Extensions of Natural Hamiltonians

2nd International Conference on Mathematical Modeling in Physical Sciences 2013

物理学;数学

Rastelli, G.^1

Cna Ortolano 7, Ronsecco, Italy^1

关键词: Configuration manifold;    Differential operators;    Hamiltonian systems;    Harmonic oscillators;    Laplace-Beltrami;    Natural number;    Polynomial first integral;    Riemannian manifold;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/490/1/012088/pdf
DOI  :  10.1088/1742-6596/490/1/012088

来源: IOP

【 摘 要 】

Given an n-dimensional natural Hamiltonian L on a Riemannian or pseudo-Riemannian manifold, we call "extension" of L the n+1 dimensional Hamiltonian H 1/2p2u+ α(u)L + β(u) with new canonically conjugated coordinates (u,pu). For suitable L, the functions α and β can be chosen depending on any natural number m such that H admits an extra polynomial first integral in the momenta of degree m, explicitly determined in the form of the m-th power of a differential operator applied to a certain function of coordinates and momenta. In particular, if L is maximally superintegrable (MS) then H is MS also. Therefore, the extension procedure allows the creation of new superintegrable systems from old ones. For m=2, the extra first integral generated by the extension procedure determines a second-order symmetry operator of a Laplace-Beltrami quantization of H, modified by taking in account the curvature of the configuration manifold. The extension procedure can be applied to several Hamiltonian systems, including the three-body Calogero and Wolfes systems (without harmonic term), the Tremblay-Turbiner-Winternitz system and n-dimensional anisotropic harmonic oscillators. We propose here a short review of the known results of the theory and some previews of new ones.