Perturbation to symmetry and adiabatic invariants of discrete nonholonomic nonconservative mechanical system

Perturbation to Noether symmetries and adiabatic invariants of discrete nonholonomic nonconservative mechanical systems on an uniform lattice are investigated. Firstly, we review Noether symmetry and conservation laws of a nonholonomic nonconservative system. Secondly, we study continuous Noether symmetry of a discrete nonholonomic system, give the Noether symmetry criterion and theorem of discrete corresponding holonomic system and nonholonomic system. Thirdly, we study perturbation to Noether symmetry of the discrete nonholonomic nonconservative system, give the criterion of perturbation to Noether symmetry for this system, and based on the definition of adiabatic invariants, we construct the theorem under which can lead to Noether adiabatic invariants for this system, and the forms of discrete Noether adiabatic invariants are given. Finally, we give an example to illustrate our results.     

Mei’s symmetry theorem for time scales nonshifted mechanical systems

We focus on Mei symmetry for time scales nonshifted mechanical systems within Lagrangian framework and its resulting new conserved quantities. Firstly, the dynamic equations of time scales nonshifted holonomic systems and time scales nonshifted nonholonomic systems are derived from the generalized Hamilton’s principle. Secondly, the definitions of Mei symmetry on time scales are given and its criterions are deduced. Finally, Mei’s symmetry theorems for time scales nonshifted holonomic conservative systems, time scales nonshifted holonomic nonconservative systems and time scales nonshifted nonholonomic systems are established and proved, and new conserved quantities of above systems are obtained. Results are illustrated with two examples.     

A field method for solving the equations of motion of nonholonomic systems

In this paper, the field method^[1] for solving the equations of motion of holonomic nonconservative systems is extended to nonholonomic systems with constant mass and with variable mass. Two examples are given to illustrate its application. The project supported by the National Natural Science Foundation of China.     

The principles of least action of variable mass nonholonomic nonconservative system in noninertial reference frames

This paper presents the generalized principles of least action of variable massnonholonomic nonconservative system in noninertial reference frame, proves theequivalence between Holder form and Suslov form, and then obtains differential equationsof motion of variable mass nonholonomic nonconservative system in noninertial referenceframe.     

Parametric equations of nonholonomic nonconservative systems in the event space and the method of their integration

In this paper, the parametric equations with multipliers of nonholonomic nonconservative systems in the event space are established, their properties are studied, and their explicit formulation is obtained. And then the field method for integrating these equations is given. Finally, an example illustrating the application of the integration method is given. The Project is supported by the National Natural Science Foundation of China.     

Lie symmetries and conserved quantities of nonconservative nonholonomic systems in phase space

The invariance and conserved quantities of the nonconservative nonholonomic systems are studied by introducing the infinitesimal transformations in phase space.The Lie’s symmetrical determining equations are establish ed.The Lie’s symmetrical structure equation is obtained.An example to illustrate the application of the result is given.     

Szebehely's problem for nonholonomic systems with a given first integral

I.IntroductionTheinverseproblemofdynamicsisoneoftheimportantsubjectsinmechanics.In1977,Szebehelysetforthaninverseproblemforthedeterminationofthet'orcefunctiontoamaterialpointintheplanefromparametricfamilyoftrajectories,andobtainedalinearfirstorderpartialdifferentialequationfortheforcefunction.Later,Erdil'l,MellsandPirast=l,MellsandBorgherol'l,BoilsandMertnsl4]extendedSzebehely'sproblemtoboththreeandndimensionalholonomicsystem.Recently,theauthorandProfessorMetFengxiangl'1studiedtheSzebehe…     

切塔耶夫型非完整系统的广义梯度表示

非定常非完整力学系统的稳定性研究是重要而又困难的问题,直接从微分方程出发来构造李雅普诺夫函数往往很难实现.本文给出了一种间接方法.提出了10类广义梯度系统的定义,并分别给出了10类广义梯度系统的微分方程.进一步研究一般切塔耶夫型非完整系统的广义梯度表示,给出该系统分别成为这10类广义梯度系统的条件,从而将切塔耶夫型非完整系统化成各类广义梯度系统.最后利用广义梯度系统的性质来研究切塔耶夫型非完整系统零解的稳定性.这种方法在直接构造李雅普诺夫函数发生困难时,显得更为有效.举例说明结果的应用.     

非完整系统稳定性的若干进展

介绍非完整系统稳定性理论的某些近代进展,包括非完整系统平衡位置关于全部变量的稳定性,平衡状态流形的稳定性,平衡位置关于部分变量稳定性及其与关于全部变量的稳定性的关系,平稳运动的稳定性,以及非完整控制系统的镇定．同时,讨论非完整系统稳定性的几个主要应用,并给出几个未来研究方向的建议      

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Lyapunov＇s first method, extended by V. V. Kozlov to nonlinear mechani- cal systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The mo- tion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin＇s series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three eases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints （V. V. Kozlov （1986）） is generalized to the case of nonlin- ear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov （1994） to nonholonomic systems with nonlinear constraints.     

Complete identification of chaos of nonlinear nonholonomic systems

The chaos of nonholonomic systems with two external nonlinear nonholonomic constraints where the magnitude of velocity is a constant and the magnitude of the velocity is a constant with a periodic disturbance, respectively, is completely identified for the first time. The scope of the chaos study is extended to nonlinear nonholonomic systems. By applying the nonlinear nonholonomic form of Lagrange’s equations, the dynamic equation is expressed. The existence of chaos in these two nonlinear nonholonomic systems is first wholly proved by all numerical criteria of chaos, i.e., the most reliable Lyapunov exponents, phase portraits, Poincaré maps, and bifurcation diagrams. Furthermore, it is found that the Feigenbaum number still holds for nonlinear nonholonomic systems.     

Structure-preserving approach for infinite dimensional nonconservative system

The current structure-preserving theory, including the symplectic method and the multisymplectic method, pays most attention on the conservative properties of the continuous systems because that the conservative properties of the conservative systems can be formulated in the mathematical form. But, the nonconservative characteristics are the nature of the systems existing in engineering. In this letter, the structure-preserving approach for the infinite dimensional nonconservative systems is proposed based on the generalized multi-symplectic method to broaden the application fields of the current structure-preserving idea. In the numerical examples,two nonconservative factors, including the strong excitation on the string and the impact on the cantilever, are considered respectively. The vibrations of the string and the cantilever are investigated by the structure-preserving approach and the good long-time numerical behaviors as well as the high numerical precision of which are illustrated by the numerical results presented.     

Applicability of Hamilton-Jacobi method to nonlinear nonholonomic systems

This paper uses Poincaré formalism to obtain a generalization of the Hamilton-Jacobi method of integrating dynamical systems moving with nonlinear nonholonomic constraints. Necessary and sufficient conditions are investigated for the applicability of this method to such systems. The method is illustrated by considering some concrete examples of nonholonomic systems.     

LAGRANGE'S THEOREM FOR A CLASS OF NONHOLONOMICSYSTEMS AND ITS APPLICATION

I.IntroductionSinceE.T.Whittaker.proposedfoestabilit}'problellll'lofnonholononlicsystemsin1904forthefirsttime,thescholarsathomeandabroad11a\'emadealotofresearchesontheequilibriunlstabilityoflinearand11olllinearnonllolollolnicsystems,andhaveobtainedaseriesofimportantresultslZ--7].Hobbled'er,theexpositionandapplicationrelatedtoLagrange'stheorenlinthestabilityanalysisfornonholonomicsystemsisseldonlseenuptonow.Althoughitwasmentionedinreference[3].aspecialdiscussionhasnotbeencarriedoutyet.Asafam…     

The method of variation on parameters for integration of a generalized Birkhoffian system

This paper focuses on studying the integration method of a generalized Birkhoffian system.The method of variation on parameters for the dynamical equations of a generalized Birkhoffian system is presented.The procedure for solving the problem can be divided into two steps:the first step,a system of auxiliary equations is constructed and its general solution is given;the second step,the parameters are varied,and the solution of the problem is obtained by using the properties of generalized canonical transformation.The method of variation on parameters for the generalized Birkhoffian system is of universal significance,and we take a nonholonomic system and a nonconservative system as examples to illustrate the application of the results of this paper.     

非完整系统的Lie对称性与Noether对称性

研究非完整系统的Lie对称性与Noether对称性及其间的关系,具体研究了Chetaev型变量质量非完整系统和非Chetaev型非完整系统的Lie对称性与Noether对称性。给出Lie对称性导致Noether对称性及Noether对称性导致Lie对称性的条件。     

非完整约束系统几何动力学研究进展:Lagrange理论及其它

近10年来, 非完整力学的发展主要集中在两个相互关联的方向上, 一个是非完整运动规划, 另一个则是非完整约束系统的几何动力学, 这两个研究方向都充分地利用了现代几何学, 如纤维丛理论、辛流形和Poisson流形结构等等.本文主要综述非完整约束系统几何动力学的外附型和内禀型Lagrange理论, 包括非定常力学系统所需要的射丛几何学的基本概念、射丛按约束的直和分解、约束流形上的水平分布、D'Alembert-Lagrange方程与Chaplygin方程的整体描述、以及Riemann-Cartan流形上的非完整力学, 文中对Chetaev条件和d-δ交换关系的几何意义作了深入讨论.除此之外, 简要评述非完整力学的Hamilton理论与赝Poisson结构、Noether对称性和Lie对称性、动量映射与对称约化、Vakonomic动力学等几个非常重要专题的研究进展.