On the geometry of Hamiltonian chaos
arXiv:physics/0701212 · doi:10.1103/PhysRevLett.98.234301
Abstract
We show that Gutzwiller's characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian can be extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model when a transition is made to the dual manifold, and the geodesics in the dual space coincide with the orbits of the Hamiltonian potential model. We therefore find a direct geometrical description of the time development of a Hamiltonian potential model. The second covariant derivative of the geodesic deviation in this dual manifold generates a dynamical curvature, resulting in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We discuss some examples of unstable Hamiltonian systems in two dimensions giving, in particular, detailed results for a potential obtained from a fifth order expansion of a Toda lattice Hamiltonian.
7 pages TeX, Figure captions, 4 figures (eps). Some clarifications, added references
References in corpus (2)
Cited by in corpus (12)
- Free motion around black holes with discs or rings: between integrability and chaos - II
- Free motion around black holes with discs or rings: between integrability and chaos - I
- An Elementary Canonical Classical and Quantum Dynamics for General Relativity
- Relativistic invariance of Lyapunov exponents in bounded and unbounded systems
- Kaluza-Klein Theory as a Dynamics in a Dual Geometry
- Hamiltonian Map to Conformal Modification of Spacetime Metric: Kaluza-Klein and TeVeS
- Controlling Effect of Geometrically Defined Local Structural Changes on Chaotic Hamiltonian Systems
- Entropy measures as geometrical tools in the study of cosmology
- Spectral bounds on the entropy flow rate and Lyapunov exponents in differentiable dynamical systems
- Geometric investigation of chaos unfolding in Hamiltonian systems
- Criterion for Stability of a Special Relativistically Covariant Dynamical System
- An Underlying Geometrical Manifold for Hamiltonian Mechanics