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Two models with linear and nonlinear second class constraints are consideredand gauged by embedding in an extended phase space. These models are the freenon-relativistic particle on a hyperplane and hyper sphere in configurationspace. For the first model we construct its gauged corresponding by thecondition of converting second class system to first class one, directly. Incontrast the first class system related to the free particle on hyper sphere isderived by the BFT embedding procedure, where its steps are infinite. We give apractical formula for gauging linear and some of the nonlinear second classsystems, based on the simplified BFT method. As a result of the gauging twomodels, we show that in the conversion of second class to the first classconstraints the minimum number of phase space degrees of freedom for bothsystems is a pair of phase space coordinate. This pair for first system is acoordinate and its momentum conjugate, but Poisson structure of embeddednon-relativistic particle on hyper sphere is a non-trivial one. We deriveinfinite correction terms for the Hamiltonian of the nonlinear constraints. Thesummation is done and constructs an interacting gauged Hamiltonian. We find anopen algebra for three first class objects of the embedded nonlinear system.

First class models from linear and nonlinear second class constraints