Horwitz Lawrence, Zion Yossi Ben, Lewkowicz Meir, Schiffer Marcelo, Levitan Jacob
Department of Physics, College of Judea and Samaria, Ariel 44837, Israel.
Phys Rev Lett. 2007 Jun 8;98(23):234301. doi: 10.1103/PhysRevLett.98.234301. Epub 2007 Jun 4.
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.
通过共形度量的定义，将根据哈密顿量结构中与黎曼度量张量相关的曲率对混沌哈密顿系统的刻画扩展到一大类标准形式的势模型。当过渡到相关流形时，测地线方程重现了原始势模型的哈密顿方程。通过这种方式，我们找到了哈密顿势模型时间演化的直接几何描述。在这个相关流形中，测地线偏差的二阶协变导数产生了（与能量有关的）不同于通常李雅普诺夫准则的不稳定行为准则。我们讨论了二维不稳定哈密顿系统的一些例子。
