A completely integrable Hamiltonian motion on the surface of a sphere

Abstract

It has been shown (Gaffet B 1996 J. Fluid. Mech. 325 113) that the Hamiltonian in three-dimensional flat space, defined by the simple equations of motion is reducible by transformations of variables, to another Hamiltonian governing the two-dimensional motion of a particle on the surface of a sphere. The equations of motion were shown to possess the Painleve property and their integration was reduced to essentially, one quadrature. The earlier analysis essentially concerned the - comparatively much simpler - case where the second integral vanishes; all such solutions were found to be described by elliptic functions. We show in the present work that, when the energy integral (denoted m) and the second integral (denoted ) are related by the general solution is again represented by elliptic functions. The separation of variables is completed, and the solution presented in detail for the case m=-3, chosen as an example.