A New type of conserved quantity deduced from Mei symmetry of nonholonomic systems in terms of quasi-coordinates

This paper studies the new type of conserved quantity which is directly induced by Mei symmetry of nonholonomic systems in terms of quasi-coordinates. A coordination function is introduced, and the conditions for the existence of the new conserved quantities as well as their forms are proposed. Some special cases are given to illustrate the generalized significance of the new type conserved quantity. Finally, an illustrated example is given to show the application of the nonholonomic system＇s results.

Symmetry of Lagrangians of holonomic systems in terms of quasi-coordinates

This paper discusses the symmetry of Lagrangians of holonomic systems in terms of quasi-coordinates. Firstly, the definition and the criterion of the symmetry are given. Secondly, the condition under which there exists a conserved quantity and the form of the conserved quantity are obtained. Finally, an example is shown to illustrate the application of the results.

Structural Equation and Mei Conserved Quantity of Mei Symmetry for Appell Equations in Holonomic Systems with Unilateral Constraints

Structural equation and Mei conserved quantity of Mei symmetry for Appell equations in holonomicsystems with unilateral constraints are investigated.Appell equations and differential equations of motion for holonomicmechanic systems with unilateral constraints are established.The definition and the criterion of Mei symmetry forAppell equations in holonomic systems with unilateral constraints under the infinitesimal transformations of groups arealso given.The expressions of the structural equation and Mei conserved quantity of Mei symmetry for Appell equationsin holonomic systems with unilateral constraints expressed by Appell functions are obtained.An example is given toillustrate the application of the results.

Mei symmetry and generalized Hojman conserved quantity for variable mass systems with unilateral holonomic constraints

This paper studies Mei symmetry which leads to a generalized Hojman conserved quantity for variable mass systems with unilateral holonomic constraints. The differential equations of motion for the systems are established, the definition and criterion of the Mei symmetry for the systems are given. The necessary and sufficient condition under which the Mei symmetry is a Lie symmetry for the systems is obtained and a generalized Hojman conserved quantity deduced from the Mei symmetry is got. An example is given to illustrate the application of the results.

Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales

This paper focuses on studying the Noether symmetry and the conserved quantity with non-standard Lagrangians, namely exponential Lagrangians and power-law Lagrangians on time scales. Firstly, for each case, the Hamilton prin- ciple based on the action with non-standard Lagrangians on time scales is established, with which the corresponding Euler-Lagrange equation is given. Secondly, according to the invariance of the Hamilton action under the infinitesimal transformation, the Noether theorem for the dynamical system with non-standard Lagrangians on time scales is established. The proof of the theorem consists of two steps. First, it is proved under the infinitesimal transformations of a special one-parameter group without transforming time. Second, utilizing the technique of time-re-parameterization, the Noether theorem in a general form is obtained. The Noether-type conserved quantities with non-standard Lagrangians in both clas- sical and discrete cases are given. Finally, an example in Friedmann-Robertson-Walker spacetime and an example about second order Duffing equation are given to illustrate the application of the results.

Weak Noether Symmetry and non-Noether Conserved Quantities for General Holonomic Systems

A new kind of weak Noether symmetry for a general holonomic system is defined in such a way that themethods to construct Hojman conserved quantity and new-type conserved quantity are given.It turns out that weintroduce a new approach to look for the conserved laws.Two examples are presented.

Mei conserved quantity of the Nielsen equation for a non-Chetaev-type non-holonomic system

The Mei symmetry and Mei conserved quantity of the Nielsen equation for a non-Chetaewtype non-holonomic non-conservative system are studied. The differential equations of motion of the Nielsen equation for the system, the definition and the criterion of Mei symmetry and the condition and the form of Mei conserved quantities deduced directly from the Mei symmetry for the system are obtained. Finally, an example is given to illustrate the application of the results.

Lie Symmetry and Conserved Quantity of Three-Order Lagrangian Equations for Non-conserved Mechanical System

基于无穷小并且一参数转变，三顺序的 Lagrangian 方程的谎言对称的问题被学习了。在谎言转变下面，使 Lagrangian 方程三顺序未改变的足够、必要的条件并且不变在这篇论文被获得。

Generalized Lutzky Conserved Quantities of Holonomic Systems with Remainder Coordinates Subjected to Unilateral Constraints

<正> This paper focuses on studying the relation between a velocity-dependent symmetry and a generalizedLutzky conserved quantity for a holonomic system with remainder coordinates subjected to unilateral constraints.Thedifferential equations of motion of the system are established,and the definition of Lie symmetry for the system is given.The conditions under which a Lie symmetry can directly lead up to a generalized Lutzky conserved quantity and theform of the new conserved quantity are obtained,and an example is given to illustrate the application of the results.

Two New Types of Conserved Quantities of Mei Symmetry for Holonomic Mechanical System

保存数量的二种新类型直接由 holonomic 的 Mei 对称推断机械系统被学习。为 holonomic 系统的 Mei 对称的定义和标准被给。协作功能被介绍， Mei 对称能直接在下面导致保存数量的二种类型和保存数量的二种类型的形式的条件被获得。一个解说性的例子被给。结果显示协作功能能根据计量器功能的需求适当地被选择，从而，计量器功能能更容易被发现。而且，后来，协作功能的选择有多形，为 holonomic 的 Mei 对称的更多保存数量机械系统能被获得。

Symmetry of Lagrangians of a holonomic variable mass system

The symmetry of Lagrangians of a holonomic variable mass system is studied. Firstly, the differential equations of motion of the system are established. Secondly, the definition and the criterion of the symmetry of the system are presented. Thirdly, the conditions under which there exists a conserved quantity deduced by the symmetry are obtained. The form of the conserved quantity is the same as that of the constant mass Lagrange system. Finally, an example is shown to illustrate the application of the result.

Unified symmetry of non-holonomic singular systems

In this paper, the unified symmetry of non-holonomic singular systems is studied. The differential equations of motion of the systems are given. The definition and the criterion of the unified symmetry for the systems are presented. The Noether conserved quantity, the Hojman conserved quantity and the Mei conserved quantity are obtained. An example is given to illustrate the application of the results.

The Lagrangian and the Lie symmetries of charged particle motion in homogeneous electromagnetic field

In this paper, a constant of motion of charged particle motion in homogeneous electromagnetic field is derived from Newton＇s equations and the characteristics of partial differential equation, the related Lagrangian is also given by means of the obtained constant of motion. By discussing the Lie symmetry for this classical system, this paper obtains the general expression of the conserved quantity, It is shown that the conserved quantity is the same as the constant of motion in essence,

Lie-form invariance of non-holonomic systems with unilateral constraints

In this paper the Lie-form invariance of the non-holonomic systems with unilateral constraints is studied. The definition and the criterion of the Lie-form invariance of the system are given. The generalized Hojman conserved quantity and a new type of conserved quantity deduced from the Lie-form invariance are obtained. Finally, an example is presented to illustrate the application of the results.

Lie-Form Invariance of a Type of Non-holonomic Singular Systems

在这篇论文， non-holonomic 的一种类型的谎言形式不变性单个系统被学习。系统的运动的微分方程被给。定义和为系统的谎言形式不变性的标准被介绍。Hojman 保存了数量和 Mei 保存数量被获得。一个例子被给说明结果的申请。

Birkhoff symmetry and Lagrange symmetry

The purpose of this paper is to study the symmetry of first order Lagrangians and the cor- responding conserved quantity. The relation between the Lagrangians and the Birkhoff＇ s functions and the Birkhoff symmetry of Birkhoffian systems are used to obtain the symmetry of first order La- grangians and the corresponding conserved quantity. Two examples are given to illustrate the appli- cation of the result.

Lie symmetries of mechanical systems with unilateral holonomic constraints

In the recent twenty years, the study on the Lie symmetries and conserved quantities of constrained mechanical systems has been making great progress[1-5]. Up to now, however, all the studies are limited to the mechanical systems with ideal bilateral constraints. This note further studies the Lie symmetries of mechanical systems with unilateral holonomic constraints, and two problems of Lie symmetries for the systems are put forward and solved, i.e. finding the conserved quantity from a Lie symmetric transformation of the system and finding the corresponding Liesymmetry from an integral of the systemi ? 梛 ? Jsymmetry from an integral of the system.

完整系统形式不变性导致的新守恒量

研究完整力学系统由形式不变性直接导出的新型守恒量.用双面理想完整约束力学系统的运动微分方程在无限小变换下的形式不变性,给出系统形式不变性的定义和判据.得到形式不变性导致守恒量的条件以及守恒量的形式,并给出三个特殊情形下的推论.举例说明结果的应用.

基于El-Nabulsi分数阶模型的广义Birkhoff系统Noether对称性研究

为了进一步揭示力学系统的对称性与守恒量之间的内在关系,基于El-Nabulsi分数阶模型提出并研究了广义Birkhoff系统的Noether定理。首先,提出分数阶广义El-Nabulsi-PfaffBirkhoff原理,建立广义El-Nabulsi-Birkhoff方程。其次,基于El-Nabulsi-Pfaff作用量在无限小变换下的不变性,给出广义Birkhoff系统Noether对称性的定义和判据。最后,提出广义Birkhoff系统的Noether定理。该文研究结果可进一步应用于完整和非完整约束系统。

一般完整系统Appell方程的Mei对称性

研究一般完整系统Appell方程的Mei对称性和Mei守恒量.建立一般完整系统的Appell方程;在群的无限小变换下,给出一般完整系统Appell方程的Mei对称性的定义和判据;讨论一般完整系统Appell方程Mei对称性和Mei守恒量的研究方法;得到Mei对称性导致的Mei守恒量的存在条件以及Mei守恒量的表达式.