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lie SYMMETRIES AND CONSERVED QUANTITY OF A BIPED ROBOT

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Acta Mechanica Solida Sinica 2004年第2期17卷 183-188页

作者： KeXianxin GongZhenbang FuJingli DepartmentofPrecisionMechanicalEngineering.ShanghaiUniversity Shanghai200072China ShanghaiInstituteofAppliedMathematicsandMechanics.ShanghaiUniversity Shanghai200072China

For a better understanding of the dynamic principles governing biped locomotion, the lie symmetries and conservation laws of a biped robot are studied. In lie theory, lie sym- metries and conservation laws can be derived from the form invariance of di?erential equations undergoing in?nitesimal transformation. By introducing in?nitesimal transformations including time and spatial coordinates, the determining equations of a biped robot are established. Then the necessary and su?cient conditions for a biped robot to have conserved quantities are obtained. For the lateral-plane dynamical model of a biped robot, a lie conserved quantity is found.

关键词： biped robot lie symmetry conserved quantity

lie symmetries and conserved quantities for generalized Birkhoff system

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Journal of Beijing Institute of Technology 2011年第3期20卷 285-288页

作者：梅凤翔崔金超 School of Aerospace Engineering Beijing Institute of Technology

To study lie symmetry and the conserved quantity of a generalized Birkhoff system with additional terms, the determining equations of the lie symmetry of the system is derived. A con- served quantity of Hojman＇ s type and a Noether＇ s conserved quantity are deduced by the lie symme- try under some conditions. One example is given to illustrate the application of the result.

关键词： generalized Birkhoff system lie symmetry Noether conserved quantity conservedquantity of Hojman' s type

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

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Applied Mathematics and Mechanics(English Edition) 2002年第10期23卷 1215-1220页

作者：方建会赵嵩卿焦志勇 Department of Applied Physics University of Petroleum Shandong P R China

The lie symmetries and the conserved quantities of a variable mass nonholonomic system of non-Chetaev's type are studied by introducing the infinitesimal transformations of groups in phase space. By using the invariance of the differential equations of motion under the infinitesmal transformations of groups, the determining equations and the restriction equations of the lie symmetries of the system are established, and the structure equations and the conserved quantities are obtained. An example is given to illustrate the application of the result.

关键词： nonholonomic system phase space analytic mechanics variable mass lie symmetry conserved quantity

lie symmetries and conserved quantities of fractional nonconservative singular systems

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International Journal of Mechanical System Dynamics 2023年第3期3卷 274-279页

作者： Mingliang Zheng Institute of Mechanical Engineering Wuxi Taihu UniversityWuxiP.R.China

In this paper,according to the fractional factor derivative method,we study the lie symmetry theory of fractional nonconservative singular Lagrange systems in a configuration ***,fractional calculus is calculated by using the fractional factor,and the fractional equations of motion are derived by using the differential variational ***,the determining equations and the limiting equations of lie symmetry under an infinitesimal group transformation are ***,the fractional conserved quantity form of singular Lagrange systems caused by lie symmetry is obtained by constructing a gauge-generating function that fulfills the structural equation,which conforms to the Noether criterion ***,we present an example of a *** results show that the lie symmetry condition of nonconservative singular Lagrange systems is more strict than conservative singular systems,but because of increased invariance restriction,the nonconservative forces do not change the form of conserved quantity;meanwhile,the fractional factor method has high natural consistency with the integral calculus,so the theory of integer-order singular systems can be easily extended to fractional singular Lagrange systems.

关键词： fractional factor nonconservative singular systems lie symmetry conserved quantity

lie Symmetrical Non-Noether Conserved Quantities of Poincaré-Chetaev Equations

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Communications in Theoretical Physics 2005年第2X期44卷 223-225页

作者： ZHANG Peng-Yu FANG Jian-Hui College of Physics and Technology China University of Petroleum Dongying 257061 China

In the present paper the lie symmetrical non-Noether conserved quantity of the Poincaré-Chetaev equations under the general infinitesimal transformations of lie groups is discussed. First, we establish the determining equations of lie symmetry of the equations. Second, the lie symmetrical non-Noether conserved quantity of the equations is deduced.

关键词： Poincaré-Chetaev equations lie symmetry non-Noether conserved quantity

Conserved Quantity for Fractional Constrained Hamiltonian System

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Wuhan University Journal of Natural Sciences 2022年第3期27卷 201-210页

作者： SONG Chuanjing WANG Jiahang School of Mathematical Sciences Suzhou University of Science and TechnologySuzhou 215009JiangsuChina College of Civil Engineering Suzhou University of Science and TechnologySuzhou 215009JiangsuChina

Singular system has great relationship with gauge field theory,condensed matter theory and some other research *** on the mixed integer and Riemann-Liouville fractional derivatives,the fractional singular system is ***,the fractional constrained Hamilton equation and the fractional inherent constraint are ***,lie symmetry and conserved quantity are analyzed,including determined equation,limited equation,additional limited equation and structural *** finally,an example is given to illustrate the methods and results.

关键词： fractional constrained Hamilton equation inherent constraint lie symmetry conserved quantity

Hojman＇s conservation theorems for generalized Raitzin canonical equations of motion

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Chinese Physics B 2005年第5期14卷 882-887页

作者：乔永芬李仁杰赵淑红 DepartmentofMechanicalEngineeringandAutomation ZhejiangUniversityofSciencesHangzhou310027China//FacultyofScienceLaiyangAgriculturalCollegeLaiyang265200China FacultyofScience LaiyangAgriculturalCollegeLaiyang265200China EngineeringCollegeofNortheastAgriculturalUniversity Harbin150030China

Using the lie symmetry under infinitesimal transformations in which the time is not variable, Hojman's conservation theorems for Raitzin's canonical equations of motion in generalized classical mechanics are studied. The generalized Raitzin's canonical equations of motion are established. The determining equations of lie symmetry under infinitesimal transformations are given. The Hojman conservation theorems of the system are established. Finally, an example is also presented to illustrate the application of the result.

关键词： generalized classical mechanics Hojman's conservation theorem Raitzin's canonical equation lie symmetry

lie symmetries and conserved quantities of Birkhoff systems with unilateral constraints

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Chinese Physics B 2002年第8期11卷 765-770页

作者：张宏彬顾书龙 DepartmentofPhysics ChaohuTeachersCollegeChaohu238000China

In this paper we study the lie symmetries of Birkhoff systems with unilateral constraints. We give the conditionsfor, and the form of, conserved quantities due to the lie symmetries of the systems, and we also study the inverseproblem of the lie symmetries of the systems. Finally, an example is given to illustrate the application of the results.

关键词： analytical mechanics Birkhoff systems with unilateral constraints lie symmetry con-served quantity inverse problem

Theory of symmetry for a rotational relativistic Birkhoff system

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Chinese Physics B 2002年第5期11卷 429-436页

作者：罗绍凯陈向炜等 InstituteofMathematicalMechanicsandMathematicalPhysics ChangshaUniversityChangsha410003China InstituteofMathematicalMechanicsandMathematicalPhysics ShangqiuTeachersCollegeShangqiu

The theory of symmetry for a rotational relativistic Birkhoff system is studied. In terms of the invariance of therotational relativistic Pfaff-Birkhoff-D'Alembert principle under infinitesimal transformations, the Noether symmetriesand conserved quantities of a rotational relativistic Birkhoff system are given. In terms of the invariance of rotationalrelativistic Birkhoff equations under infinitesimal transformations, the lie symmetries and conserved quantities of therotational relativistic Birkhoff system are given.

关键词： rotational relativity Birkhoff system Noether symmetry lie symmetry