Lie symmetries,symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems

For a nonlinear nonholonomic constrained mechanical system with the action of small forces of perturbation, Lie symmetries, symmetrical perturbation, and a new type of non-Noether adiabatic invariants are presented in general infinitesimal transformation of Lie groups. Based on the invariance of the equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, constraints restriction equations, additional restriction equations, and exact invariants of the system are given. Then, under the action of small forces of perturbation, the determining equations, constraints restriction equations, and additional restriction equations of the Lie symmetrical perturbation are obtained, and adiabatic invariants of the Lie symmetrical perturbation, the weakly Lie symmetrical perturbation, and the strongly Lie symmetrical perturbation for the disturbed nonholonomic system are obtained, respectively. Furthermore, a set of non-Noether exact invariants and adiabatic invariants are given in the special infinitesimal transformations. Finally, one example is given to illustrate the application of the method and results.     

Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems

For a generalized Birkhoffian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants are presented. On the basis of the invariance of disturbed generalized Birkhoffian system under general infinitesimal transformation of group, the determining equation of Lie symmetrical perturbation of the system is constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of non-Noether adiabatic invariants of a disturbed generalized Birkhoffian system is obtained by investigating the Lie symmetrical perturbation. Then, a new type of exact invariants of non-Noether type is given, furthermore adiabatic invariants and exact invariants of non-Noether type are obtained under the special infinitesimal transformation of group. Finally, an example is given to illustrate the application of the method and results.     

A new Lie symmetrical method of finding conserved quantity for Birkhoffian systems

For a Birkhoffian system, a new Lie symmetrical method to find a conserved quantity is given. Based on the invariance of the equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. Then, several important relationships which reveal the interior properties of the Birkhoffian system are given. By using these relationships, a new Lie symmetrical conservation law for the Birkhoffian system is presented. The new conserved quantity is constructed in terms of infinitesimal generators of the Lie symmetry and the system itself without solving the structural equation which may be very difficult to solve. Furthermore, several deductions are given in the special infinitesimal transformations and the results are reduced to a Hamiltonian system. Finally, one example is given to illustrate the method and results of the application.     

Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems

Based on Riemann-Liouville fractional derivatives, conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems are investigated. Firstly, fractional generalized Birkhoff equations are obtained by studying fractional generalized Pfaff-Birkhoff principle. Secondly, the definition of fractional generalized quasi-symmetry is given, the criteria of fractional generalized quasi-symmetry and the corresponding conserved quantity are achieved for fractional generalized Birkhoffian systems. Thirdly, perturbation to symmetry and adiabatic invariants for disturbed fractional generalized Birkhoffian systems are presented. Finally, an example is given to illustrate the results.     

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.     

Perturbation to Noether symmetries and adiabatic invariants for disturbed Hamiltonian systems based on El-Nabulsi nonconservative dynamics model

This paper focuses on studying the perturbation to the Noether symmetries and the adiabatic invariants for nonconservative dynamic systems in phase space under nonconservative dynamics model presented by El-Nabulsi. First of all, the El-Nabulsi dynamics model for a nonconservative system is introduced and the El-Nabulsi–Hamilton canonical equations are established. Secondly, the basic formulae for the variation of El-Nabulsi–Hamilton action in phase space are deduced, the definition and criterion of the Noether quasi-symmetric transformation are given, and the exact invariant led directly by the Noether symmetry is obtained. Finally, based upon the concept of high-order adiabatic invariant of a mechanical system, the relationship between the perturbation to the Noether symmetry and the adiabatic invariant after the action of a small disturbance is studied and the conditions that the perturbation of symmetry leads to the adiabatic invariant and its formulation are given. At the end of the paper, two examples are given to illustrate the application of the method and results.     

Hamilton系统的一类新型守恒律

研究Hamilton系统的Lie对称性与守恒律。根据微分方程在无限小群变换下的不变性理论，建立了Hamilton系统仅依赖于正则变量的无限小群变换的Lie对称变换，给出了Lie对称性的确定方程，并直接由系统的Lie对称性得到了系统的一类新型定恒律。文末，举例说明结果的应用。     

Perturbation of symmetries of Birkhoff system and adiabatic invariants

The perturbation of symmetries of the free Birkhoff system under small excitation is discussed. The concept of high-order adiabatic invariant is presented, and the form of adiabatic invariants and the conditions for their existence are given. Then these results are generalized to the constrained Birkhoff system. One example is presented to illustrate these results. The project supported by the National Natural Science Foundation (19972010) and the Doctoral Program Foundation of Institution of Higher Education of China and the Natural Science Foundation of Henan Province     

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.     

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

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Lie symmetries and conserved quantities of rotational relativistic systems

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ1979,Ｒ.ＢｅｎｇｔｓｓｏｎａｎｄＳ.Ｆｒａｎｅｎｄｏｒｆａｃｃｕｒａｔｌｙｍｅａｓｕｒｅｄｔｈｅｍａｘｉｍｕｍｖａｌｕｅｓｏｆｔｈｅｓｐｉｎｖｅｌｏｃｉｔｙｏｆ14ｋｉｎｄｓｏｆｎｕｃｌｅｏｎｓ,ａｎｄｔｈｅｒｅｓｕｌｔｓｓｈｏｗｅｄｔｈａｔｔｈｅｍａｘｉｍｕｍｖａｌｕｅｏｆｔｈｅｓｐｉｎｖｅｌｏｃｉｔｙｏｆｏｎｅｎｕｃｌｅｏｎｗａｓｄｉｆｆｅｒｅｎｔｔｏｔｈｏｓｅｏｆｔｈｅｏｔｈｅｒｓ[1].Ｗｉｔｈｔｈｅｄｅｖｅｌｏｐｍｅｎｔｏｆｓｃｉｅｎｃｅａｎｄｔｅｃｈｎｏｌｏｇｙ,…     

相空间中有二阶线性单面约束的非完整系统的Lie对称性与守恒量

研究相空间中有二阶线性单面约束的非完整系统的Lie对称性与守恒量。首先根据微分方程在无限小变换下的不变性建立Lie对称性所满足的确定方程和限制方程，给出结构方程和守恒量；其次讨论系统的Lie对称性逆问题。最后举一实例说明结果的应用。     

非Четаев型非完整系统的Lie对称性与守恒量

研究非Четаев型非完整系统的Ｌｉｅ对称性．首先利用微分方程在无限小变换下的不变性建立Ｌｉｅ对称所满足的确定方程和限制方程，给出结构方程并求出守恒量；其次研究上述问题的逆问题：根据已知积分求相应的Ｌｉｅ对称性；最后举例说明结果的应用．     

Conformal invariance for the nonholonomic constrained mechanical system of non-Chetaev’s type

The conformal invariance and conserved quantity for the nonholonomic system of non-Chetaev’s type are studied. Firstly, by introducing a one-parameter infinitesimal transformation group and its infinitesimal generator vector, the definition of conformal invariance and determining equation for the holonomic system which corresponds to a nonholonomic system of non-Chetaev’s type are provided, and the relationship between the system’s conformal invariance and Lie symmetry are discussed. Secondly, the conformal invariance of weak and strong Lie symmetry for the nonholonomic system of non-Chetaev’s type is given using restriction equations and additional restriction equations. Thirdly, the system’s corresponding conserved quantity is derived with the aid of a structure equation that the gauge function satisfies. Lastly, an example is given to illustrate the application of the method and its result.     

Lagrange-Maxwell系统的Lie对称性与守恒量

由微分方程在无限小主为换下的不变性,定义Ｌａｇｒａｎｇｅ－Ｍａｘｗｅｌｌ方程元限元小变失生成元,给出Ｌｉｅ对称性的确定方程,得到结构方程和守恒量。     

具有可积微分约束的力学系统的Lie对称性

研究具有可积微分约束的力学系统的Ｌｉｅ对称性与守恒量。采用两种方法：一是用不可积微分约束系统的方法;另一是用积分后降阶系统的方法,研究两种方法之间的关系。     

Group theoretic methods for approximate invariants and Lagrangians for some classes of y″+εF(t)y′+y=f(y,y′)

Some recent results on the Lie symmetry generators of equations with a small parameter and the relationship between symmetries and conservation laws for such equations are used to construct first integrals and Lagrangians for autonomous weakly non-linear systems, y″+εF(t)y′+y=f(y,y′). An adaptation of a theorem that provides the point symmetry generators that leave the invariant functional involving a Lagrangian for such equations is presented. A detailed example to illustrate the method is given (and other examples are discussed). The (approximate) symmetry generators, invariants and Lagrangians maintain the perturbation order of the ‘small parameter’ stipulated in the equation — first order in this case.     

Lie symmetries and conserved quantities of holonomic variable mass systems

In this paper, the Lie symmetries and the conserved quantities of the holonomic variable mass systems are studied. By using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the conserved quantities are given. And an example is given to illustrate the application of the result. Foundation item: the National Natural Science Foundation of China (19572038)     

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.     

Invariants of a Remarkable Family of Nonlinear Equations

In classical literature, invariants of families of differentialequations were considered for linear equations only, e.g. the renownedLaplace invariants for linear hyperbolic partial differential equationsand invariants of linear ordinary differential equations with variablecoefficients. The restriction to linear equations was essential inpioneering works of Cockle, Laguerre, Halphen, andForsyth for tackling the problem of invariants of differentialequations. Lie regretted that these authors did not use advantagesprovided by his theory of infinite continuous groups, but he himself didnot undertake further developments in this direction.Recently, the present author considered the possibility hinted byLie's remark and introduced the infinitesimal technique in thetheory of invariants of families of differential equations thatwas lacking in old methods. In consequence, a simple unifiedapproach was developed for calculation of invariants of algebraicand differential equations independent on the assumption oflinearity of equations. It was employed recently for calculationof Laplace type invariants for parabolic equations. Here, themethod is applied to calculation of invariants for the family ofnonlinear equations appearing in the problem on linearization ofnonlinear ordinary differential equations.