Conformal Invariance of Higher-Order Lagrange Systems by Lie Point Transformation

摘要

Conformal invariance and conserved quantities for a higher-order Lagrange system by Lie point transformation of groups are studied. The differential equation of motion for the higher-order Lagrange system is introduced. The definition of conformal invariance for the system together with its determining equations and conformal factor are provided. The necessary and sufficient condition that the system's conformal invariance would be Lie symmetry by the infinitesimal one-parameter point transformation group is deduced. The conserved quantity of the system is derived using the structural equation satisfied by the gauge function. An example of a higher-order mechanical system is offered to illustrate the application of the result.%Conformal invariance and conserved quantities for a higher-order Lagrange system by Lie point transformation of groups are studied.The differential equation of motion for the higher-order Lagrange system is introduced.The definition of conformal invariance for the system together with its determining equations and conformal factor are provided.The necessary and sufficient condition that the system's conformal invariance would be Lie symmetry by the infinitesimal one-parameter point transformation group is deduced.The conserved quantity of the system is derived using the structural equation satisfied by the gauge function.An example of a higher-order mechanical system is offered to illustrate the application of the result.Since the Noether theorem was published in 1918,[1] the symmetry and conserved quantity for a dynamical system play important roles in the fields of modern science and technology,and some important results have been gained so far.[2-21] Conformal invariance is a modern method for finding conserved quantities.In 1997,Galiullin etal.[22] studied conformal invariance of Birkhoff systems under special infinitesimal transformations.In recent years,we have discussed the conformal invariance of Lie symmetry for Lagrange systems,general holonomic mechanical systems and nonholonomic systems.[23-26] The authors of Refs.[27-33] have studied the conformal invariance of Lie symmetry for first-order differential equations,generalized Birkhoff equations,generalized Hamilton systems,mechanico-electrical systems,and so on.