A physical interpretation for certain Hamiltonian on the tangent bundle of spheres

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What would be  an interpretation for the  Hamiltonian described in this question in terms of Hamiltonian mechanics?

asked Nov 4, 2016
edited Nov 7, 2016

Please, could you import the main question ?

@igael: The link in the question now points to the post imported from MO.

@ArnoldNeumaier  Prof. Neumaier, Thank you very much for your revision.

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