Lie Symmetries and Conserved Quantities for Super-Long Elastic Slender Rod

DNA is a nucleic acid molecule with double-helical structures that are special symmetrical structures attracting great attention of numerous researchers. The super-long elastic slender rod, an important structural model of DNA and other long-train molecules, is a useful tool in analysing the symmetrical properties and the stabilities of DNA. We study the Lie symmetries of a super-long elastic slender rod by using the methods of infinitesimal transformation. Based on Kirchhoff＇s analogue, generalized Hamilton canonical equations are analysed. The infinitesimal transformations with respect to the radian coordinate, the generalized coordinate, and the quasimomentum of the model are introduced. The Lie symmetries and conserved quantities of the model are presented.     

Perturbation of symmetries for super-long elastic slender rods

This paper analyses perturbations of Noether symmetry, Lie symmetry, and form invariance for super-long elastic slender rod systems. Criterion and structure equations of the symmetries after disturbance are proposed. Considering perturbation of all infinitesimal generators, three types of adiabatic invariants induced by perturbation of symmetries for the system are obtained.     

Perturbation of symmetries for super-long elastic slender rods

This paper analyses perturbations of Noether symmetry, Lie symmetry, and form invariance for super-long elastic slender rod systems. Criterion and structure equations of the symmetries after disturbance are proposed. Considering perturbation of all infinitesimal generators, three types of adiabatic invariants induced by perturbation of symmetries for the system are obtained.     

超细长弹性杆的Mei对称性及其Noether守恒量

基于Kirchhoff的动力学比拟,用动力学的概念和方法研究圆截面弹性杆的Hamilton函数和方程,并给出弹性杆的Mei对称性定义和定理以及定理的证明,最后给出弹性杆动力学系统的Mei对称性导致Noether守恒量的条件及定理,并给出算例. 关键词： 超细长弹性杆 Mei对称性 Noether守恒量     

Noether symmetries of discrete nonholonomic dynamical systems

In this letter, we investigate Noether symmetries and conservation laws of discrete dynamical systems on an uniform lattice with the nonholonomic constraints. Based on the quasi-invariance of discrete Hamiltonian action of the systems under the infinitesimal transformation with respect to the time and generalized coordinates, we give the discrete analogue of generalized variational formula of the systems. From this formula we derive the discrete analogue of generalized Noether-type identity, and then we present the generalized quasi-extremal equations of the systems. We also obtain the discrete analogue of Noether theorems and the discrete analogue of Noether conservation laws of the systems. Finally, an example is discussed to illustrate these results.     

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.     

黏性流体中超细长弹性杆的动力学不稳定性

基于坐标基矢摄动的方法研究了黏性流体中超细长弹性杆动力学稳定性判据与失稳后的模态选择,推导出了黏性介质中超细长弹性杆Kirchoff动力学方程的一阶摄动表示,即线性的二阶偏微分方程组.以平面扭转DNA环为例,说明了以上结果的应用,得到了平面扭转DNA环的稳定性判据及其稳定的临界区域,讨论了其失稳后的模态选择及黏性阻力对其的影响.     

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     

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 conserved quantities and Lie point symmetries for difference nonholonomic Hamiltonian systems in irregular lattices

The Noether conserved quantities and the Lie point symmetries for difference nonholonomic Hamiltonian systems in irregular lattices are studied. The generalized Hamiltonian equations of the systems are given on the basis of the transformation operators in the space of discrete Hamiltonians. The Lie transformations acting on the lattice, as well as the equations and the determining equations of the Lie symmetries are obtained for the nonholonomic Hamiltonian systems. The discrete analogue of the Noether conserved quantity is constructed by using the Lie point symmetries. An example is discussed to illustrate the results.     

Conserved Quantities and Conformal Mechanico-Electrical Systems

The conformal meehanico-electrical systems are presented by infinitesimal point transformations of time and generalized coordinates. The necessary and suflleient conditions that the eonformal meehanieo-eleetrieal systems possess Lie symmetry are given. The Noether conserved quantities of the eonformal meehanieo-eleetrieal systems are obtained from Lie symmetries.     

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.     

Noether symmetry and conserved quantities of the analytical dynamics of a Cosserat thin elastic rod

In this paper, we investigate the Noether symmetry and Noether conservation law of elastic rod dynamics with two independent variables： time t and arc coordinate s. Starting from the Lagrange equations of Cosserat rod dynamics, the criterion of Noether symmetry with Lagrange style for rod dynamics is given and the Noether conserved quantity is obtained. Not only are the conservations of generalized moment and generalized energy obtained, but also some other integrals.     

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Disturbed Nonholonomic Systems

For a nonholonomic mechanics system with the action of small disturbance, the Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type are studied under general infinitesimal transformations of groups in which the generalized coordinates and time are variable. On the basis of the invariance of disturbed nonholonomic dynamical equations under general infinitesimal transformations, the determining equations, the constrained restriction equations and the additional restriction equations of Lie symmetries of the system are constructed, which only depend on the variables t, qs and q^.s. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for a nonholonomic system with the action of small disturbance is investigated, and the Lie symmetrical adiabatic invariants, the weakly Lie symmetrical adiabatic invariants and the strongly Lie symmetrical adiabatic invariants of generalized Hojman type of disturbed nonholonomic systems are obtained. An example is given to illustrate applications of the results.     

The Lie symmetries and Noether conserved quantities of discrete mechanical systems with variable mass

This paper studies the Lie symmetries and Noether conserved quantities of discrete mechanical systems with variable mass. The discrete Euler-Lagrange equation and energy evolution equation are derived by using a total variational principle. The invariance of discrete equations under infinitesimal transformation groups is defined to be Lie symmetry. The condition of obtaining the Noether conserved quantities from the Lie symmetries is also presented. An example is discussed for applications of the results.     

Affine symmetry,geodesics, and homogeneous spacetimes

We show that the conservation laws for the geodesic equation which are associated to affine symmetries can be obtained from symmetries of the Lagrangian for affinely parametrized geodesics according to Noether’s theorem, in contrast to claims found in the literature. In particular, using Aminova’s classification of affine motions of Lorentzian manifolds, we show in detail how affine motions define generalized symmetries of the geodesic Lagrangian. We compute all infinitesimal proper affine symmetries and the corresponding geodesic conservation laws for all homogeneous solutions to the Einstein field equations in four spacetime dimensions with each of the following energy–momentum contents: vacuum, cosmological constant, perfect fluid, pure radiation, and homogeneous electromagnetic fields.     

Noether-type theorem for discrete nonconservative dynamical systems with nonregular lattices

We investigate Noether symmetries and conservation laws of the discrete nonconserved 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 nonconserved forces under the infinitesimal transformations with respect to the time and generalized coordinates, we give the discrete analog of generalized variational f...     

El-Nabulsi动力学模型下Birkhoff系统Noether对称性的摄动与绝热不变量

基于El-Nabulsi动力学模型,研究了小扰动作用下Birkhoff系统Noether对称性的摄动与绝热不变量问题.首先,将El-Nabulsi提出的在分数阶微积分框架下基于Riemann-Liouville分数阶积分的非保守系统动力学模型拓展到Birkhoff系统,建立El-Nabulsi-Birkhoff方程;其次,基于在无限小变换下El-Nabulsi-Pfaff作用量的不变性,给出Noether准对称性的定义和判据,得到了Noether对称性导致的精确不变量;再次,引入力学系统的绝热不变量概念,研究El-Nabulsi动力学模型下受小扰动作用的Birkhoff系统Noether对称性的摄动与绝热不变量之间的关系,得到了对称性摄动导致的绝热不变量的条件及其形式.作为特例,给出了El-Nabulsi动力学模型下相空间中非保守系统和经典Birkhoff系统的Noether对称性的摄动与绝热不变量.以著名的Hojman-Urrutia问题为例,研究其在El-Nabulsi动力学模型下的Noether对称性,得到了相应的精确不变量和绝热不变量.     

A Set of Lie Symmetrical Conservation Law for Rotational Relativistic Hamiltonian Systems

For the rotational relativistic Hamiltonian system, a new type of the Lie symmetries and conserved quantitiesare given. On the basis of the theory of invariance of differential equations under infinitesimal transformations, andintroducing infinitesimal transformations for generalized coordinates qs and generalized momentums ps, the determiningequations of Lie symmetrical transformations of the system are constructed, which only depend on the canonical variables.A set of non-Noether conserved quantities are directly obtained from the Lie symmetries of the system. An example isgiven to illustrate the application of the results.     

A Set of Lie Symmetrical Conservation Law for Rotational Relativistic Hamiltonian Systems

For the rotational relativistic Hamiltonian system, a new type of the Lie symmetries and conserved quantities are given. On the basis of the theory of invariance of differential equations under infinitesimal transformations, and introducing infinitesimal transformations for generalized coordinates qs and generalized momentums ps, the determining equations of Lie symmetrical transformations of the system are constructed, which only depend on the canonical variables.A set of non-Noether conserved quantities are directly obtained from the Lie symmetries of the system. An example is given to illustrate the application of the results.