A New Type of Non-Noether Adiabatic Invariants for Disturbed Lagrangian Systems： Adiabatic Invariants of Generalized Lutzky Type

For a Lagrangian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariant are presented in general infinitesimal transformation groups. On the basis of the invariance of disturbed Lagrangian systems under general infinitesimal transformations, the determining equations of Lie symmetries of the system are constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariant, i.e. generalized Lutzky adiabatic invariants, of a disturbed Lagrangian system are obtained by investigating the perturbation of Lie symmetries t＇or a Lagrangian system with the action of small disturbance. Finally, an example is given to illustrate the application of the method and results.     

Symmetries and variational calculation of discrete Hamiltonian systems

We present a numerical simulation method of Noether and Lie symmetries for discrete Hamiltonian systems. The Noether and Lie symmetries for the systems are proposed by investigating the invariance properties of discrete Lagrangian in phase space. The numerical calculations of a two-degree-of-freedom nonlinear harmonic oscillator show that the difference discrete variational method preserves the exactness and the invariant quantity.     

Adiabatic invariants induced by perturbation to form invariance of mechanico-electrical systems

This Letter focuses on perturbation to form invariance of mechanico-electrical systems. Criteria and structural equations for form invariance after disturbance are proposed. Considering perturbation of all the infinitesimal generators, a new type of adiabatic invariant induced by perturbation to form invariance of the system is reported.     

相空间中离散力学系统对称性的摄动与Hojman型绝热不变量

研究相空间中离散力学系统对称性的摄动与绝热不变量.列出相空间中未受扰离散力学系统的特殊Lie对称性导致的Hojman型精确不变量.基于相空间中力学系统的高阶绝热不变量的定义，研究在小扰动作用下系统Lie对称性的摄动，得到了相空间中离散力学系统的一类新的绝热不变量——Hojman型绝热不变量.举例说明结果的应用. 关键词： 相空间 Lie对称性 摄动 绝热不变量     

Lagrange系统对称性的摄动与Hojman型绝热不变量

研究Lagrange系统对称性的摄动与绝热不变量.列出未受扰Lagrange系统的Lie对称性导致的Hojman守恒量；基于力学系统的高阶绝热不变量的定义，研究在小扰动作用下Lagrange系统Lie对称性的摄动，得到了系统的一类Hojman形式的绝热不变量.并举例说明结果的应用. 关键词： Lagrange系统 对称性 摄动 绝热不变量     

Lagrange系统一类新型的非Noether绝热不变量——Lutzky型绝热不变量

研究了Lagrange系统的Lie对称性摄动与新型的非Noether绝热不变量. 列出了未受扰Lagrange系统的Lie对称性导致的Lutzky型精确不变量；基于力学系统的高阶绝热不变量的定义，研究在小扰动作用下Lagrange系统Lie对称性的摄动，得到了系统的一类Lutzky形式的绝热不变量.举例说明方法和结果的应用.     

Conformal invariance in perturbation theory

Asymptotic conformal invariance is proved to be true at all orders in perturbation theory. The correct Ward Identities for broken conformal invariance are derived: they are the extension of the Callan Symanzik equation from scale to conformal transformations.     

General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas

We show that the Lagrangian for interacting nonrelativistic particles can be coupled to an external gauge field and metric tensor in a way that exhibits a nonrelativistic version of general coordinate invariance. We explore the consequences of this invariance on the example of the degenerate Fermi gas at infinite scattering length, where conformal invariance also plays an important role. We find the most general effective Lagrangian consistent with both general coordinate and conformal invariance to leading and next-to-leading orders in the momentum expansion. At the leading order the Lagrangian contains one phenomenological constant and reproduces the results of the Thomas-Fermi theory and superfluid hydrodynamics. At the next-to-leading order there are two additional constants. We express various physical quantities through these constants.     

Hamilton形式的离散变分原理和第一积分

本文的研究表明：离散Hamilton系统的运动方程的第一积分可以通过研究其相空间离散拉格朗日函数的不变性来确定，提出一个类似连续情况的Hamilton形式的离散诺特定理。     

First integrals of the discrete nonconservative and nonholonomic systems

In this paper we show that the first integrals of the discrete equation of motion for nonconservative and nonholonomic mechanical systems can be determined explicitly by investigating the invariance properties of the discrete Lagrangian. The result obtained is a discrete analogue of the generalized theorem of Noether in the Calculus of variations.     

Perturbation to Mei symmetry and Mei adiabatic invariants for discrete generalized Birkhoffian system

Based on the concept of discrete adiabatic invariant,this paper studies the perturbation to Mei symmetry and Mei adiabatic invariants of the discrete generalized Birkhoffian system.The discrete Mei exact invariant induced from the Mei symmetry of the system without perturbation is given.The criterion of the perturbation to Mei symmetry is established and the discrete Mei adiabatic invariant induced from the perturbation to Mei symmetry is obtained.Meanwhile,an example is discussed to illustrate the application of the results.     

事件空间中保守完整系统的离散变分原理和第一积分

本文的研究表明：事件空间中完整保守系统的离散运动方程的第一积分可以通过研究其离散拉格朗日函数的不变性来确定，得到一个类似连续情况的离散诺特定理。最后给出两个例子用以说明本文结果的应用。     

Conformal invariance and generalized Hojman conserved quantities of mechanico-electrical systems

This paper studies conformal invariance and generalized Hojman conserved quantities of mechanico-electrical systems. The definition and the determining equation of conformal invariance for mechanico-electrical systems are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry under the infinitesimal single-parameter transformation group. The generalized Hojman conserved quantities from the conformal invariance of the system are given. An example is given to illustrate the application of the result.     

El-Nabulsi动力学模型下Birkhoff系统Noether对称性的摄动与绝热不变量

基于El-Nabulsi动力学模型,研究了小扰动作用下Birkhoff系统Noether对称性的摄动与绝热不变量问题.首先,将El-Nabulsi提出的在分数阶微积分框架下基于Riemann-Liouville分数阶积分的非保守系统动力学模型拓展到Birkhoff系统,建立El-Nabulsi-Birkhoff方程;其次,基于在无限小变换下El-Nabulsi-Pfaff作用量的不变性,给出Noether准对称性的定义和判据,得到了Noether对称性导致的精确不变量;再次,引入力学系统的绝热不变量概念,研究El-Nabulsi动力学模型下受小扰动作用的Birkhoff系统Noether对称性的摄动与绝热不变量之间的关系,得到了对称性摄动导致的绝热不变量的条件及其形式.作为特例,给出了El-Nabulsi动力学模型下相空间中非保守系统和经典Birkhoff系统的Noether对称性的摄动与绝热不变量.以著名的Hojman-Urrutia问题为例,研究其在El-Nabulsi动力学模型下的Noether对称性,得到了相应的精确不变量和绝热不变量.     

Perturbation of symmetries for super-long elastic slender rods

This paper analyses perturbations of Noether symmetry, Lie symmetry, and form invariance for super-long elastic slender rod systems. Criterion and structure equations of the symmetries after disturbance are proposed. Considering perturbation of all infinitesimal generators, three types of adiabatic invariants induced by perturbation of symmetries for the system are obtained.     

Perturbation of symmetries for super-long elastic slender rods

This paper analyses perturbations of Noether symmetry, Lie symmetry, and form invariance for super-long elastic slender rod systems. Criterion and structure equations of the symmetries after disturbance are proposed. Considering perturbation of all infinitesimal generators, three types of adiabatic invariants induced by perturbation of symmetries for the system are obtained.     

Conformal invariance and conserved quantities of Birkhoff systems under second-class Mei symmetry

This paper proposes a new concept of the conformal invariance and the conserved quantities for Birkhoff systems under second-class Mei symmetry.The definition about conformal invariance of Birkhoff systems under second-class Mei symmetry is given.The conformal factor in the determining equations is found.The relationship between Birkhoff system’s conformal invariance and second-class Mei symmetry are discussed.The necessary and sufficient conditions of conformal invariance,which are simultaneously of second-class symmetry,are given.And Birkhoff system’s conformal invariance may lead to corresponding Mei conserved quantities,which is deduced directly from the second-class Mei symmetry when the conformal invariance satisfies some conditions.Lastly,an example is provided to illustrate the application of the result.     

Conformal invariance and conserved quantities of general holonomic systems in phase space

This paper studies the conformal invariance and conserved quantities of general holonomic systems in phase space. The definition and the determining equation of conformal invariance for general holonomic systems in phase space are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The relationship between the conformal invariance and the Lie symmetry is discussed, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry of the system under the infinitesimal single-parameter transformation group is deduced. The conserved quantities of the system are given. An example is given to illustrate the application of the result.     

Gauge fixing problem in the conformal QED

The gauge fixing problem in the conformal (spinor and scalar) QED is examined. For the analysis, we generalize Dirac's manifestly conformal-covariant formalism. It is shown that the (vector and matter) fields must obey a certain mixed (conformal and gauge) type of transformation law in order to fix the local gauge symmetry preserving the conformal invariance in the Lagrangian.     

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Disturbed Nonholonomic Systems

For a nonholonomic mechanics system with the action of small disturbance, the Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type are studied under general infinitesimal transformations of groups in which the generalized coordinates and time are variable. On the basis of the invariance of disturbed nonholonomic dynamical equations under general infinitesimal transformations, the determining equations, the constrained restriction equations and the additional restriction equations of Lie symmetries of the system are constructed, which only depend on the variables t, qs and q^.s. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for a nonholonomic system with the action of small disturbance is investigated, and the Lie symmetrical adiabatic invariants, the weakly Lie symmetrical adiabatic invariants and the strongly Lie symmetrical adiabatic invariants of generalized Hojman type of disturbed nonholonomic systems are obtained. An example is given to illustrate applications of the results.