DYNAMICAL BEHAVIOR OF LAGRANGIAN SYSTEMS ON FINSLER MANIFOLDS

Tn this paper we develop a theoretical framework devoted to a geometri cal description of the behavior of dynamical systems and their chaotic properties. The underground manifold is a Finsler space whose feature s permit the description of a wide class of dynamical systems such as those with potentials depending on thr time and velocities for which t he Riemannian approach is unsuitable. Another appealing feature of thi s more general setting relies on its very origin: Finsler spaces arise in a direct way on imposing the invariance for time reparametrization to a standard variational problem. A Finsler metric is a generalizati on of the well-known Jacobi and Eisenhart-metrics for conservative dyn amical systems. We use this geometry to derive the main geometrical in variants and related expressions that are needed to establish the tran sition to chaos in very general Lagrangian systems. In order to point out the versatility and the effectiveness of this extension of the geo metrical approach, we suggest the introduction of this formalism to so me interesting dynamical systems for which the Finsler metric is much more suitable than the Riemannian one. In particular, we present the f ollowing: (i) an exhaustive description and numerical results for a re sonant oscillator with a time-dependent potential, (ii) an exact descr iption (without any approximation) of the dynamics of Bianchi type-IX cosmological models, and (iii) a geometrical description of the restri cted three-body problem chose effective potential depends Linearly on the velocities. In the first case, the numerical integration of the ge odesics and geodesic deviation equations shows that in the geometrical picture the source of the exponential instability of trajectories rel ies on the mechanism of parametric resonance and does not originate fr om the negativity of curvature.