Extensions of natural Hamiltonians

Abstract

Given an n-dimensional natural Hamiltonian L on a Riemannian or pseudo-Riemannian manifold, we call "extension" of L the n+1 dimensional Hamiltonian $H=\frac{1}{2}{p}_u^2+\alpha (u)L+\beta (u)$ with new canonically conjugated coordinates $(u,p_u)$. For suitable L, the functions $\alpha$ and $\beta$ 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.

4 pages. Talk presented at the International Conference on Mathematical Modeling in Physical Sciences September 1-5, 2013 Prague, Czech Republic

References in corpus (5)

• Superintegrable 3-body systems on the line
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• Classical and quantum higher order superintegrable systems from coalgebra symmetry
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