Geometry of Hamiltonian chaos.
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.
Department of Physics, College of Judea and Samaria, Ariel 44837, Israel.
Lawrence
L
Horwitz
Yossi Ben
YB
Zion
Meir
M
Lewkowicz
Marcelo
M
Schiffer
Jacob
J
Levitan