Symmetry Of Lagrangians And Conserved Quantity For Constrained Lagrangians Systems

The symmetry principle is one of the top-level principles in physics, it has importanttheoretical and pratical significance not only in physics but also in mathmatics to discusssymmetry of mechanical dynamical equations, and the scientists keeped enthusisam to thesymmetry theory of mechanics which was estabished by either I. Newton or J. Lagrange.While symmetry theory is one of major branches in mordern analytical mechanics, and itreveals much information about the dynamical system. There are mainly three types ofsymmetry, i.e., Noether Symmetry, Lie Symmetry and Mei Symmetry, to find conservedquantities in analytical mechanics, and the theories of these three symmetres have becomealmost perfect. We should find some new symmetres to study and get many more quantities.The symmetry of Lagrangians is one of symmetry which is different from the above threesymmetries, in this article we will mainly discuss the symmetry of Lagrangians. First of all,we give the criterions of the Symmetry of Lagrangians for a non-holonomic variable masssystem, a holonomic non-conservative system in event space, a nonholonomic relativisticsystem and the system of unified dynamic equations in configuration spcace, in which thegroup thansformations and matrix theory play fatal roles. Secondly, the conditions underwhich there exist conserved quantities and the forms of the conserved quantities are obtained.And we show three examples to illustrate the application of the symmetry of Lagrangians.Finally, we generalize the thory of the symmetry of Lagrangians and give some advises forthe next developing trend of the symmetry.