含时滞的非保守系统动力学的Noether对称性

提出并研究含时滞的非保守系统动力学的Noether对称性与守恒量. 首先,建立含时滞的非保守系统的Hamilton原理,得到含时滞的Lagrange方程;其次,基于含时滞的Hamilton作用量在依赖于广义速度的无限小群变换下的不变性,定义系统的Noether对称变换和准对称变换,建立Noether对称性的判据;最后,研究对称性与守恒量之间的关系,建立含时滞的非保守系统的Noether理论. 文末举例说明结果的应用. 关键词： 时滞系统 非保守力学 Noether对称性 守恒量     

Noether symmetry and conserved quantity for a Hamilton system with time delay

In this paper, the Noether symmetries and the conserved quantities for a Hamilton system with time delay are discussed. Firstly, the variational principles with time delay for the Hamilton system are given, and the Hamilton canonical equations with time delay are established. Secondly, according to the invariance of the function under the infinitesimal transformations of the group, the basic formulas for the variational of the Hamilton action with time delay are discussed,the definitions and the criteria of the Noether symmetric transformations and quasi-symmetric transformations with time delay are obtained, and the relationship between the Noether symmetry and the conserved quantity with time delay is studied. In addition, examples are given to illustrate the application of the results.     

离散差分序列变质量Hamilton系统的Lie对称性与Noether守恒量

本文研究离散差分序列变质量Hamilton系统的Lie对称性与Noether守恒量. 构建了离散差分序列变质量Hamilton系统的差分动力学方程, 给出了离散差分序列变质量Hamilton系统差分动力学方程在无限小变 换群下的Lie对称性的确定方程和定义, 得到了离散力学系统Lie对称性导致Noether守恒量的条件及形式, 举例说明结果的应用. 关键词： 离散力学 Hamilton系统 Lie对称性 Noether守恒量     

Noether Symmetry and Noether Conserved Quantity of Nielsen Equation for Dynamical Systems of Relative Motion

Noether symmetry of Nielsen equation and Noether conserved quantitydeduced directly from Noether symmetry for dynamical systems of the relative motion are studied. The definition and criteria of Noether symmetry of a Nielsen equation under the infinitesimal transformations of groups are given. Expression of Noether conserved quantity deduced directly from Noether symmetry of Nielsen equation for the system are obtained. Finally, an example is given to illustrate the application of the results.     

事件空间中Birkhoff系统的Noether理论

研究事件空间中Birkhoff系统的Noether对称性与Noether守恒量.首先,建立了事件空间中Birkhoff系统的参数方程;其次,基于Pfaff作用量在无限小变换下的不变性,给出了事件空间中Birkhoff系统的Noether定理及其逆定理;最后,举例说明结果的应用. 关键词： 事件空间 Birkhoff系统 Pfaff作用量 Noether对称性     

Hamilton formalism and Noether symmetry for mechanico—electrical systems with fractional derivatives

This paper presents extensions to the traditional calculus of variations for mechanico-electrical systems containing fractional derivatives.The Euler-Lagrange equations and the Hamilton formalism of the mechanico-electrical systems with fractional derivatives are established.The definition and the criteria for the fractional generalized Noether quasisymmetry are presented.Furthermore,the fractional Noether theorem and conseved quantities of the systems are obtained by virtue of the invariance of the Hamiltonian action under the infinitesimal transformations.An example is presented to illustrate the application of the results.     

Noether's theorems of a fractional Birkhoffian system within Riemann–Liouville derivatives

The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann–Liouville fractional derivatives. Firstly, the fractional Birkhoff's equations and the corresponding transversality conditions are given. Secondly, from special to general forms, Noether's theorems of a standard Birhoffian system are given, which provide an approach and theoretical basis for the further research on the Noether symmetry of the fractional Birkhoffian system. Thirdly, the invariances of the fractional Pfaffian action under a special one-parameter group of infinitesimal transformations without transforming the time and a general one-parameter group of infinitesimal transformations with transforming the time are studied, respectively, and the corresponding Noether's theorems are established. Finally, an example is given to illustrate the application of the results.     

Noether Symmetry Can Lead to Non-Noether Conserved Quantity of Holonomic Nonconservative Systems in General Lie Transformations

For the holonomic nonconservative system, by using the Noether symmetry, a non-Noether conserved quantity is obtained directly under general infinitesimal transformations of groups in which time is variable. At first, the Noether symmetry, Lie symmetry, and Noether conserved quantity are given. Secondly, the condition under which the Noether symmetry is a Lie symmetry under general infinitesimal transformations is obtained. Finally, a set of non-Noether conserved quantities of the system are given by the Noether symmetry, and an example is given to illustrate the application of the results.     

Fractional Noether Theorem Based on Extended Exponentially Fractional Integral

Based on the new type of fractional integral definition, namely extended exponentially fractional integral introduced by EI-Nabulsi, we study the fractional Noether symmetries and conserved quantities for both holonomic system and nonholonomic system. First, the fractional variational problem under the sense of extended exponentially fractional integral is established, the fractional d’Alembert-Lagrange principle is deduced, then the fractional Euler-Lagrange equations of holonomic system and the fractional Routh equations of nonholonomic system are given; secondly, the invariance of fractional Hamilton action under infinitesimal transformations of group is also discussed, the corresponding definitions and criteria of fractional Noether symmetric transformations and quasi-symmetric transformations are established; finally, the fractional Noether theorems for both holonomic system and nonholonomic system are explored. What’s more, the relationship between the fractional Noether symmetry and conserved quantity are revealed.     

离散差分变分Hamilton系统的Lie对称性与Noether守恒量

研究离散差分Hamilton系统的Lie对称性与Noether守恒量. 根据扩展的时间离散力学变分原理构建Hamilton系统的差分动力学方程.定义离散系统运动差分方程在无限小变换群下的不变性为Lie对称性, 导出由Lie对称性得到系统离散Noether守恒量的判据. 举例说明结果的应用. 关键词： 离散力学 差分Hamilton系统 Lie对称性 Noether守恒量     

Noether symmetries of discrete mechanico-electrical systems

This paper focuses on studying Noether symmetries and conservation laws of the discrete mechanico-electricM systems with the nonconservative and the dissipative forces. Based on the invariance of discrete Hamilton action of the systems under the infinitesimal transformation with respect to the generalized coordinates, the generalized electrical quantities and time, it presents the discrete analogue of variational principle, the discrete analogue of Lagrange-Maxwell equations, the discrete analogue of Noether theorems for Lagrange Maxwell and Lagrange mechanico-electrical systems. Also, the discrete Noether operator identity and the discrete Noether-type conservation laws are obtained for these systems. An actual example is given to illustrate these results.     

The approximate conserved quantity of the weakly nonholonomic mechanical-electrical system

We study the approximate conserved quantity of the weakly nonholonomic mechanical-electrical system.By means of the Lagrange-Maxwell equation,the Noether equality of the weakly nonholonomic mechanical-electrical system is obtained.The multiple powers-series expansion of the parameter of the generators of infinitesimal transformations and the gauge function is put into a generalized Noether identity.Using the Noether theorem,we obtain an approximate conserved quantity.An example is provided to prove the existence of the approximate conserved quantity.     

一般完整系统Mei对称性的逆问题

动力学逆问题是星际航行学、火箭动力学、规划运动学理论的基本问题. Mei对称性是力学系统的动力学函数在群的无限小变换下仍然满足系统原来的运动微分方程的一种新的不变性. 本文研究广义坐标下一般完整系统的Mei对称性以及与Mei对称性相关的动力学逆问题. 首先, 给出系统动力学正问题的提法和解法. 引入时间和广义坐标的无限小单参数变换群, 得到无限小生成元向量及其一次扩展. 讨论由n个广义坐标确定的一般完整力学系统的运动微分方程, 将其Lagrange函数和非势广义力作无限小变换, 给出系统运动微分方程的Mei对称性定义, 在忽略无限小变换的高阶小量的情况下得到Mei对称性的确定方程, 借助规范函数满足的结构方程导出系统Mei对称性导致的Noether守恒量. 其次, 研究系统Mei对称性的逆问题. Mei对称性的逆问题的提法是: 由已知守恒量来求相应的Mei对称性. 采取的方法是将已知积分当作由Mei对称性导致的Noether守恒量, 由Noether逆定理得到无限小变换的生成元, 再由确定方程来判断所得生成元是否为Mei对称性的. 然后, 讨论生成元变化对各种对称性的影响. 结果表明, 生成元变化对Noether和Lie对称性没有影响, 对Mei 对称性有影响, 但在调整规范函数时, 若满足一定条件, 生成元变化对Mei对称性也可以没有影响. 最后, 举例说明结果的应用.     

Noether symmetry and non-Noether conserved quantity of the relativistic holonomic nonconservative systems in general Lie transformations

For a relativistic holonomic nonconservative system, by using the Noether symmetry, a new non-Noether conserved quantity is given under general infinitesimal transformations of groups. On the basis of the theory of invariance of differential equations of motion under general infinitesimal transformations, we construct the relativistic Noether symmetry, Lie symmetry and the condition under which the Noether symmetry is a Lie symmetry under general infinitesimal transformations. By using the Noether symmetry, a new relativistic non-Noether conserved quantity is given which only depends on the variables $t$, $q_s $ and $\dot {q}_s $. An example is given to illustrate the application of the results.     

Noether-type theory for discrete mechanico-electrical dynamical systems with nonregular lattices

We investigate Noether symmetries and conservation laws of the discrete mechanico-electrical systems with nonregular lattices.The operators of discrete transformation and discrete differentiation to the right and left are introduced for the systems.Based on the invariance of discrete Hamilton action on nonregular lattices of the systems with the dissipation forces under the infinitesimal transformations with respect to the time,generalized coordinates and generalized charge quantities,we work out the discrete analog of the generalized variational formula.From this formula we derive the discrete analog of generalized Noether-type identity,and then we present the generalized quasi-extremal equations and properties of these equations for the systems.We also obtain the discrete analog of Noether-type conserved laws and the discrete analog of generalized Noether theorems for the systems.Finally we use an example to illustrate these results.     

非完整系统的Noether对称性与Hojman守恒量

研究非完整力学系统的Noether对称性导致的非Noether守恒量——Hojman守恒量. 在时间不变的特殊无限小变换下，给出系统的特殊Noether对称性与守恒量，并给出特殊Noether对称性导致特殊Lie对称性的条件. 由系统的特殊Noether对称性，得到相应完整系统的Hojman守恒量以及非完整系统的弱Hojman守恒量和强Hojman守恒量. 给出一个例子说明本结果的应用 关键词： 分析力学 非完整系统 Noether对称性 非Noether守恒量 Hojman守恒量     

The Hamilton Cartan formalism for rth-order Lagrangians and the integrability of the KdV and modified KdV equations

The Hamilton Cartan formalism for rth order Lagrangians is presented in a form suited to dealing with higher-order conserved currents. Noether's Theorem and its converse are stated and Poisson brackets are defined for conserved charges. An isomorphism between the Lie algebra of conserved currents and a Lie algebra of infinitesimal symmetries of the Cartan form is established. This isomorphism, together with the commutativity of the Bäcklund transformations for the KdV and modified KdV equations, allows a simple geometric proof that the infinite collections of conserved charges for these equations are in involution with respect to the Poisson bracket determined by their Lagrangians. Thus, the formal complete integrability of these equations appears as a consequence of the properties of their Bäcklund transformations.It is noted that the Hamilton Cartan formalism determines a symplectic structure on the space of functionals determined by conserved charges and that, in the case of the KdV equation, the structure is the same as that given by Miura et al. [5].     

Generalized Conformal Symmetries and Its Application of Hamilton Systems

In this paper the generalized conformal symmetries and conserved quantities by Lie point transformations of Hamilton systems are studied. The necessary and sufficient conditions of conformal symmetry by the action of infinitesimal Lie point transformations which are simultaneous Lie symmetry are given. This kind type determining equations of conformal symmetry of mechanical systems are studied. The Hojman conserved quantities of the Hamilton systems under infinitesimal special transformations are obtained. The relations between conformal symmetries and the Lie symmetries are derived for Hamilton systems. Finally, as application of the conformal symmetries, an illustration example is introduced.     

Conformal invariance and conserved quantities of non-conservative Lagrange systems by point transformations

This paper studies the conformal invariance by infinitesimal point transformations of non-conservative Lagrange systems. It gives the necessary and sufficient conditions of conformal invariance by the action of infinitesimal point transformations being Lie symmetric simultaneously. Then the Noether conserved quantities of conformal invariance are obtained. Finally an illustrative example is given to verify the results.     

准坐标下完整力学系统的非Noether守恒量——Hojman定理的推广

利用时间不变的无限小变换下的Lie对称性，研究准坐标下完整力学系统的一类新守恒量.建立系统的运动微分方程，给出无限小变换下的Lie对称性确定方程.将Hojman定理推广，并举例说明结果的应用. 关键词： 准坐标 完整力学系统 Lie对称性 非Noether守恒量 Hojman定理