We study the first integral and the solution of electromagnetic field by Lie symmetry technique and the differential invariant method.The definition and properties of differential invariants are introduced and the inf...We study the first integral and the solution of electromagnetic field by Lie symmetry technique and the differential invariant method.The definition and properties of differential invariants are introduced and the infinitesimal generators of Lie symmetries and the differential invariants of electromagnetic field are obtained.The first integral and the solution of electromagnetic field are given by the Lie symmetry technique and the differential invariants method.A typical example is presented to illustrate the application of our theoretical results.展开更多
In this paper we give a new method to investigate Noether symmetries and conservation laws of nonconservative and nonholonomic mechanical systems on time scales , which unifies the Noether's theories of the two ca...In this paper we give a new method to investigate Noether symmetries and conservation laws of nonconservative and nonholonomic mechanical systems on time scales , which unifies the Noether's theories of the two cases for the continuous and the discrete nonconservative and nonholonomic systems. Firstly, the exchanging relationships between the isochronous variation and the delta derivatives as well as the relationships between the isochronous variation and the total variation on time scales are obtained. Secondly, using the exchanging relationships, the Hamilton's principle is presented for nonconservative systems with delta derivatives and then the Lagrange equations of the systems are obtained. Thirdly, based on the quasi-invariance of Hamiltonian action of the systems under the infinitesimal transformations with respect to the time and generalized coordinates, the Noether's theorem and the conservation laws for nonconservative systems on time scales are given. Fourthly, the d'Alembert-Lagrange principle with delta derivatives is presented, and the Lagrange equations of nonholonomic systems with delta derivatives are obtained. In addition, the Noether's theorems and the conservation laws for nonholonomic systems on time scales are also obtained. Lastly, we present a new version of Noether's theorems for discrete systems. Several examples are given to illustrate the application of our results.展开更多
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 qu...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.展开更多
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 invarian...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.展开更多
基金National Natural Science Foundation of China(No.11872335)。
文摘We study the first integral and the solution of electromagnetic field by Lie symmetry technique and the differential invariant method.The definition and properties of differential invariants are introduced and the infinitesimal generators of Lie symmetries and the differential invariants of electromagnetic field are obtained.The first integral and the solution of electromagnetic field are given by the Lie symmetry technique and the differential invariants method.A typical example is presented to illustrate the application of our theoretical results.
基金supported by the National Natural Science Foundations of China (Grant Nos.11072218 and 11272287)the Natural Science Foundations of Zhejiang Province of China (Grant No.Y6110314)
文摘In this paper we give a new method to investigate Noether symmetries and conservation laws of nonconservative and nonholonomic mechanical systems on time scales , which unifies the Noether's theories of the two cases for the continuous and the discrete nonconservative and nonholonomic systems. Firstly, the exchanging relationships between the isochronous variation and the delta derivatives as well as the relationships between the isochronous variation and the total variation on time scales are obtained. Secondly, using the exchanging relationships, the Hamilton's principle is presented for nonconservative systems with delta derivatives and then the Lagrange equations of the systems are obtained. Thirdly, based on the quasi-invariance of Hamiltonian action of the systems under the infinitesimal transformations with respect to the time and generalized coordinates, the Noether's theorem and the conservation laws for nonconservative systems on time scales are given. Fourthly, the d'Alembert-Lagrange principle with delta derivatives is presented, and the Lagrange equations of nonholonomic systems with delta derivatives are obtained. In addition, the Noether's theorems and the conservation laws for nonholonomic systems on time scales are also obtained. Lastly, we present a new version of Noether's theorems for discrete systems. Several examples are given to illustrate the application of our results.
基金supported by the National Natural Science Foundation of China (Grant No. 11072218)
文摘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.
基金supported by the National Natural Science Foundation of China (Grant Nos.10672143 and 11072218)
文摘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.
