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.     

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 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.     

Noether conserved quantities and Lie point symmetries of difference Lagrange--Maxwell equations and lattices

This paper presents a method to find Noether-type conserved quantities and Lie point symmetries for discrete mechanico-electrical dynamical systems, which leave invariant the set of solutions of the corresponding difference scheme. This approach makes it possible to devise techniques for solving the Lagrange--Maxwell equations in differences which correspond to mechanico-electrical systems, by adapting existing differential equations. In particular, it obtains a new systematic method to determine both the one-parameter Lie groups and the discrete Noether conserved quantities of Lie point symmetries for mechanico-electrical systems. As an application, it obtains the Lie point symmetries and the conserved quantities for the difference equation of a model that represents a capacitor microphone.     

Symmetries and variational calculation of discrete Hamiltonian systems

We present a numerical simulation method of Noether and Lie symmetries for discrete Hamiltonian systems. The Noether and Lie symmetries for the systems are proposed by investigating the invariance properties of discrete Lagrangian in phase space. The numerical calculations of a two-degree-of-freedom nonlinear harmonic oscillator show that the difference discrete variational method preserves the exactness and the invariant quantity.     

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     

Discrete variational principle and first integrals for Lagrange--Maxwell mechanico-electrical systems

This paper presents a discrete variational principle and a method to build first-integrals for finite dimensional Lagrange--Maxwell mechanico-electrical systems with nonconservative forces and a dissipation function. The discrete variational principle and the corresponding Euler--Lagrange equations are derived from a discrete action associated to these systems. The first-integrals are obtained by introducing the infinitesimal transformation with respect to the generalized coordinates and electric quantities of the systems. This work also extends discrete Noether symmetries to mechanico-electrical dynamical systems. A practical example is presented to illustrate the results.     

Noether symmetry and Lie symmetry of discrete holonomic systems with dependent coordinates

The Noether symmetry, the Lie symmetry and the conserved quantity of discrete holonomic systems with dependent coordinates are investigated in this paper. The Noether symmetry provides a discrete Noether identity and a conserved quantity of the system. The invariance of discrete motion equations under infinitesimal transformation groups is defined as the Lie symmetry, and the condition of obtaining the Noether conserved quantity from the Lie symmetry is also presented. An example is discussed to show the applications of the results.     

广义非完整力学系统的Lie对称性与Noether守恒量

研究广义非完整力学系统的Lie对称性与Noether守恒量,建立Lie对称性的确定方程、限制方程和附加限制方程,给出结构方程和Noether守恒量的形式,研究Lie对称性的逆问题,并举算例说明结果的应用.     

相空间中单面完整约束力学系统的对称性与守恒量

在增广相空间中研究单面完整约束力学系统的对称性与守恒量.建立了系统的运动微分方程；给出了系统的Norther对称性，Lie对称性和Mei对称性的判据；研究了三种对称性之间的关系；得到了相空间中单面完整约束力学系统的Noether守恒量以及两类新守恒量——Hojman守恒量和Mei守恒量，研究了三种对称性和三类守恒量之间的内在关系.文中举例说明研究结果的应用. 关键词： 分析力学 单面约束 对称性 守恒量 相空间     

相空间中离散完整系统的Noether对称性和Mei对称性

研究相空间中离散完整系统的Noether对称性、Mei对称性及其导致的守恒量.利用差分离散变分方法,给出相空间中离散完整系统的差分离散变分原理,建立系统的离散正则方程和能量演化方程;给出系统Noether对称性和Mei对称性的判定条件,得到系统离散形式的Noether守恒量和Mei守恒量及其存在的条件.举例说明结果的应用. 关键词： 相空间 离散完整系统 对称性 守恒量     

Symmetry and conserved quantities of discrete generalized Birkhoffian system

The Noether symmetry,the Mei symmetry and the conserved quantities of discrete generalized Birkhoffian system are studied in this paper.Using the difference discrete variational approach,the difference discrete variational principle of discrete generalized Birkhoffian system is derived.The discrete equations of motion of the system are established.The criterion of Noether symmetry and Mei symmetry of the system is given.The discrete Noether and Mei conserved quantities and the conditions for their existence are obtained.Finally,an example is given to show the applications of the results.     

Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems

This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of 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.     

Non-Noether symmetries and conserved quantities of the Lagrange mechano-electrical systems

This paper focuses on studying non-Noether symmetries and conserved quantities of Lagrange mechano-electrical dynamical systems. Based on the relationships between the motion and Lagrangian, we present conservation laws on non-Noether symmetries for Lagrange mechano-electrical dynamical systems. A criterion is obtained on which non-Noether symmetry leads to Noether symmetry of the systems. The work also gives connections between the non-Noether symmetries and Lie point symmetries, and further obtains Lie invariants to form a complete set of non-Noether conserved quantity. Finally, an example is discussed to illustrate these 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.     

离散Lagrange系统的Lie对称性

研究离散Lagrange系统的Lie对称性. 根据离散变分原理建立离散系统的运动方程. 给出离散运动方程Lie对称性的定义和确定方程. 举例说明结果的应用. 关键词： 离散Lagrange系统 离散变分原理 Lie对称性 确定方程     

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.     

Symmetries and Mei Conserved Quantities of Nonholonomic Controllable Mechanical Systems

This paper concentrates on studying the symmetries and a new type of conserved quantities called Mei conserved quantity. The criterions of the Mei symmetry, the Noether symmetry and the Lie symmetry are given. The conditions and the forms of the Mei conserved quantities deduced from these three symmetries are obtained. An example is given to illustrate the application of the result.     

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.