Form invariance and noether symmetrical conserved quantity of relativistic Birkhoffian systems

IntroductionIn 1 92 7,theAmericanmathematicianG .D .BirkhoffmadeprimaryresearchesonBirkhoffiandynamics[1].In 1 983,theAmericanphysicistR .M .SantillistudiedthetransformationtheoryofBirkhoffequationsandgeneralizationofGalileirelativity ,andsummarizedcomprehensivelytheoriginofBirkhoffequationsandthelaterstudiesonthem[2 ].Since 1 992 ,theChinesemechanicianMeiFeng_xianganditsco_workershaveconstructedthedynamicsofBirkhoffiansystemonthebasisofRefs.[1 ,2 ] ,andgavethebasictheoreticalframe[3 - …     

Form invariance and Lie symmetry of variable mass nonholonomic mechanical system

IntroductionThereisacloserelationbetweenthesymmetryandtheconservedquantityinamechanicalsystem .ModernmethodstofindconservedquantityofamechanicalsystemaremainlyNoethersymmetrymethod[1]andLiesymmetrymethod[2 ].NoethersymmetryisaninvarianceoftheHamiltonactionundertheinfinitesimaltransformations.Liesymmetryisaninvarianceofthedifferentialequationsundertheinfinitesimaltransformations.Inthepasttenyears,aseriesofimportresultshavebeenobtainedonthestudyoftheNoethersymmetryandLiesymmetry[3～12 ].Thefo…     

Lie symmetries and conserved quantities of second-order nonholonomic mechanical system

The Lie symmetries and the conserved quantities of the second-order nonholonomic mechanical system are studied. Firstly, by using the invariance of the differential equation of motion under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the system are established, and the structure equation and the conservative quantities of the Lie symmetries are obtained. Secondly , the inverse problems of the Lie symmetries are studied . Finally , an example is given to illustrate the application of the result.     

Conformal invariance for nonholonomic system of Chetaev’s type with variable mass

Conformal invariance and conserved quantities for a nonholonomic system of Chetaev’s type with variable mass are studied. The conformal factor expressions are derived. The necessary and sufficient conditions are obtained to make the system’s conformal invariance Lie symmetrical. The conformal invariance of the weak and strong Lie symmetries for the system is given. The corresponding conserved quantities of the system are derived. Finally, an application of the result is shown with an example.     

Algebraic structure and Poisson method for a weakly nonholonomic system

The algebraic structure and the Poisson method for a weakly nonholonomic system are studied. The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed. The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system. An example is given to illustrate the application of the result.     

THE LIE SYMMETRIES AND CONSERVED QUANTITIES OF VARIABLE-MASS NONHOLONOMIC SYSTEM OF NON-CHETAEV''''S TYPE IN PHASE SPACE

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有多余坐标的完整系统形式不变性导致的新守恒量

研究有多余坐标的完整力学系统由形式不变性直接导出的新型守恒量。用有多余坐标的双面理想完整约束力学系统的运动微分方程和约束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，并给出三种特殊情形下的推论。举例说明结果的应用。     

Lie symmetry and approximate Hojman conserved quantity of Appell equations for a weakly nonholonomic system

For a weakly nonholonomic system, the Lie symmetry and approximate Hojman conserved quantity of Appell equations are studied. Based on the Appell equations for a weakly nonholonomic system under special infinitesimal transformations of a group in which the time is invariable, the definition of the Lie symmetry of the weakly nonholonomic system and its first-degree approximate holonomic system are given. With the aid of the structure equation that the gauge function satisfies, the exact and approximate Hojman conserved quantities deduced directly from the Lie symmetry are derived. Finally, an example is given to study the exact and approximate Hojman conserved quantity of the system.     

Special Mei symmetry and approximate conserved quantity of Appell equations for a weakly nonholonomic system

The weakly nonholonomic system is a nonholonomic system whose constraint equations contain a small parameter. The special Mei symmetry and approximate conserved quantity of Appell equations for a weakly nonholonomic system are studied. Appell equations for a weakly nonholonomic system are established and the definition and the criterion of the special Mei symmetry of the system are given. The structure equation of the special Mei symmetry for a weakly nonholonomic system and approximate conserved quantity deduced from the special Mei symmetry of the system are obtained. Finally, special approximate conserved quantity issues of Appell equations for a two freedom degrees weakly nonholonomic system are investigated using the results of this paper.     

A FORM INVARIANCE OF CONSTRAINED BIRKHOFFIAN SYSTEM

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Fractional conformal invariance method for finding conserved quantities of dynamical systems

For a dynamical system that can be transformed into fractional Birkhoffian representation, under a more general kind of fractional infinitesimal transformation of Lie group, we present the fractional conformal invariance method and it is found that, using the new method, we can find a new kind of non-Noether conserved quantity; and we find that, as a special case, an autonomous fractional Birkhoffian system possesses more conserved quantities. Also, as the fractional conformal invariance method’s applications, we, respectively, explore the conformal invariance and conserved quantities of a fractional Lotka biochemical oscillator and a fractional Hojman–Urrutia model. This work constructs a basic theoretical framework of fractional conformal invariance method, and provides a general method for finding conserved quantities of an actual fractional dynamical system that is related to science and engineering.     

Special Lie symmetry and Hojman conserved quantity of Appell equations for a Chetaev nonholonomic system

A special Lie symmetry and Hojman conserved quantity of the Appell equations for a Chetaev nonholonomic system are studied. The differential equations of motion and Appell equations of the Chetaev nonholonomic system are established. Under the special Lie symmetry group transformations in which the time is invariable, the determining equation of the special Lie symmetry of the Appell equations for a Chetaev nonholonomic system is given, and the expression of the Hojman conserved quantity is deduced directly from the Lie symmetry. Finally, an example is given to illustrate the application of the results.     

Elastoplastic analysis of cylindrically orthotropic composite thick-walled tube

In the present paper, Tsai-Hill Yield Criterion in a state of plane strain is derived for incompressible and compressible materials respectively. the elastoplastic stress and displacement fields of cylindrically orthotropic composite thick-walled tube under uniform radial pressure are studied. The formulas of the yield pressure, limit pressure and region of shakedown are obtained.     

Birkhoff系统的一般Lie对称性和非Noether守恒量

研究Birkhoff系统的一般Lie对称性导致的非Noether守恒量。得到非Noether守恒量的存在定理,举例说明结果的应用。     

Formulation and solution for inverse problem of nonholonomic dynamics

This paper presents a formulation and solution for the inverse problem ofnonholonomic dynamics:to find the form of nonholonomic constraints when someintegrals are given and to find the generalized reactive forces of constraint acting onthe system when the expression of the kinetic energy is given.An example is given toillustrate the application of the result.     

Form invariance, Noether symmetry and Lie symmetry of Hamiltonian systems in phase space

For the holonomic and non-holonomic Hamiltonian systems in phase space, the definitions and criterions of the form invariance of both Hamilton and generalized Hamilton canonical equations are given. The relations among the form invariance, Noether symmetry and Lie symmetry are studied. Two examples are given to illustrate these results.     

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…     

基于浸入与不变原理的四旋翼姿态系统反步滑模控制

针对较大幅度外部不确定扰动下的四旋翼姿态稳定问题,设计了一种基于浸入与不变原理(ⅠⅠ)的自适应反步滑模控制器(ABSMC)。首先建立了未知大扰动下四旋翼姿态系统动力学模型,然后以横滚角子系统搭建为例,设计并应用了反步法和基于趋进率的滑模控制策略。在扰动估计误差流型设计中,融合了ⅠⅠ原理,即自适应率的选取实现了误差流型的不变和吸引,确保估计误差收敛到0。最后,对系统进行了稳定性分析和数字仿真。结果表明,在较大未知扰动情况下,融合ⅠⅠ原理方法后,经10 s所测跟踪误差平方的累加和仅为传统ABSMC方法的11.2%,控制精度大幅提高。     

FIRST INTEGRALS AND INTEGRAL INVARIANTS FOR VARIABLE MASS NONHOLONOMIC SYSTEM IN NONINERTIAL REFERENCE FRAMES

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