Study Of Some Problems About The Symmetries And Conserved Quantities Of Several Constrained Mechanical Systems

The researches on the symmetry theory and the conservation laws of mechanical systems have important theoretical and practical significances. If some dynamical systems possess a kind of symmetry, it means that there are some properties corresponding to this kind of symmetry, and because there are some close relationships between symmetries and invariants of dynamical systems, the symmetrical theory is a useful tool for integral equations of motion too. In this paper, we will study some problems about symmetry and conserved quantity for four kinds of constrained mechanical systems, including the conserved quantities directly deduced from the Lie symmetry and Mei symmetry for the general holonomic system,the mechanico-electrical system,the Vacco dynamical system and the unilateral constrained mechanical system, and the conserved quantities indirectly deduced from the Lie symmetry and Mei symmetry for the above systems. First, the definitions of the Lie symmetry and Mei symmetry for the above systems are established, and then we obtain the criterions of Lie symmetry and Mei symmetry; Then, we study the relationship between the Lie symmetry and the Mei symmetry of the above systems, and give the sufficient condition one symmetry is another symmetry; Last, the generalized Hojman conserved quantity,the generalized Lutzky conserved quantity,the generalized Mei conserved quantity and the Mei conserved quantity directly deduced from the Lie symmetry and Mei symmetry are got, and the generalized Lutzky conserved quantity,the generalized Mei conserved quantity and the Mei conserved quantity indirectly deduced from the Lie symmetry and Mei symmetry are also obtained.