Symmetry And Conserved Quantity Research For Nonconservative Dynamics With Nonstandard Lagrangians

In1978,Non-standard Lagrangians were introuduced by Arnold in?Mathematical Methods of Classical Mechanics?.However,the topic was ignored until it back into people's vision in the string theory.Non-standard Lagrangians has more advantages than standard Lagrangians in describing nonlinear dynamical systems,the study of dynamical systems with non-standard Lagrangians can find some characteristics that standard Lagrangians do not have.In this paper,Routh method of reduction,Whittaker method of reduction,Lie symmetries and Mei symmetries as well as Noether symmetries and Lie symmetries based on El-Nabulsi models of dynamical systems with non-standard Lagrangians contained exponential Lagrangians and power law Lagrangians are studied.In the first part,firstly,based on non-standard Lagrangians,the Hamilton action of dynamical systems are defined,corresponding the Hamilton principle are established,the d'Alembert-Lagrange principle and Lagrange equations of the systems based on non-standard Lagrangians are obtained;secondly,the form and the condition under which the generalized energy integral and the cyclic integral exists are established by using Lagrange equations of the systems;Finally,the famous Routh method of reduction and the Whittaker method of reduction are extended,the Routh equations and the Whittaker equations for the dynamic systems with non-standard Lagrangians are obtained.In the second part,Introducing infinitesimal transformation and its generating vectors,according to Lagrange equations of the systems,the definitions of Lie symmetries,the definitions and the criterions of Mei symmetries for dynamical systems with non-standard Lagrangians are given;the structure equations and corresponding conserved quantities are established by invariance of generalized acceleration and dynamical functions.In the third part,firstly,according to El-Nabulsi modeling method,variational problem with non-standard Lagrangians are established,the Lagrange equations of the systems are obtained;secondly,the definitions and the criterions of Noether(quasi-)symmetrical transformations and the definitions of Lie symmetry of dynamical systems are obtained in terms of invariance of Hamilton action and generalized acceleration;the Noether theorems and the structure equations and conserved quantities correspond to Lie symmetries of systems are established.The article mainly studies integral methods and symmetries for dynamical systemswith non-standard Lagrangians.For integral methods,Routh method and Whittaker method based on exponential Lagrangians and power law Lagrangians are given.For symmetries,First of all,the relationship between Lie symmetries,Mei symmetries and conserved quantities are studied.After,the relationship between Noether symmetries,Lie symmetries and conserved quantities of dynamical systems with non-standard Lagrangians in El-Nabulsi models below are further discussed.the study of integral methods and symmetries for dynamical systems with non-standard Lagrangians can provide new ideas for the nonlinear problem-solving with analytical mechanics methods.At the same time,it also reveals that the relationship between the conserved quantities with non-standard Lagrangians and its symmetries of the intrinsic.