This paper proposes a new concept of the conformal invariance and the conserved quantities for Birkhoff systems under second-class Mei symmetry.The definition about conformal invariance of Birkhoff systems under second-class Mei symmetry is given.The conformal factor in the determining equations is found.The relationship between Birkhoff system's conformal invariance and second-class Mei symmetry are discussed.The necessary and sufficient conditions of conformal invariance,which are simultaneously of second-class symmetry,are given.And Birkhoff system's conformal invariance may lead to corresponding Mei conserved quantities,which is deduced directly from the second-class Mei symmetry when the conformal invariance satisfies some conditions.Lastly,an example is provided to illustrate the application of the result.
The Noether and Lie symmetries as well as the conserved quantities of Hamiltonian system with fractional derivatives are es-tablished. The definitions and criteria for the fractional symmetrical transformations and quasi-symmetrical transformations inthe Noether sense of Hamiltonian system are first discussed. Then, using the invariance of Hamiltonian action under the infini-tesimal transformations with respect to time, generalized coordinates and generalized momentums, the fractional Noethertheorem of the system is obtained. Further, the Lie symmetry and conserved quantity of the system are acquired. Two exam-ples are presented to illustrate the application of the results.
This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of the results.
The theory of velocity-dependent symmetries(or Lie symmetry) and non-Noether conserved quantities are presented corresponding to both the continuous and discrete electromechanical systems.Firstly,based on the invariance of Lagrange-Maxwell equations under infinitesimal transformations with respect to generalized coordinates and generalized charge quantities,the definition and the determining equations of velocity-dependent symmetry are obtained for continuous electromechanical systems;the Lie's theorem and the non-Noether conserved quantity of this symmetry are produced associated with continuous electromechanical systems.Secondly,the operators of transformation and the operators of differentiation are introduced in the space of discrete variables;a series of commuting relations of discrete vector operators are defined.Thirdly,based on the invariance of discrete Lagrange-Maxwell equations under infinitesimal transformations with respect to generalized coordinates and generalized charge quantities,the definition and the determining equations of velocity-dependent symmetry are obtained associated with discrete electromechanical systems;the Lie's theorem and the non-Noether conserved quantity are proved for the discrete electromechanical systems.This paper has shown that the discrete analogue of conserved quantity can be directly demonstrated by the commuting relation of discrete vector operators.Finally,an example is discussed to illustrate the results.
FU JingLiCHEN BenYongFU HaoZHAO GangLingLIU RongWanZHU ZhiYan
We obtain a new type of conserved quantity of Mei symmetry for the motion of mechanico-electrical coupling dynamical systems under the infinitesimal transformations. A criterion of Mei symmetry for the mechanico-electrical coupling dynamical systems is given. Simultaneously, the condition of existence of the new conserved quantity of Mei symmetry for mechanico-electrical coupling dynamical systems is obtained. Finally, an example is given to illustrate the application of the results.
