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In this paper, we investigate the dynamics of both free particle andisotropic harmonic oscillator constrained to move on a spheroidal surface usingtwo consecutive projections: a projection onto a sphere surface followed by thegnomonic projection onto a tangent plane to the spheroid. We obtain theHamiltonian of the aforementioned systems in terms of the Cartesian coordinatesof the tangent plane and then quantize it in the standard way. It is shown thatthe effect of non-sphericity of the surface can be treated as the appearance ofan effective potential. By using the perturbation theory up to the first orderin second eccentricity of the spheroid, we approximately calculate theeigenfunctions and eigenvalues of the free particle, as well as the isotropicharmonic oscillator on the spheroidal surface. We find that the deviation fromthe sphericity plays an important role in splitting the energy levels of theisotropic oscillator on a sphere, and lifting the degeneracy.

Free particle and isotropic harmonic oscillator on a spheroidal surface: the Higgs-like approach