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Register a new userExtensions of natural Hamiltonians
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Authors
Giovanni Rastelli
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Given an n-dimensional natural Hamiltonian L on a Riemannian or pseudo-Riemannian manifold, we call extension of L the n+1 dimensional Hamiltonian $H=frac 12 p_u^2+alpha(u)L+beta(u)$ 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.
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The concept of extended Hamiltonian systems allows the geometrical interpretation of several integrable and superintegrable systems with polynomial first integrals of degree depending on a rational parameter. Until now, the procedure of extension has been applied only in the case of natural Hamiltonians. In this article, we give several examples of application to non-natural Hamiltonians, such as the two point-vortices, the Lotka-Volterra and some quartic in the momenta Hamiltonians, obtaining effectively extended Hamiltonians in some cases and failing in others. We briefly discuss the reasons of these results.
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