CURVATURE OF GRAVITATIONALLY BOUND MECHANICAL SYSTEMS

In the present work mathematical aspects of determining the local inst ability parameters are focused on by using invariant characteristics o f the internal Riemannian geometry with the Jacobi metric (in principl e, for Hamiltonian dynamical systems with the natural Lagrangian). Fir st, it is shown that the Ricci scalar indeed measures the sectional cu rvature averaged upon all two-directions. Second, necessary and suffic ient criteria for non-negativity and of nonpositivity of the sectional curvature for any system with the natural Lagrangian are given. Third , analytical formulas allowing us to compute the separation rate of ne arby trajectories are given. Fourth, it is shown that for any collisio nless problem of n gravitationally bounded bodies, the sectional curva ture in every direction is negative if n tends to infinity.