2007/01/31 by L. P. Horwitz, Lawrence Horwitz, Jacob Levitan +4 · 2 citations
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Chaotic #Classical mechanics #Conformal map #Covariant Hamiltonian field theory #Covariant transformation #Curvature #Geodesic #Geometry #Hamiltonian (control theory) #Hamiltonian mechanics #Hamiltonian system #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Physics #Quantum chaos and dynamical systems #Quantum mechanics #Riemann curvature tensor #gr-qc #nlin.CD #physics.class-ph #physics.gen-ph
paper · pdf · doi:10.1103/physrevlett.98.234301
published as Phys.Rev.Lett.98:234301,2007 · 7 pages TeX, Figure captions, 4 figures (eps). Some clarifications, added references
arxiv created 2007/04/17 · openalex publication_date 2007/06/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian is 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 an associated manifold. We find, in this way, a direct geometrical description of the time development of a Hamiltonian potential model. The second covariant derivative of the geodesic deviation in this associated manifold results in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We discuss some examples of unstable Hamiltonian systems in two dimensions.