By introducing a tangential space to the manifold of all possible positions of a mechanical system of equations, its motions are written in the form of a single vector equation, which has the form of Newton's second law. From this equation, written for ideal non-linear time-dependent non-holonomic first-order constraints, the Poincaré- Chetayev- Rumyantsev equations, as well as other fundamental types of equations of motion, are obtained.