Gauge-invariant counterparts and quantization of systems under holonomic constraints

Systems under holonomic constraints are classified within the generalized Hamiltonian framework as second-class constraints systems. We show that each system of point particles with holonomic constraints has a hidden gauge symmetry which allows to quantize it in the original phase space as a first-class constraints system. The proposed method is illustrated with quantization of a point particle moving on an (n−1)-dimensional sphere Sn−1 as well as its field theory analog the O(n) nonlinear sigma model.