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Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales

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Chinese Physics B 2017年第8期26卷 201-209页

作者：宋静张毅 College of Mathematics and Physics Suzhou University of Science and TechnologySuzhou 215009China College of Civil Engineering Suzhou University of Science and TechnologySuzhou 215011China

This paper focuses on studying the Noether symmetry and the conserved quantity with non-standard Lagrangians, namely exponential Lagrangians and power-law Lagrangians on time scales. Firstly, for each case, the Hamilton prin- ciple based on the action with non-standard Lagrangians on time scales is established, with which the corresponding Euler-Lagrange equation is given. Secondly, according to the invariance of the Hamilton action under the infinitesimal transformation, the Noether theorem for the dynamical system with non-standard Lagrangians on time scales is established. The proof of the theorem consists of two steps. First, it is proved under the infinitesimal transformations of a special one-parameter group without transforming time. Second, utilizing the technique of time-re-parameterization, the Noether theorem in a general form is obtained. The Noether-type conserved quantities with non-standard Lagrangians in both clas- sical and discrete cases are given. Finally, an example in Friedmann-Robertson-Walker spacetime and an example about second order Duffing equation are given to illustrate the application of the results.

关键词： time scale non-standard Lagrangian Noether symmetry conserved quantity

Form invariance and conserved quantity for weakly nonholonomic system

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Applied Mathematics and Mechanics(English Edition) 2014年第10期35卷 1293-1300页

作者：吴惠彬梅凤翔 School of Mathematics Beijing Institute of Technology

The form invariance and the conserved quantity for a weakly nonholonomic system （WNS） are studied. The WNS is a nonholonomic system （NS） whose constraint equations contain a small parameter. The differential equations of motion of the system are established. The definition and the criterion of form invariance of the system are given. The conserved quantity deduced from the form invariance is obtained. Finally, an illustrative example is shown.

关键词： weakly nonholonomic system （WNS） form invariance conserved quantity

Symmetry of Hamiltonian and conserved quantity for a system of generalized classical mechanics

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Chinese Physics B 2011年第3期20卷 289-294页

作者：张毅 College of Civil Engineering Suzhou University of Science and Technology

This paper focuses on a new symmetry of Hamiltonian and its conserved quantity for a system of generalized classical mechanics. The differential equations of motion of the system are established. The definition and the criterion of the symmetry of Hamiltonian of the system are given. A conserved quantity directly derived from the symmetry of Hamiltonian of the generalized classical mechanical system is given. Since a Hamilton system is a special case of the generalized classical mechanics, the results above are equally applicable to the Hamilton system. The results of the paper are the generalization of a theorem known for the existing nonsingular equivalent Lagrangian. Finally, two examples are given to illustrate the application of the results.

关键词： symmetry of Hamiltonian generalized classical mechanics conserved quantity

Noether symmetry and conserved quantity for a Hamilton system with time delay

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Chinese Physics B 2014年第5期23卷 339-346页

作者：金世欣张毅 College of Mathematics and Physics Suzhou University of Science and Technology College of Civil Engineering Suzhou University of Science and Technology

In this paper, the Noether symmetries and the conserved quantities for a Hamilton system with time delay are dis-cussed. Firstly, the variational principles with time delay for the Hamilton system are given, and the Hamilton canonical equations with time delay are established. Secondly, according to the invariance of the function under the infinitesimal transformations of the group, the basic formulas for the variational of the Hamilton action with time delay are discussed, the definitions and the criteria of the Noether symmetric transformations and quasi-symmetric transformations with time delay are obtained, and the relationship between the Noether symmetry and the conserved quantity with time delay is studied. In addition, examples are given to illustrate the application of the results.

关键词： time delay Hamilton system Noether symmetry conserved quantity

A type of conserved quantity of Mei symmetry of Nielsen equations for a holonomic system

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Chinese Physics B 2012年第8期21卷 14-17页

作者：崔金超韩月林贾利群 School of Science Jiangnan University School of Astronautics Beijing Institute of Technology

A type of structural equation and conserved quantity which are directly induced by Mei symmetry of Nielsen equations for a holonomic system are studied. Under the infinitesimal transformation of the groups, from the definition and the criterion of Mei symmetry, a type of structural equation and conserved quantity for the system by proposition 2 are obtained, and the inferences in two special cases are given. Finally, an example is given to illustrate the application of the results.

关键词： Nielsen equation Mei symmetry structural equation conserved quantity

A New type of conserved quantity deduced from Mei symmetry of nonholonomic systems in terms of quasi-coordinates

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Chinese Physics B 2009年第8期18卷 3150-3154页

作者：庞婷方建会张明江蔺鹏路凯 College of Physics Science and Technology China University of Petroleum

This paper studies the new type of conserved quantity which is directly induced by Mei symmetry of nonholonomic systems in terms of quasi-coordinates. A coordination function is introduced, and the conditions for the existence of the new conserved quantities as well as their forms are proposed. Some special cases are given to illustrate the generalized significance of the new type conserved quantity. Finally, an illustrated example is given to show the application of the nonholonomic system＇s results.

关键词： Mei symmetry conserved quantity quasi-coordinate nonholonomic system

Energetics and conserved quantity of an axially moving string undergoing three-dimensional nonlinear vibration

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Acta Mechanica Sinica 2008年第2期24卷 215-221页

作者： Liqun Chen C. W. Lim Hu Ding Department of Mechanics Shanghai University Shanghai 200444 China Department of Construction and Building City University of Hong Kong Hong Kong China Shanghai Institute of Applied Mathematics and Mechanics Shanghai 200072 China

Nonlinear three-dimensional vibration of axially moving strings is investigated in the view of energetics. The governing equation is derived from the Eulerian equation of motion of a continuum for axially accelerating strings. The time-rate of the total mechanical energy associated with the vibration is calculated for the string with its ends moving in a prescribed way. For a string moving in a constant axial speed and constrained by two fixed ends, a conserved quantity is proved to remain unchanged during three-dimensional vibration, while the string energy is not conserved. An approximate conserved quantity is derived from the conserved quantity in the neighborhood of the straight equilibrium configuration. The approximate conserved quantity is applied to verify the Lyapunov stability of the straight equilibrium configuration. Numerical simulations are performed for a rubber string and a steel string. The results demonstrate the variation of the total mechanical energy and the invariance of the conserved quantity.

关键词： Nonlinear free vibration Axial movingstring Energetics conserved quantity Stability

Conformal invariance and conserved quantity of Hamilton systems

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Chinese Physics B 2008年第9期17卷 3170-3174页

作者：蔡建乐罗绍凯梅凤翔 College of Science Hangzhou Normal University College of Science Beijing Institute of Technology Institute of Mathematical Mechanics and Mathematical Physics Zhejiang Sci-Tech University

This paper studies conformal invariance and conserved quantities of Hamilton system. The definition and the determining equation of conformal invariance for Hamilton system are provided. The relationship between the conformal invariance and the Lie symmetry are discussed, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry of the system under the infinitesimal one-parameter transformation group is deduced. It gives the conserved quantities of the system and an example for illustration.

关键词： Hamilton system conformal invariance determining equation conserved quantity

A type of new conserved quantity deduced from Mei symmetry for Appell equations in a holonomic system with unilateral constraints

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Chinese Physics B 2010年第11期19卷 48-52页

作者：贾利群解银丽张耀宇杨新芳 School of Science Jiangnan University Electric and Information Engineering College Pingdingshan University

A type of new conserved quantity deduced from Mei symmetry of Appell equations for a holonomic system with unilateral constraints is investigated. The expressions of new structural equation and new conserved quantity deduced from Mei symmetry of Appell equations for a holonomic system with unilateral constraints expressed by Appell functions are obtained. An example is given to illustrate the application of the results.

关键词： unilateral constraint Appell equation Mei symmetry conserved quantity