Symmetries And Conserved Quntities Of Controllable Mechanical Systems

It is significant to study the symmetries and conserved quantities of dynamic systems not only in mathematics but also in physics. The research on the controllable mechanical system becomes more important and has the important theoretical and practical significance. In the paper, we study the symmetries and conserved quantities of controllable mechanical systems. We give equations of motion of the systems, then Noether symmetry, Lie symmetry and form invariance (also is called Mei symmetry) to the holonomic controllable mechanical system, the nonholonomic controllable mechanical system and the controllable mechanical system in phase space are studied. Respectly, the Noether equation and the Killing equation of Noether symmetry, the determinining equation of Lie symmetry and form invariance are given. The conditions and the forms of the Noether conserved quantity, the Hojman conserved quantities and a new conserved quantity (also is called Mei conserved quantity) of the three symmetries leads to directly are obtained. Based on the three symmetries above, the unified symmetry of the holonomic controllable mechanical system, the nonholonomic controllable mechanical system, the controllable mechanical system with variable mass, the controllable mechanical system in phase space are studied. The determining equations of these four systrms are given, the conditions and the forms of three conserved quantities are obtained directly from unified symmetry. On the bases of the symmetries and conserved quantities of controllable mechanical systems, we prospect the foreground of controllable mechanical systems.