The law, of a motion of mechanical system, represented in the vector form, is applied to the solution of the mixed problem of dynamics. The essence of the problem is to find additional generalized forces such that the program constraints, given in the form of system of differential equations of order n ≥ 3, are satisfied. The notion of generalized control force is introduced. The fact is proved that if the number of program constraints is equal to the number of generalized control forces, then the latter can be found as the time functions from the system of differential equations in generalized coordinates and these forces. The conditions, under which this system of equations has a unique solution, are determined. The conditions are also obtained under which for the constraints of any order the motion control is realized according to Gauss' principle. Thus, the theory is constructed with the help of which a new class of control problems can be solved. This theory is used to consider two problems connected with the dynamics of spacecraft motion. In the first problem a radial control force, providing the motion of spacecraft with modulo constant acceleration, is determined as a time function. In the second problem we seek the law, of varying in time the radial and tangential control forces, by which a smooth passage of spacecraft from one circular orbit to another occur.