Curvature and mechanics

Abstract

Classical 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 ( M h , g h ) relative to a total energy value h, where M h is an admissible configuration space and g h 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 ∂M h = { x ε M: V( x) = h} and in a neighborhood of a critical point of the potential function V. Also we construct m = ( n 2 ) (n = dim M) functions defined globally on M h , called curvature functions which characterize the sign of the h-mechanical curvature. Applications are made to the Kepler problem and the three-body problem.