Chaos indicator and integrability conditions from geometrodynamics

Abstract

Stability and chaoticity in conservative Hamiltonian systems are analyzed using an indicator based on a generalization of the virtual work principle (VWP) for Riemannian manifolds. The geometrodynamic formalism obtained in this way is applied to define a mechanical manifold using the Jacobi metric, where the system trajectories are geodesics. The VWP for static mechanical equilibrium in Euclidean spaces is generalized and applied to trajectories in this manifold through geodesic equations derived from a Weyl transformation to this metric. We further interpret each trajectory of the system as a curve representing a non-stretchable string under tension derived from a potential function with constant length in this mechanical manifold, and analyze its stability through the fluctuation of an observable defined from the previous analysis. In this way, we can define a practical chaos indicator and find a sufficiency condition for a conservative dynamical system to have a regular dynamics. Several benchmark cases in two and three dimensions are presented as illustrations.