Quantization of the motion of a particle on an n-dimensional sphere

We develop here a simple formalism that converts second-class constraints into first-class ones for a particle moving on an n-dimensional sphere. The Poisson algebra generated by the Hamiltonian and the constraints closes and by quantization transforms into a Lie algebra. The observable of the theory is given by the Casimir operator of this algebra and coincides with the square of the angular momentum.