One kind motion of controllable constrained Birkhoffian system: the absence of constraints

This paper is devoted to discuss the motion of controllable constrained Birkhoffian system along with its absence of constraints. The first step is to establish the autonomous and non-autonomous differential equations of motion of the system, based on Pfaff–Birkhoff principle. Secondly, the existence of constraint multipliers are exhaustively discussed. Thirdly, the definition of one kind motion of the system, called free motion, is given, which is described and analyzed by the absence of constraints that are determined by constraint multipliers. Lemma 2 illustrates that one system can realize its free motion by selecting proper control parameters. In particular, theorem 2 provides that one system can naturally realize free motion when we consider the integral of the unconstrained Birkhoffian system as the constraints of constrained Birkhoffian system. Finally, the results obtained are illustrated by several examples.