Noether's theorems of a fractional Birkhoffian system within Riemann-Liouville derivatives

The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann–Liouville fractional derivatives. Firstly, the fractional Birkhoff＇s equations and the corresponding transversality conditions are given. Secondly, from special to general forms, Noether＇s theorems of a standard Birhoffian system are given, which provide an approach and theoretical basis for the further research on the Noether symmetry of the fractional Birkhoffian system. Thirdly, the invariances of the fractional Pfaffian action under a special one-parameter group of infinitesimal transformations without transforming the time and a general one-parameter group of infinitesimal transformations with transforming the time are studied, respectively, and the corresponding Noether＇s theorems are established. Finally, an example is given to illustrate the application of the results.

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅲ)—NOETHER'S THEOREM

The existing various couple stress theories have been carefully restudied.The purpose is to propose a coupled Noether's theorem and to reestablish rather complete conservation laws and balance equations for couple stress elastodynamics. The new concrete forms of various conservation laws of couple stress elasticity are derived. The precise nature of these conservation laws which result from the given invariance requirements are established. Various special cases are reduced and the results of micropolar continua may be naturally transited from the results presented in this paper.

NOETHER'S THEOREM FOR NONHOLONOMIC SYSTEMS OF NON-CHETAEV'S TYPE WITH UNILATERAL CONSTRAINTS IN EVENT SPACE

To study the Noether's theorem of nonholonomic systems of non_Chetaev's type with unilateral constraints in event space, firstly, the principle of D'Alembert_Lagrange for the systems with unilateral constraints in event space is presented, secondly, the Noether's theorem and the Noether's inverse theorem for the nonholonomic systems of non_Chetaev's type with unilateral constraints in event space are studied and obtained, which is based upon the invariance of the differential variational principle under the infinitesimal transformations of group, finally, an example is given to illustrate the application of the result.

Noether's theorem for non-conservative Hamilton system based on El-Nabulsi dynamical model extended by periodic laws

This paper focuses on the Noether symmetries and the conserved quantities for both holonomic and nonholonomic systems based on a new non-conservative dynamical model introduced by E1-Nabulsi. First, the E1-Nabulsi dynamical model which is based on a fractional integral extended by periodic laws is introduced, and E1-Nabulsi-Hamilton＇s canoni- cal equations for non-conservative Hamilton system with holonomic or nonholonomic constraints are established. Second, the definitions and criteria of E1-Nabulsi-Noether symmetrical transformations and quasi-symmetrical transformations are presented in terms of the invariance of E1-Nabulsi-Hamilton action under the infinitesimal transformations of the group. Fi- nally, Noether＇s theorems for the non-conservative Hamilton system under the E1-Nabulsi dynamical system are established, which reveal the relationship between the Noether symmetry and the conserved quantity of the system.

广义Birkhoff系统分数阶最优控制问题的Noether定理

探讨了广义Birkhoff系统分数阶最优控制问题的Noether定理。首先,基于Caputo分数阶导数提出广义Birkhoff系统的分数阶最优控制问题,并运用分数阶变分方法得到了广义Birkhoff系统的分数阶最优控制问题的极值条件。然后,运用分数阶微积分和分数阶变分方法讨论了泛函在无穷小变换群作用下的不变性,分别给出了其时间不变和时间变化情况下的分数阶最优控制问题的Noether定理。最后,应用实例显示方法的有效性。

Several Fundamental Theorems and Conservative Integrals in Quasicrystal Elasticity Theory

Aim To extend several fundamental theorems of conventional elasticity theory to quasicrystalelasticity theory. Methods The basic governing equations of quasicrystal elasticity theory and Gauss's theorem were applied in the derivation. Results and Conclusion The principle of virtual work, Betti's reciprocal theorem and the uniqueness theorem of quasicrystal elasticity theory are proud, and some conservative integrals in quasicrystal elasticty theory are obtained.

The Noether's Theory of Constrained Birkhoffian System

By using the synmeby and the qusi-symmetry of the infinitesimal transformation of the transformation group G1 and by imposing restrictions of constraints on the transformation, the Noether's theory of constrained Birkhoffian system has been established. The theory includes the generalized Noether's theorem obtaining the first integrals from the known symmetry and quasi-symmetry and its inverse obtaining the corresponding symmetry and quasi-symmetry from the known first integrals for the systerm.

三类非线性偏微分方程(组)的守恒律

非线性偏微分方程在多个学科领域中扮演着关键角色,其守恒律的求解是研究的核心之一。基于Lie对称性,并结合Maple等数学软件,应用对称-共轭对称‘对’方法和Ibragimov新守恒定理,推导不同方程的守恒律。通过比较这2种方法在同一方程上的应用结果,总结了各自的优势和适用性,从而为实际问题选择合适的求解策略提供了有价值的结论。

两种高阶非线性偏微分方程组守恒律的构造

利用Ibragimov新守恒定理和对称-共轭对称‘对’方法构造了3阶形变Boussinesq型方程、耦合KdV方程等2个高阶非线性偏微分方程组的局部守恒律,并对所求得的守恒律进行比较,揭示了2种方法在求解高阶非线性偏微分方程组守恒律中的等价关系。

Integrating factors and conservation theorems of constrained Birkhoffian systems

In this paper the conservation theorems of the constrained Birkhoffian systems are studied by using the method of integrating factors. The differential equations of motion of the system are written. The definition of integrating factors is given for the system. The necessary conditions for the existence of the conserved quantity for the system are studied. The conservation theorem and its inverse for the system are established. Finally, an example is given to illustrate the application of the results.

Power Balance Theorem of Frequency Domain and Its Application

This paper proves a power balance theorem of frequency domain. It becomes another circuit law concerning power conservation after Tellegen’s theorem. Moreover the universality and importance worth of application of the theorem are introduced in this paper. Various calculation of frequency domain in nonlinear circuit possess fixed intrinsic rule. There exists the mutual influence of nonlinear coupling among various harmonics. But every harmonic component must observe individually KCL, KVL and conservation of complex power in nonlinear circuit. It is a lossless network that the nonlinear conservative system with excited source has not dissipative element. The theorem proved by this paper can directly be used to find the main harmonic solutions of the lossless circuit. The results of solution are consistent with the balancing condition of reactive power, and accord with the traditional harmonic analysis method. This paper demonstrates that the lossless network can universally produce chaos. The phase portrait is related closely to the initial conditions, thus it is not an attractor. Furthermore it also reveals the difference between the attractiveness and boundedness for chaos.

Integrating Factors and Conservation Theorems of Nonholonomic Dynamical System of Relative Motion

The integrating factors and conservation theorems of nonholonomic dynamical system of relative motion are studied. First, the dynamical equations of relative motion of system are written. Next, the definition of integrating factors is given, and the necessary conditions for the existence of the conserved quantities are studied in detail. Then, the conservation theorem and its inverse of system are established. Finally, an example is given to illustrate the application of the result.

Integrating Factors and Conservation Theorems of Lagrangian Equations for Nonconservative Mechanical System in Generalized Classical Mechanics

The conservation theorems of the generalized Lagrangian equations for nonconservative mechanical system are studied by using method of integrating factors. Firstly, the differential equations of motion of system are given, and the definition of integrating factors is given. Next, the necessary conditions for the existence of the conserved quantity are studied in detail. Finally, the conservation theorem and its inverse for the system are established, and an example is given to illustrate the application of the result.

NOETHER'S THEORY OF MECHANICAL SYSTEMS WITH UNILATERAL CONSTRAINTS

Noether's theory of dynamical systems with unilateral constraints by introducing the generalized quasi_symmetry of the infinitesimal transformation for the transformation group G r is presented and two examples to illustrate the application of the result are given.

GENERALIZED NOETHER’S THEOREM OF NONHOLONOMIC NONPOTENTIAL SYSTEM IN NONINERTIAL REFERENCE FRAMES

The new Lagrangian of the relative motion of mechanical system is constructed, the varialional principles oj Jourdain's form of nonlinear nonlwlonomic nonpotential system in noninertial reference frame are established, the generalized Noether's theorem of the system above is presented and proved, and the conserved quantities of system are studied.

The approximate conserved quantity of the weakly nonholonomic mechanical-electrical system

We study the approximate conserved quantity of the weakly nonholonomic mechanical-electrical system. By means of the Lagrange-Maxwell equation, the Noether equality of the weakly nonholonomic mechanical-electrical system is obtained. The multiple powers-series expansion of the parameter of the generators of infinitesimal transformations and the gauge function is put into a generalized Noether identity. Using the Noether theorem, we obtain an approximate conserved quantity. An example is provided to prove the existence of the approximate conserved quantity.

Framing Noether’s Theorem

We discuss Noether’s theorem from a new perspective and show that the spatial continuous symmetries of a system are on one hand symmetries of the space and on the other hand are dictated by the system’s potential energy. The Noether’s charges arising from an infinitesimal motion, or a Killing vector field, of the space, are conserved if the Lie derivative of the potential energy by this vector field vanishes. The possible spatial symmetries of a mechanical system are listed according to the potential energy of the external forces.

二阶非完整Vacco动力学及其Noether定理

研究非完整约束Vacco动力学及其Noether对称性。建立二阶非完整系统的Vacco型变分原理,导出Vacco动力学方程。利用全变分与等时变分之间的关系,导出复合作用量的全变分公式。通过复合作用量在无限小变换下的不变性,定义Noether对称与准对称变换,给出对称性判据的Noether等式。建立二阶非完整约束Vacco动力学系统的Noether定理。如果约束是一阶非完整的,则定理退化为一阶非完整Vacco动力学的Noether定理。文末给出一个算例以演示定理之应用。

若干非线性PDEs守恒律的构造

利用对称-共轭对称‘对’方法和Ibragimov新守恒定理分别构造出Euler-Lagrange-type方程、CauchyKovalevskaya方程组及(2+1)维热方程中的诸多守恒律,同时通过对比已得守恒律发现两种守恒律方法的等价关系。结果对深入研究给定PDEs相关属性方面具有重要的理论意义。

Ito方程组的对称分析与守恒律

基于Lie群分析方法,得到了Ito方程组的一维子代数最优系统和相似约化系统,并借助Ibragimov的新守恒定理成功构造出该方程组的3个守恒律。结果表明:这些揭示了Ito方程组对称与守恒律之间的内在联系,而且克服了Nöether定理的局限性,对研究Ito方程组的相关属性方面具有重要意义。