Curvature of gravitationally bound mechanical systems

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Abstract
In the present work mathematical aspects of determining the local instability parameters are focused on by using invariant characteristics of the internal Riemannian geometry with the Jacobi metric (in principle, for Hamiltonian dynamical systems with the natural Lagrangian). First, it is shown that the Ricci scalar indeed measures the sectional curvature averaged upon all two‐directions. Second, necessary and sufficient 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 nearby trajectories are given. Fourth, it is shown that for any collisionless problem of n gravitationally bounded bodies, the sectional curvature in every direction is negative if n tends to infinity.