Symmetry And Conserved Of Variable Mass Nonholonomic Systems In Phase Space

The researches on the symmetry theory and conservationion laws of mechanical systems have import theoretical andsignificance.The modern methods to find conserved quantitiesare the followings:Noether symmetry, Lie symmetry and Meisymmetry.In the paper,we study the symmetry and conserved ofvariable mass nonholonomic systems in phase space. Thefirst-order and second-order differential equations of the systemare established. We also study Noether symmetry, Lie symmetryand Mei symmetry of variable mass nonholonmic mechanicalsystems and the relation between Noether symmetry and Liesymmetry in phase space. Noether equation and Killing equationof Noether symmetry and the criterion equation of Lie symmetryand Mei symmetry are given. The conditions and specific formof Noether conserved quantity and Mei conserved quantitywhich Noether symmetry and Mei symmetry lead to are given.The paper discusses Noether-Mei, Lie-Mei symmetry and twotypes of generalized conserved quantities of variable massnonholonomic systems in phase space, based on three kinds of single-symmetry theory. The criterion equation of Noether-Meisymmetry and Lie-Mei symmetry of variable mass nonholonomicsystems in phase space are given. Two types of generalizedconserved quantity are obtained. We study the definition andcriterion of Mei symmetry of second-order variable massnonholonomic systems in phase space, and get the form ofstructure equations and conserved quantities of Mei symmetry.Meanwhile, the paper also gives examples.