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.     

Lie symmetries and conserved quantities of rotational relativistic systems

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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。     

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。     

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．     

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。     

Mei symmetries and conserved quantities for non-conservative Hamiltonian difference systems with irregular lattices

Noether conserved quantities and Mei symmetries for non-conservative Hamiltonian difference systems with irregular lattices are studied. The generalized Hamiltonian equations of the systems are given on the basis of the transformation operators in the space of discrete Hamiltonians. The Lie point transformations acting on the lattice, as well as the difference equations, and the determining equations of Mei symmetries are obtained for the systems. The discrete versions of Noether conserved quantity are constructed by the Mei symmetries. An example is presented to illustrate the results.     

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 SYMMETRIES AND CONSERVED QUANTITY OF A BIPED ROBOT

I. INTRODUCTION It is well known there are close relationships between the symmetries and conservation laws inmechanical systems. The symmetric principles are among the key issues in mechanics. Two e?ectivemethods of studying the symmetries and conservation laws of mechanical are Noether’s method[1] andLie’s method. The approach to Lie symmetries was reported in the 19th century, but no applicationin mechanics appeared until 1979[2]. In recent years, studies of Lie’s method have be…     

Noether symmetries and conserved quantities for Birkhoffian systems with time delay

The Noether symmetries and conserved quantities for Birkhoffian systems with time delay are proposed and studied. First, the Pfaff–Birkhoff principle with time delay is proposed, and Birkhoff’s equations with time delay are obtained. Second, based on the invariance of the Pfaff action with time delay under a group of infinitesimal transformations, the Noether symmetric transformations and the Noether quasisymmetric transformations of the system are defined, and the criteria of the Noether symmetries are established. Finally, the relationship between the symmetries and the conserved quantities are studied, and the Noether theorems for Birkhoffian systems with time delay are established. Some examples are given to illustrate the application of the results.     

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.     

A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems

For a generalized Hamiltonian system with the action of small forces of perturbation, the Lie symmetries, symmetrical perturbation, and adiabatic invariants is presented. Based on the invariance of equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, and exact invariants of the system are given. Then the determining equations of Lie symmetrical perturbation and adiabatic invariants of the disturbed systems are obtained. Furthermore, in the special infinitesimal transformations, two deductions are given. At the end of the paper, one example is given to illustrate the application of the method and result.     

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.     

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.     

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。     

On the Noether symmetry and Lie symmetry of mechanical systems

The Noether symmetry is an invariance of Hamilton action under infinitesimal transformations of time and the coordinates. The Lie symmetry is an invariance of the differential equations of motion under the transformations. In this paper, the relation between these two symmetries is proved definitely and firstly for mechanical systems. The results indicate that all the Noether symmetries are Lie symmetries for Lagrangian systems meanwhile a Noether symmetry is a Lie symmetry for the general holonomic or nonholonomic systems provided that some conditions hold. The project supported by the National Natural Science Foundation of China (19972010)     

Application of Lie transformation group methods to classical linear theories of rods and plates

In the present paper, a class of partial differential equations governing various rod and plate theories of Bernoulli–Euler and Poisson–Kirchhoff type is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived and the general statement of the associated group-classification problem is given. A simple relation is deduced allowing to recognize easily the variational symmetries among the “ordinary” symmetries of a self-adjoint equation of the class examined. Explicit formulae for the conserved currents of the corresponding (via Bessel-Hagen’s extension of Noether’s theorem) conservation laws are suggested. Solutions of group-classification problems are given for subclasses of equations of the foregoing type governing stability and vibration of rods, fluid conveying pipes and plates resting on variable elastic foundations. The obtained group-classification results are used to derive conservation laws and group-invariant solutions readily applicable in rod dynamics and plate statics and dynamics. New generalized symmetries and conservation laws for the theories of Timoshenko beams, Reissner–Mindlin plates and three-dimensional elastostatics are presented.     

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.     

Exact solutions to the unsteady equations of perfect gases through Lie group analysis and substitution principles

In this paper, we consider the unsteady equations that govern two- and three-dimensional flows of a perfect gas. We explicitly characterize various classes of exact solutions by introducing some invertible transformations suggested by the invariance with respect to Lie groups of point symmetries and using suitable transformations known in literature as substitution principles.     

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。