SYSTEMS

The paper studies the geometry of integrable Hamiltonian system. The basic concept of a Hamiltonian system of differential equations forms the basis of much of the more advanced work in classical mechanics, including motions of rigid bodies, celestial mechanics, quantization theory and so on. More recently, Hamiltonian methods have become increasingly important in the study of the equations of continuum mechanics, including fluids, plasmas and elastic media. It is shown that the level surfaces are three-dimensional surfaces with the positive or negative Gauss curvature at the point, where it is depend on which canonical form it has. All surfaces can be characterized by their Gaussian curvature, which describes the geometry of the surface: Euclidean, elliptic, or hyperbolic. Furthermore, curvature dictates how a vector may be transported across the surface or within the space.