Lie symmetries and conserved quantities of constrained mechanical systems

Summary The Lie symmetries and conserved quantities of constrained mechanical systems are studied. Using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the systems are established. The structure equation and the form of conserved quantities are obtained. We find the corresponding conserved quantity from a known Lie symmetry, that is a direct problem of the Lie symmetries. And then, the inverse problem of the Lie symmetries-finding the corresponding Lie symmetry from a known conserved quantity-is studied. Finally, the relation between the Lie symmetry and the Noether symmetry is given.     

Conformal invariance of Mei symmetry for discrete Lagrangian systems

The conformal invariance of the Mei symmetry and the conserved quantities are investigated for discrete Lagrangian systems under the infinitesimal transformation of the Lie group. The difference Euler–Lagrange equations on regular lattices of the discrete Lagrangian systems are presented via the transformation operators in the space of the discrete variables. The conformal invariance of the Mei symmetry is defined for the discrete Lagrangian systems. The criterion equations and the determining equations are proposed. The conserved quantities of the systems are derived from the structure equation governing the gauge function. Two examples are given to illustrate the application of the results.     

A new Lie symmetrical method of finding a conserved quantity for a dynamical system in phase space

For a dynamical system in phase space, a new Lie symmetrical method to find a conserved quantity is presented in a general infinitesimal transformation of Lie groups. Based on the invariance of the differential equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. Then, an important relationship that reveals the interior properties of a dynamical system in phase space is given. By using the relationship, a Lie symmetrical basic integral variable relation and a new Lie symmetrical conservation law for the dynamical system in phase space are given. The new conserved quantity is constructed in terms of the infinitesimal generators of Lie symmetry and the system itself without solving the structural equation. Furthermore, the method is applied in the Hamiltonian system, the nonconservative Hamiltonian system and the nonholonomic Hamiltonian system. Finally, one example is given to illustrate the method and results of the application.     

Undefined Jacobi last multiplier? Complete symmetry group!

The Jacobi last multiplier has an intimate association with the symmetries of differential equations. We present four examples in which the inability to obtain a last multiplier using the method of Lie itself provides us with information. We show that the three representations of the complete symmetry group of the linear harmonic oscillator can be obtained by searching for Jacobi last multipliers: they correspond to the Lie point symmetries that have zero determinant, thus far regarded as a useless case. The point is emphasized using the examples of the Volterra–Verhulst–Pearl equation, the Kepler problem and a scalar equation of fourth order.     

Conformal invariance and conserved quantity of Mei symmetry for higher-order nonholonomic system

The conformal invariance and conserved quantity of Mei symmetry for a higher-order nonholonomic mechanical system are presented. Introducing an infinitesimal transformation group and infinitesimal generator vector, the definition of conformal invariance of Mei symmetry and the determining equation for the holonomic system which corresponds to a higher-order nonholonomic system are provided, and the relationship between Mei symmetry and conformal invariance of the system is discussed. The basis of restriction equations and additional restriction equations, the conformal invariance of weak and strong Mei symmetry for the higher-order nonholonomic mechanical system is constructed. With the aid of a structure equation that the gauge function satisfies, the system’s corresponding conserved quantity is derived. Finally, an example is given to illustrate the application of the method and its result.     

Lie symmetries and exact solutions of a new generalized Hirota–Satsuma coupled KdV system with variable coefficients

A new generalized Hirota–Satsuma coupled KdV system with variable coefficients is examined for Lie symmetry group and admissible forms of the coefficients with the help of the symmetry method based on the Fréchet derivative of the differential operators. An optimal system, of non-equivalent (non-conjugate) one dimensional sub-algebras of the symmetry algebra of the KdV system, having ten basic fields is determined. Using the non-equivalent Lie ansätze, for each essential vector field, the nonlinear system is reduced to systems of ordinary differential equations, and some special exact solutions of the KdV system are constructed.     

A Lie symmetrical basic integral variable relation and a new conservation law for generalized Hamiltonian systems

For a generalized Hamiltonian system, a new Lie symmetrical method to find a conserved quantity is presented in a general infinitesimal transformation of Lie groups. Based on the invariance of the differential equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. And a number of important relationships of the Lie symmetrical method for a generalized Hamiltonian system are investigated, which reveal the interior properties of the system. By using the relationships, a Lie symmetrical basic integral variable relation and a new Lie symmetrical conservation law for generalized Hamiltonian systems is presented. The basic integral variable relation not only can be used for a linear dynamic system but can also be used for a nonlinear dynamic system. The new conserved quantity is constructed in terms of infinitesimal generators of Lie symmetry and the interior properties of the system itself without solving the structural equation that may be very difficult to solve. Then, the method is applied in the generalized Hamiltonian system of even dimensions and the Hamiltonian system, respectively. Furthermore, the relationship between the generalized Hamiltonian system and a Birkhoffian system is studied, and a Lie symmetrical conservation law for a semiautonomous (or autonomous) Birkhoffian system is obtained. Finally, one example is given to illustrate the method and results of the application.     

The inverse problem of a mixed Liénard-type nonlinear oscillator equation from symmetry perspective

In this paper, we discuss the inverse problem for a mixed Liénard-type nonlinear oscillator equation \({\ddot{x}+f(x)\dot{x}^2+g(x)\dot{x}+h(x)=0}\), where \({f(x), g(x)}\) and h(x) are arbitrary functions of x. Very recently, we have reported the Lie point symmetries of this equation. By exploiting the interconnection between Jacobi last multiplier, Lie point symmetries and Prelle–Singer procedure, we construct a time-independent integral for the case exhibiting maximal symmetry from which we identify the associated conservative nonstandard Lagrangian and Hamiltonian functions. The classical dynamics of the nonlinear oscillator is also discussed, and certain special properties including isochronous oscillations are brought out.     

On the first integral of the non-linear shallow-membrane equation via Noether's theorem

In this paper, using the invariance identities of Rund [2], involving the Lagrangian and the generators of the infinitesimal Lie group and then using Noether's theorem, the first integral of the shallow-membrane equation has been obtained. Further, through the repeated application of invariance under the transformation obtained its exact solution, for certain particular values of the parameters involved, is obtained by reducing the first integral to quadrature.     

Restricted Lie point symmetries and reductions for ideal magnetohydrodynamics equilibria

Restricted Lie point symmetries are derived for the axially symmetric steady solutions to the ideal magnetohydrodynamics equations. The symmetries transform vectors of magnetic field B and plasma velocity V linearly with coefficients depending on a function u(z, r). A reduction of the eight MHD equilibrium equations to a single second-order partial differential equation for the function u(z, r) is obtained. Analogous Lie point symmetries and reduction are derived for the translationally invariant MHD equilibria. Applications of the symmetry transforms are indicated.     

非线性包装系统自由振动特性的对称性解法

目的基于Lie积分法精确解析求解包装系统的非线性自由振动响应。方法考虑到弹簧恢复力三次多项式形式的非线性关系,依据分析力学准则建立系统单自由度含阻尼动力学模型;首次运用微分方程Lie群变换理论求解系统的对称性和2个首次积分,证明在结构设计参数满足一定关系时,包装系统自由振动的精确解是一类椭圆积分函数。结果实际算例仿真计算表明,系统的自共振频率随着初始振幅条件的增大而增大,非线性系数项使得位移响应振幅的衰减变快。结论从推演过程可看出,将Lie对称性理论应用到包装系统非线性动力学特性研究中,系统的非线性系数以及阻尼系数无需满足小参数假设,因此适用范围更广。     

A note on the integrability of a remarkable static Euler–Bernoulli beam equation

It has recently been shown that the fourth-order static Euler–Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, in the maximal case has three symmetries. This corresponds to the negative fractional power law y ^?5/3, and the equation has the nonsolvable algebra ${sl(2, \mathbb{R})}$ . We obtain new two- and three-parameter families of exact solutions when the equation has this symmetry algebra. This is studied via the symmetry classification of the three-parameter family of second-order ordinary differential equations that arises from the relationship among the Noether integrals. In addition, we present a complete symmetry classification of the second-order family of equations. Hence the admittance of ${sl(2, \mathbb{R})}$ remarkably allows for a three-parameter family of exact solutions for the static beam equation with load a fractional power law y ^?5/3.     

基于数据增强的滚动轴承智能故障诊断方法

目的 针对包装机械设备中滚动轴承应用场景多且有效故障数据难采集而导致的智能诊断方法诊断准确率较低的问题,提出一种基于数据增强的滚动轴承智能诊断方法.方法 首先根据轴承振动信号的故障特征,提出一种数据增强方法,有效扩充训练数据样本多样性.然后采用卷积神经网络对原始样本和增强样本进行故障诊断训练,从而大幅度提高诊断模型的诊断性能.为了验证所提方法的有效性,建立滚动轴承故障试验台并采集轴承故障数据.结果 实验结果表明,在标签训练样本不充足的情况下,提出的方法与不使用数据增强方法相比,模型在诊断准确率方面取得了较大的提高,能够准确地识别各类轴承故障.结论 该方法实现了准确地对稀缺标记样本下滚动轴承故障的诊断,为保证包装机械滚动轴承故障诊断的诊断精度提供了可靠的方法.     

Lie Group Classification for a Generalised Coupled Lane-Emden System in Dimension One

In this article, we discuss the generalised coupled Lane-Emden system $u^{′′}+H(v)=0$, $v^{′′}+G(u)=0$ that applies to several physical phenomena. The Lie group classification of the underlying system shows that it admits a ten-dimensional equivalence Lie algebra. We also show that the principal Lie algebra in one dimension has several possible extensions, and obtain an exact solution for an interesting particular case via Noether integrals.     

Exactly and quasi-exactly solvable 'discrete' quantum mechanics

A brief introduction to discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators and dynamical symmetry algebras, including the q-oscillator algebra and the Askey-Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable (QES) Hamiltonians in one-dimensional 'discrete' quantum mechanics is presented. It reproduces all the known Hamiltonians whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. Several new exactly and QES Hamiltonians are constructed. The sinusoidal coordinate plays an essential role.     

Internal symmetry groups for perfect elastoplasticity under axial‐torsional loadings

Abstract

In the material modeling of experimental axial‐torsional strain control tests, the hoop and radial strains are always unknown, a priori, and hence can not be viewed as inputs. This greatly complicates constitutive model analyses because the resulting differential equations become highly nonlinear. To tackle this problem, we demonstrate two new formulations. By using the two‐integrating factors idea we derive two Lie type systems in the product space M ¹⁺¹?M ¹⁺¹. The Lie algebra is the direct sum so(1, 1)?so(1, 1), and correspondingly the symmetry group is the direct product SO_o (1, 1) ?SO_o (1, 1). Then, by using the one‐integrating factor idea we convert the nonlinear constitutive equations to a Lie type system X=A(X, t)X with A?sl(2, 1, R), a Lie algebra of the special orthochronous pseudo‐linear group SL(2, 1, R). The underlying space is a cone in the pseudo‐Riemann manifold. Consistent numerical methods are also developed according to these Lie symmetries.     

On symmetries of the Fokker–Planck equation

This paper is devoted to Lie point symmetries of the Fokker–Planck (FP) equation. It describes the relation between symmetries and first integrals of stochastic differential equations (SDEs) and symmetries of the associated FP equations. This relation is illustrated on symmetries of (1 + 1)-dimensional FP equations specified by Lie group classification of the scalar SDE. Further, it is used to find symmetries of (1 + 2)-dimensional FP equations specified by Lie group classification of the system of two SDEs.     

含自旋—轨道相互作用的碱金属原子的精确量子论

根据新量子变换理论 ,结合创新的SU(1,1)Lie代数理论及SO(3)Lie代数理论 ,对含自旋—轨道相互作用的碱金属原子的量子力学问题进行系统的研究。结果表明 ,碱金属原子的能级为双能级结构 ,并且是精确的     

Non-autonomous normal forms for Hamiltonian systems

A method for calculating normal forms for non-autonomous periodically perturbed Hamiltonian systems is developed. The solution for an autonomous Hamiltonian normal form is well known, and involves the solution of a homological equation on the vector space of homogeneous scalar polynomials. An algorithm is presented for generating an analogous non-autonomous homological equation using Lie transforms. Solution of this equation will generate a normal form for the non-autonomous Hamiltonian. Although this equation is defined on an infinite-dimensional space, it is shown that the problem can be reduced to an equivalent one on a finite-dimensional space. A solution can then be found in an analogous way to the solution for the autonomous problem. It is also shown that the normal form satisfies invariance properties. Finally, an example problem is presented to illustrate the solution technique.     

Equivalence groups for second order balance equations

The groups of equivalence transformations for a family of second order balance equations involving arbitrary number of independent and dependent variables are investigated. Equivalence groups are much more general than symmetry groups in the sense that they map equations containing arbitrary functions or parameters onto equations of the same structure but with different functions or parameters. Our approach to attack this problem is based on exterior calculus. The analysis is reduced to determine isovector fields of an ideal of the exterior algebra over an appropriate differentiable manifold dictated by the structure of the differential equations. The isovector fields induce point transformations, which are none other than the desired equivalence transformations, via their orbits which leave that particular ideal invariant. The general scheme is applied to a one-dimensional nonlinear wave equation and hyperelasticity. It is shown that symmetry transformations can be deduced directly from equivalence transformations.