Symmetry And Conserved Quantity Of Second-Order Lagrange System On Time Scales

This paper focuses on the second-order Lagrange system on time scales and researches on its symmetry and conserved quantities.Give the Noether theorem and Mei symmetry of the system;For the second-order Lagrange non-conservative system,the Lie symmetry and corresponding conserved quantities of the system are based on the knowledge of time scale theory and the principle of invariance of differential equations.1.The Noether symmetry of the second-order Lagrange system on time scales is studied.Based on the motion equation of the second-order Lagrange system on time scales,the Noether equation of the system is given,and then definition and criterion of generalized Noether symmetric transformation and generalized Noether quasi-symmetric transformation for the system are obtained.Finally the Noether theorem corresponding to the system is given.2.We studied the Mei symmetry of the second-order Lagrange system on time scales.Based on the second-order Lagrange system motion differential equation under the infinitesimal transformation of the group,the Mei symmetry of the second-order Lagrange system on time scales is defined,and Mei symmetry decision equation is derived.Then the condition of conserved quantity caused by the system on time scales is established;Finally,the symmetry problem of the second-order Lagrange system in continuous and discrete time are given.3.The Lie symmetry and conserved quantity of the second-order Lagrange non-conservative system on time scales are studied,and the motion equation of the second-order Lagrange non-conservative system on time scales is established.Based on the principle of invariance of differential equations under infinitely small transformations,the Lie symmetry deterministic equation of the second-order Lagrange non-conservative system is established on time scales,and the structural equation of the system and the conserved quantity are given.