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.     

Mei Symmetries and Lie Symmetries for Nonholonomic Controllable Mechanical Systems with Relativistic Rotational Variable Mass

The Mei symmetries and the Lie symmetries for nonholonomic controllable mechanical systems with relativistic rotational variable mass are studied. The differential equations of motion of the systems are established. The definition and criterion of the Mei symmetries and the Lie symmetries of the system are studied respectively. The necessary and sufficient condition under which the Mei symmetry is Lie symmetry is given. The condition under which the Mei symmetries can be led to a new kind of conserved quantity and the form of the conserved quantity are obtained. An example is given to illustrate the application of the results.     

Dynamics on the cone: Closed orbits and superintegrability

The generalization of Bertrand’s theorem to the case of the motion of point particle on the surface of a cone is presented. The superintegrability of such models is discussed. The additional integrals of motion are analysed for the case of Kepler and harmonic oscillator potentials.     

仅含第二类约束的约束Hamilton系统的Lie对称性

研究仅含第二类约束的约束Hamilton系统的Lie对称性.建立Lie对称性的确定方程、限制方程和附加限制方程,给出由Lie对称性导致守恒量的条件及守恒量的形式 关键词： 奇异系统 正则变量 约束 Lie对称性 守恒量     

单面非Chetaev型非完整约束系统的非Noether守恒量

研究单面非Chetaev型非完整约束力学系统的对称性与非Noether守恒量.建立了系统的运动微分方程；给出了系统的Lie对称性和Mei对称性的定义和判据；对于单面非Chetaev型非完整系统，证明了在一定条件下，由系统的Lie对称性可直接导致一类新守恒量——Hojman守恒量，由系统的Mei对称性可直接导致一类新守恒量——Mei守恒量；研究了对称性和新守恒量之间的相互关系.文末，举例说明结果的应用. 关键词： 分析力学 单面约束 非完整系统 对称性 Hojman守恒量 Mei守恒量     

Recursion operators, higher-order symmetries and superintegrability in quantum mechanics

A connection between the theory of superintegrable quantum-mechanical systems, which admit a maximal number of integrals of motion, and the standard Lie group theory is established. It is shown that the flows generated by first- and second-order Lie symmetries of the bidimensional Schrödinger equation can be classified and interpreted as quantum-mechanical operators which commute with integrable or superintegrable Hamiltonians. In this way, all known superintegrable potentials in the plane are naturally obtained and slightly more general integrals of motion are found.     

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.     

一维减幅-增幅谐振子的守恒量与对称性

从一维减幅-增幅谐振子的运动微分方程出发得到系统的运动积分常数,从而得到系统的Lagrange函数和Hamilton函数,再根据Hamilton函数的形式假定守恒量的形式,由Poisson括号的性质得到了系统的三个守恒量,并讨论与三个守恒量相应的无限小变换的Noether对称性与Lie对称性.还对守恒量与对称性的物理意义作了合理的解释. 关键词： 一维减幅-增幅谐振子 守恒量 Noether对称性 Lie对称性     

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.     

Generalized Lie Symmetries and the Integrability of Generalized Emden-Fowler Equations

Generalized Lie symmetries and the integrability of generalized Emden-Fowler equations (GEFEs) are considered. It is shown that the constraint which the variable-coefficient functions must satisfy for the GEFEs to have infinite-dimensional symmetry algebras is precisely the same as this in order that the equation may be transformed into the integrable Emden-Fowler equation. fiom the nature of the symmetry vector fields one can write down the integrals of motion for the above systems. The structure of the symmetry algebras is also presented.     

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.     

Quasi-integrable and superintegrable systems from a ‘deformed-mass’ two-photon algebra

A quantum deformation of the two-photon (or Schrödinger) Lie algebra is introduced in order to construct newn-dimensional classical Hamiltonian systems which have (n?2) functionally independent integrals of motion in involution; we say that such Hamiltonians define quasi-integrable systems. Furthermore, Hopf subalgebras of this quantum two-photon algebra (quantum extended Galilei and harmonic oscillator algebras) provide another set of (n?1) integrals of motion for Hamiltonians defined on these Hopf subalgebras, so that they lead to superintegrable systems.     

耦合Burgers系统的Painlevé性质分析，对称和对称约化

本文给出了耦合Burgers系统的Painlevé性质，逆强对称算子，无穷多对称和李对称约化。通过把强对称和逆强对称算子重复多次作用到耦合Burgers模型的一些平庸对称，如恒等变换，空间平移变换和标度变换上，我们得到了三族无穷多对称。这些对称构成了无穷维李代数。用其中的有限维子代数——点李代数对模型进行对称约化，得到了模型的群不变解。     

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.     

Symmetries and symmetry reductions of the combined KP3 and KP4 equation

To find symmetries,symmetry groups and group invariant solutions are fundamental and significant in nonlinear physics.In this paper,the finite point symmetry group of the combined KP3 and KP4(CKP34) equation is found by means of a direct method.The related point symmetries can be obtained simply by taking the infinitesimal form of the finite point symmetry group.The point symmetries of the CKP34 equation constitute an infinite dimensional KacMoody-Virasoro algebra.The point symmetry invariant so...     

Symmetry groups and spiral wave solution of a wave propagation equation

We study a third-order nonlinear evolution equation, which can be transformed to the modified KdV equation, using the Lie symmetry method. The Lie point symmetries and the one-dimensional optimal system of the symmetry algebras are determined. Those symmetries are some types of nonlocal symmetries or hidden symmetries of the modified KdV equation. The group-invariant solutions, particularly the travelling wave and spiral wave solutions, are discussed in detail, and a type of spiral wave solution which is smooth in the origin is obtained.     

相对论性转动变质量非完整可控力学系统的非Noether守恒量

研究相对论性转动变质量非完整可控力学系统的非Noether守恒量——Hojman守恒量. 建立了系统的运动微分方程, 给出了系统在特殊无限小变换下的Mei对称性(形式不变性) 和Lie对称性的定义和判据, 以及系统的Mei对称性是Lie对称性的充分必要条件. 得到了系统Mei对称性导致非Noether守恒量的条件和具体形式. 举例说明结果的应用. 关键词： 相对论性转动 可控力学系统 变质量 非Noether守恒量     

Mei Symmetry and Lie Symmetry of Relativistic Hamiltonian System

The Mei symmetry and the Lie symmetry of the relativistic Hamiltonian system are studied. The definition and criterion of the Mei symmetry and the Lie symmetry of the relativistic Hamiltonian system are given. The relationship between them is found. The conserved quantities which the Mei symmetry and the Lie symmetry lead to are obtained.An example is given to illustrate the application of the result.     

Mei Symmetry and Lie Symmetry of Relativistic Hamiltonian System

The Mei symmetry and the Lie symmetry of the relativistic Hamiltonian system are studied. The definition and criterion of the Mei symmetry and the Lie symmetry of the reoativistic Hamiltonian system are given. The relationship between them is found. The conserved quantities which the Mei symmetry and the Lie symmetry lead to are obtained. An example is given to illustrate the application of the result.     

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.