This paper is devoted to studying the conformal invariance and Noether symmetry and Lie symmetry of a holonomic mechanical system in event space. The definition of the conformal invariance and the corresponding conformal factors of the holonomic system in event space are given. By investigating the relation between the conformal invariance and the Noether symmetry and the Lie symmetry, expressions of conformal factors of the system under these circumstances are obtained, and the Noether conserved quantity and the Hojman conserved quantity directly derived from the conformal invariance are given. Two examples are given to illustrate the application of the results.
Three kinds of symmetries and their corresponding conserved quantities of a generalized Birkhoffian system are studied. First, by using the invariance of the Pfaffian action under the infinitesimal transformations, the Noether theory of the generalized Birkhoffian system is established. Secondly, on the basis of the invariance of differential equations under infinitesimal transformations, the definition and the criterion of the Lie symmetry of the generalized Birkhoffian system are established, and the Hojman conserved quantity directly derived from the Lie symmetry of the system is given. Finally, by using the invariance that the dynamical functions in the differential equations of the motion of mechanical systems still satisfy the equations after undergoing the infinitesimal transformations, the definition and the criterion of the Mei symmetry of the generalized Birkhoffian system are presented, and the Mei conserved quantity directly derived from the Mei symmetry of the system is obtained. Some examples are given to illustrate the application of the results.
The problem on the stability of motion for a generalized Birkhoffian system was studied. The disturbed equations of motion and their first approximation for the system were established. The criterion of stability of motion for the system was set up by using Liapunov's first approximation theory. Based on the theory of Noether symmetry,the Liapunov's function was constructed,and the criterion of stability of motion for the system was also set up by using Liapunov's direct method. Two examples were given to illustrate the application of the results.
For a Birkhoffian system in the event space, this paper presents the Routh method of reduction. The parametric equations of the Birkhoffian system in the event space are established, and the definition of cyclic coordinates for the system is given and the corresponding cyclic integral is obtained. Through the cyclic integral, the order of the system can be reduced. The Routh functions for the Birkhoffian system in the event space are constructed, and the Routh method of reduction is successfully generalized to the Birkhoffian system in the event space. The results show that if the system has a cyclic integral, then the parametric equations of the system can be reduced at least by two degrees and the form of the equations holds. An example is given to illustrate the application of the results.
