Noether-Symmetry Analysis using Alternative Lagrangian Representations

We have sought to work with an approach to Noether symmetry analysis which uses the properties of infinitesimal point transformations in the space-time (q, t) variable to establish the association between symmetries and conservation laws of a dynamical system. In this approach symmetries are expressed in the form of generators. We have studied the variational or Noether symmetries of two uncoupled Harmonic oscillators and two such oscillators coupled by an interaction. Both these systems can have alternative Lagrangian representations. We have studied in detail how the association between symmetries and conservation laws changes as one alters the analytic or Lagrangian representation. This analysis is carried out with a view to explicitly demonstrate that the correlation between symmetry transformation and corresponding invariant quantity depends crucially on the choice of the analytic representation. PACS 45.20.Jj, 45.20.df, 45.20.dh     

General transformation theory of Lagrangian mechanics and the Lagrange group

The general transformation theory of Lagrangian mechanics is revisited from a group-theoretic point of view. After considering the transformation of the Lagrangian function under local coordinate transformations in configuration spacetime, the general covariance of the formalism of Lagrange is discussed. Next, the group of Lagrange (for alln-dimensional Lagrangian systems) is introduced, and some important features of this group, as well as of its action on the set of Lagrangians, are briefly examined. Only finite local transformations of coordinates are considered here, and no variational transformation of the action is required in this study. Some miscellaneous examples of the formalism are included.     

The Theory of Equivalent Lagrangians Revisited: A Symplectic Approach

The existence of different Lagrangian functions for the same dynamical vector field is studied using the methods of symplectic mechanics. The concept of Lagrangeoid transformation is introduced and its relation with the theory of bi-Hamiltonian systems analyzed. The relation between equivalent (non-gauge equivalent) Lagrangian formulations in TQ and their associated Hamiltonian dynamical systems in T*Q is developed and, finally, the Noether theorem is considered.     

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     

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Abstract

We classify the Noether point symmetries of the generalized Lane-Emden equation y″+ ny′/x+ f(y)?=?0 with respect to the standard Lagrangian L = xⁿy′²/2 — xⁿ ∫f(y)dy for various functions f(y). We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f(y). One of them is f(y) = αy^r , where r ≠ 0,1. The case r?=?5 was considered previously and only a one-parameter family of solutions was presented. Here we provide a complete integration not only for r?= 5 but for other r values. We also give the Lie point symmetries for each case. In two of the new cases, the single Noether symmetry is also the only Lie point symmetry.     

Quantization of the Schwarzschild Black Hole: A Noether Symmetry Approach

We study the canonical formalism of a spherically symmetric space-time. In the context of the 3+1 decomposition with respect to the radial coordinate r, we set up an effective Lagrangian in which a couple of metric functions play the role of independent variables. We show that the resulting r-Hamiltonian yields the correct classical solutions which can be identified with the space-time of a Schwarzschild black hole. The Noether symmetry of the model is then investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generators of the desired symmetry. According to the Noether symmetry approach, we also quantize the model and show that the existence of a Noether symmetry yields a general solution to the Wheeler-DeWitt equation which exhibits a good correlation with the classical regime. We use the resulting wave function in order to (qualitatively) investigate the possibility of the avoidance of classical singularities.     

均匀磁场中二维各向同性带电谐振子的守恒量与对称性研究

由牛顿第二定律得到二维各向同性带电谐振子在均匀磁场中运动的运动微分方程,通过对运动微分方程的直接积分得到系统的两个积分(守恒量).利用Legendre变换建立守恒量与Lagrange函数间的关系,从而求得系统的Lagrange函数,并讨论与守恒量相应的无限小变换的Noether对称性与Lie对称性,最后求得系统的运动学方程.     

Absolutely anticommuting (anti-)BRST symmetry transformations for topologically massive Abelian gauge theory

We demonstrate the existence of the nilpotent and absolutely anticommuting Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations for the four (3+1)-dimensional (4D) topologically massive Abelian U(1) gauge theory that is described by the coupled Lagrangian densities (which incorporate the celebrated (B∧F) term). The absolute anticommutativity of the (anti-) BRST symmetry transformations is ensured by the existence of a Curci–Ferrari type restriction that emerges from the superfield formalism as well as from the equations of motion which are derived from the above coupled Lagrangian densities. We show the invariance of the action from the point of view of the symmetry considerations as well as superfield formulation. We discuss, furthermore, the topological term within the framework of superfield formalism and provide the geometrical meaning of its invariance under the (anti-)BRST symmetry transformations.     

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 theorem in higher-order Lagrangian mechanics

We study higher-order Lagrangian mechanics on thek-velocity manifold. The variational problem gives rise to new concepts, such as main invariants, Zermelo conditions, higher-order energies, and new conservation laws. A theorem of Noether type is proved for higher-order Lagrangians. The invariants to the infinitesimal symmetries are explicitly written. All this construction is a natural extension of classical Lagrangian mechanics.     

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.     

On the Symmetry Structures of the Minkowski Metric and a Weyl Re-Scaled Metric

Symmetries of spacetime manifolds which are given by Killing vectors are compared with the symmetries of a Lagrangian constructed from a Weyl re-scaled metric used in discussing disorder operators in Gauge theories. We find the point generators of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian (Noether symmetries). It is shown that the Noether symmetries obtained by considering the Lagrangian provide additional symmetries which are not provided by the Killing vectors. New conservation law/s are determined.     

Hidden Galilean symmetry, conservation laws and emergence of spin current in the soliton sector of chiral helimagnet

An appearance of the transport spin current in chiral helimagnet is mathematically justified based on the symmetry arguments. Although the starting Lagrangian of the chiral magnet with the Berry phase term and the parity-violating Dzyaloshinskii-Morya coupling is not manifestly Galilean invariant, the Lie point group symmetry analysis and the variational symmetry analysis elucidate the hidden Galilean symmetry and the existence of the linear momentum as a conserved Noether current, respectively.     

A unified approach to conservation laws in general relativity,gauge theories,and elementary particle physics

A unified treatment of conservation laws in general relativity, gauge theories, and elementary particle physics is formulated in the setting of principal fiber bundles. The group AUT(P) is introduced as the general gauge transformation group that covers space-time coordinate transformations. A set of master equations is exhibited for any Lagrangian density generally covariant with respect to AUT(P). The symmetry group for elementary particle theory is shown to be the structure group of the bundle only in the special case when the gauge potential is flat and the space-time is simply connected. In the general case, the symmetry group is reduced to the symmetry group of the gauge potential. This natural mechanism for a reduction of the symmetry group is speculated on as a model for spontaneous symmetry breaking.This essay received an honorable mention from the Gravity Research Foundation for the year 1981-Ed.Partially supported by a grant from the National Science Foundation.     

An Investigation of the Effect of the Canonical Transformation on the Lagrangian

The transformation of the Euler-Lagrange derivative under the point transformation is explicitly stated, and from this view point, the canonical transformation is reinvestigated. In our arguments, the canonical transformations are discussed strictly separately from the canonical equations. A proof is given that the Lagrangian can be restored after any infinitesimal canonical transformation. Some identities are obtained giving relations between canonically transformed and untransformed Lagrangians. Using the identities, the relation between the Noether charge and the generator of the canonical transformation is investigated. The chiral gauge, Galilei and scale transformations are considered as applications to field theory.     

The KT-BRST Complex of a Degenerate Lagrangian System

Quantization of a Lagrangian field system essentially depends on its degeneracy and implies its BRST extension defined by sets of non-trivial Noether and higher-stage Noether identities. However, one meets a problem how to select trivial and non-trivial higher-stage Noether identities. We show that, under certain conditions, one can associate to a degenerate Lagrangian L the KT-BRST complex of fields, antifields and ghosts whose boundary and coboundary operators provide all non-trivial Noether identities and gauge symmetries of L. In this case, L can be extended to a proper solution of the master equation.      

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.     

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.     

Consequences of the inertial equivalence of energy

The usual macroscopic theory of relativistic mechanics and electromagnetism is formulated so that all assumptions but one are consistent with both special relativity and Newtonian mechanics, the distinguishing assumption being that to any energyE, whatever its form, there corresponds an inertial massE/c ² . The speed of light enters this formulation only as a consequence of the inertial equivalent of energy1/c ² . While, for1/c ² >0 the resulting theory has symmetry under the Poincaré group, including Lorentz transformations, all its physical consequences can be derived and tested in any one inertial frame. In particular, an account is given in one inertial frame for the dynamic causes of relativistic effects for simple accelerated clocks and roads.     

Symmetries and conservation laws of the damped harmonic oscillator

We work with a formulation of Noether-symmetry analysis which uses the properties of infinitesimal point transformations in the space-time variables to establish the association between symmetries and conservation laws of a dynamical system. Here symmetries are expressed in the form of generators. We have studied the variational or Noether symmetries of the damped harmonic oscillator representing it by an explicitly time-dependent Lagrangian and found that a five-parameter group of transformations leaves the action integral invariant. Amongst the associated conserved quantities only two are found to be functionally independent. These two conserved quantities determine the solution of the problem and correspond to a two-parameter Abelian subgroup.