Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces

Abstract

We formulate the necessary conditions for the maximal super-integrability of a certain family of classical potentials defined in the constant curvature two-dimensional spaces. We give examples of homogeneous potentials of degree -2 on $E^2$ as well as their equivalents on $S^2$ and $H^2$ for which these necessary conditions are also sufficient. We show explicit forms of the additional first integrals which always can be chosen polynomial with respect to the momenta and which can be of an arbitrary high degree with respect to the momenta.

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• Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization
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• First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator
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• General Nth order integrals of the motion
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• Computing necessary integrability conditions for planar parametrized homogeneous potentials
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• Extensions of natural Hamiltonians
• Superintegrability of the Post-Winternitz system on spherical and hyperbolic spaces