A unified algebraic construction of the classical Smorodinsky-Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N+1), ISO(N), and SO(N,1) is presented. Firstly, general expressions for the Hamiltonian and its integrals of motion are given in a linear ambient space $R^{N+1}$, and secondly they are expressed in terms of two geodesic coordinate systems on the ND spaces themselves, with an explicit dependence on the curvature as a parameter. On the sphere, the potential is interpreted as a superposition of N+1 oscillators. Furthermore each Lie algebra generator provides an integral of motion and a set of 2N-1 functionally independent ones are explicitly given. In this way the maximal superintegrability of the ND Euclidean Smorodinsky-Winternitz system is shown for any value of the curvature.
Maximal superintegrability of the generalized Kepler--Coulomb system on N-dimensional curved spaces
最近，通过找到一个附加的（隐藏的）动量四次积分，在三维欧几里得空间上叠加有三个离心项的开普勒-库仑势被证明是最大超可积的[Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902]。在本文中，我们提出的推广，这一结果的ND球面，双曲和欧几里德空间，利用统一的对称性的方法，利用曲率参数。由此产生的哈密顿量，形成的（弯曲）开普勒-库仑势连同N离心条款，被赋予（2N-1）功能独立的运动积分：其中一个是四次和其余的是二次。从适当的开普勒-库仑势，其相关的二次Laplace-Runge-Lenz N-向量，广义系统的过渡得到了充分的描述。在所有这些系统中的球形，非线性（立方），和余代数对称性的作用是突出的。
This paper has studied the three-dimensional Dunkl oscillator models in a generalization of superintegrable Euclidean Hamiltonian systems to curved ones. These models are defined based on curved Hamiltonians, which depend on a deformation parameter of underlying space and involve reflection operators. Their symmetries are obtained by the Jordan-Schwinger representations in the family of the Cayley-Klein orthogonal algebras using the creation and annihilation operators of the dynamical $sl_{-1}(2)$ algebra of the one-dimensional Dunkl oscillator. The resulting algebra is a deformation of $so_{κ…
The superintegrability of several Hamiltonian systems defined on three-dimensional configuration spaces of constant curvature is studied. We first analyze the properties of the Killing vector fields, Noether symmetries and Noether momenta. Then we study the superintegrability of the Harmonic Oscillator, the Smorodinsky-Winternitz (S-W) system and the Harmonic Oscillator with ratio of frequencies 1:1:2 and additional nonlinear terms on the 3-dimensional sphere $S^3$ ($\kp>0)$ and on the hyperbolic space $H^3$ ($\kp<0$). In the second part we present a study first of the Kepler problem and then …
We study four particular 3-dimensional natural Hamiltonian systems defined in conformally Euclidean spaces. We prove their superintegrability and we obtain, in the four cases, the maximal number of functionally independent integrals of motion. The two first systems are related to the 3-dimensional isotropic oscillator and the superintegrability is quadratic. The third system is obtained as a continuous deformation of an oscillator with ratio of frequencies 1:1:2 and with three additional nonlinear terms of the form $k_2/x^2$, $k_3/y^2$ and $k_4/z^2$, and the fourth system is obtained as a defo…
The rank-$1$ Racah algebra $R(3)$ plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the $3$-parameter system on the $2$-sphere from which all second-order conformally flat superintegrable models in $2$D can be obtained by means of suitable limits and contractions. A higher rank generalization of $R(3)$, the so-called rank $n-2$ Racah algebra $R(n)$, has been considered recently and showed to be the symmetry algebra of the general superintegrable model on the $(n-1)$-sphere. In the present work, we show that such an algebraic structure naturall…
A Hamiltonian dynamics defined on the two-dimensional hyperbolic plane by coupling the Morse and Rosen-Morse potentials is analyzed. It is demonstrated that orbits of all bounded motions are closed iff the product of the parameter $\tilde a$ of the Morse potential and the square root of the absolute value of the curvature is a rational number. This property of trajectories equivalent to the maximal superintegrability is confirmed by explicit construction of polynomial superconstant of motion.
Distribution of the time at which N vicious walkers reach their maximal height
研究了一维单位时间内N个不相交的布朗运动（恶性步行者）的极值统计。利用路径积分技术，我们精确地计算了最大值M和到达最大值的时间τ_M的联合分布。我们特别关注不相交的布朗桥（“西瓜没有墙”）和不相交的布朗游览（“西瓜有墙”）。我们详细讨论了这种恶性步行者模型在西瓜配置和随机增长模型在弯曲的几何形状，一方面和定向聚合物在无序介质（DPRM）与一个自由端点的另一方面之间的关系。我们还检查我们的结果使用戴森的布朗运动的数值模拟和面对他们的多核生长模型（PNG）和离散晶格上的DPRM模型的数值模拟。这里展示的一些结果是在最近的一封信中宣布的[J. Rambeau和G. Schehr，Europhys. Lett. 91，60006（2010）]。
