Curvature and mechanics

AbstractClassical or Newtonian Mechanics is put in the setting of Riemannian Geometry as a simple mechanical system (M, K, V), where M is a manifold which represents a configuration space, K and V are the kinetic and potential energies respectively of the system. To study the geometry of a simple mechanical system, we study the curvatures of the mechanical manifold (Mh, gh) relative to a total energy value h, where Mh is an admissible configuration space and gh the Jacobi metric relative to the energy value h. We call these curvatures h-mechanical curvatures of the simple mechanical system.Results are obtained on the signs of h-mechanical curvature for a general simple mechanical system in a neighborhood of the boundary ∂Mh = {xεM: V(x) = h} and in a neighborhood of a critical point of the potential function V. Also we construct m = (n2) (n = dim M) functions defined globally on Mh, called curvature functions which characterize the sign of the h-mechanical curvature. Applications are made to the Kepler problem and the three-body problem