Curvature-induced noncommutativity of two different components of momentum for a particle on a hypersurface

Abstract As a nonrelativistic particle constrained to remain on an ( N ? 1)-dimensional (( ? 2)) hypersurface embedded in -dimensional Euclidean space, two different components p i and j i, = 1, 2, 3,… ) of the Cartesian momentum are not mutually commutative, explicitly commutation relations [ p ˆ i , j stretchy="false">] ? 0 depend products positions momenta uncontrollable ways. The generalized Dupin indicatrix hypersurface, local analysis technique, is utilized explore dependence noncommutativity curvatures around point hypersurface. first finding that can be grouped into categories; one product sectional curvature angular momentum, another principal momentum. second that, for small circle lying tangential plane covering , leads rotation operator amount angle anholonomy; along each normal curves centering given translation plus additional half change arc.