Extensions of Natural Hamiltonians

G Rastelli

doi:10.1088/1742-6596/490/1/012088

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 = ½p2u + 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.

Highlights

We propose here a short review of the known results of the theory and some previews of new ones
Natural Hamiltonian systems are the mathematical models of those physical systems for which the energy is constant, for example harmonic oscillators or the Kepler system
As in the previous two examples, more quantities are constants of the motion: angular momentum, Laplace-Runge-Lentz vector, etc. These constants are expressed by quadratic polynomials in the momenta or, for quantum systems, by second-order differential operators

Summary

Introduction

Natural Hamiltonian systems are the mathematical models of those physical systems for which the energy is constant, for example harmonic oscillators or the Kepler system. VJC = k (x − y)−2 + (x − z)−2 + (y − z)−2 , VW = k (x + y − 2z)−2 + (x + z − 2y)−2 + (y + z − 2x)−2 , respectively (we do not consider here the harmonic oscillator terms) and they have essentially the same dynamics [1] Both the resulting natural Hamiltonians in E3 admit one linear and one quadratic in the momenta constants of the motion, making the systems Liouville-integrable and solvable by separation of variables (see [1] and references therein). Let Q be a n-dimensional (pseudo-)Riemannian manifold with metric tensor g

The natural Hamiltonian admits an extension H in

XL m