Method of variation of parameters for solving a constrained Birkhoffian system

For an in-depth study on the integration problem of the constrained mechanical systems the method of integration for the Birkhoffian system with constraints is discussed and the method of variation of parameters for solving the dynamical equations of the constrained Birkhoffian system is provided.First the differential equations of motion for the constrained Birkhoffian system as well as for the corresponding free Birkhoffian system are established.Secondly a system of auxiliary equations is constructed and the general solution of the equations is found.Finally by varying the parameters and utilizing the properties of the generalized canonical transformation of the Birkhoffian system the solution of the problem can be obtained.The proposed method reveals the inherent relationship between the solution of a free Birkhoffian system and that of a constrained Birkhoffian system. The research results are of universal significance which can be further used in a variety of constrained mechanical systems such as non-conservative systems and nonholonomic systems etc.

Integrating factors and conservation theorems of constrained Birkhoffian systems

In this paper the conservation theorems of the constrained Birkhoffian systems are studied by using the method of integrating factors. The differential equations of motion of the system are written. The definition of integrating factors is given for the system. The necessary conditions for the existence of the conserved quantity for the system are studied. The conservation theorem and its inverse for the system are established. Finally, an example is given to illustrate the application of the results.

A FORM INVARIANCE OF CONSTRAINED BIRKHOFFIAN SYSTEM

The form invariance of constrained Birkhoffian system is a kind of invariance of the constrained Birkhoffian equations under infinitesimal transformations. The definition and criteria of the form invariance of constrained Birkhoffian system are given, and the relation of the form invariance and the Noether symmetry is studied.

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.