Mathematics – Logic
Scientific paper
Jun 1997
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1997phrve..55.6448b&link_type=abstract
Physical Review E, vol. 55, Issue 6, pp. 6448-6458
Mathematics
Logic
9
Scientific paper
In this paper we develop a theoretical framework devoted to a geometrical description of the behavior of dynamical systems and their chaotic properties. The underground manifold is a Finsler space whose features permit the description of a wide class of dynamical systems such as those with potentials depending on the time and velocities for which the Riemannian approach is unsuitable. Another appealing feature of this 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 generalization of the well-known Jacobi and Eisenhart-metrics for conservative dynamical systems. We use this geometry to derive the main geometrical invariants and related expressions that are needed to establish the transition to chaos in very general Lagrangian systems. In order to point out the versatility and the effectiveness of this extension of the geometrical approach, we suggest the introduction of this formalism to some interesting dynamical systems for which the Finsler metric is much more suitable than the Riemannian one. In particular, we present the following: (i) an exhaustive description and numerical results for a resonant oscillator with a time-dependent potential, (ii) an exact description (without any approximation) of the dynamics of Bianchi type-IX cosmological models, and (iii) a geometrical description of the restricted three-body problem whose effective potential depends linearly on the velocities. In the first case, the numerical integration of the geodesics and geodesic deviation equations shows that in the geometrical picture the source of the exponential instability of trajectories relies on the mechanism of parametric resonance and does not originate from the negativity of curvature.
Bari Maria Di
Boccaletti Dino
Cipriani Piero
Pucacco Giuseppe
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