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Generalized Centripetal Force Law and Quantization of Motion Constrained on 2D Surfaces

For a particle moves on a 2D surface f(x)=0 embedded in 3D Euclidean space, the geometric momentum and potential are simultaneously admissible within the Dirac canonical quantization scheme for constrained motion. In our approach, not the full scheme but the symmetries indicated by classical brackets [x,H]_{D} and [p,H]_{D} in addition to the fundamental ones [x,x]_{D}, [x,p]_{D} and [p,p]_{D} are utilized, where the subscript D stands for the Dirac bracket. The generalized centripetal force law p=[p,H]_{D} for particle on the 2D surface play the key role, and there is no simple relationship between the force on a point of the surface and its curvatures of the point, in sharp contrast to the motion on a curve.