For general nonholonomically constrained systems, variation identity is used to define three kinds of unfree variations, i.e., nonholonomic variations: the vakonomic, the Hlder and the Suslov by means of vector fields on the extended configuration manifold. The relations among the three kinds variations are discussed and a necessary and sufficient condition for the variations to become free ones is obtained. The nonholonomic variations and the corresponding integral variational principles are utilized to derive the two kinds of dynamical equations: vakonomic equations and Routh's equations or Chaplygins equations. By comparing vakonomic equations with Rouths equations and Chaplygins equations respectively, two necessary and sufficient conditions for the two kinds of equations to have common solutions are obtained, which are not integrable conditions of the constraints.
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