Some Studies On Theory Of Symmetry And Conserved Quantity For Constrained Mechanical Systems

This paper focuses on the theme of some studies on the theory of the symmetry andconserved quantity for constrained dynamical systems, and the main research is on theproblems of the symmetry and conserved quantity in the three mechanical systems (Nielsensystem, Appell system and Lagrange system).There are three main types of symmetry: Meisymmetry, Lie symmetry and Noether symmetry. Conserved quantities mainly are Meiconserved quantity, Hojman conserved quantity and Noether conserved quantity. This articleis emphatical for the specific researches on the Mei symmetry and Lie symmetry andconserved quantities deduced from the two symmetries.The study of symmetry and conserved quantity in Nielsen system was more mature, butwhich mainly is for the situation about the bilateral constraints, and little work has been doneto the system about the unilateral constraints and new conserved quantity.The research achievements of symmetry and conserved quantity for Appell system arerelatively little. In2008, Jia Li-Qun, Xie Jia-Fang and Zheng Shi-Wang studied Meisymmetry of Appell equation, and the structure equation and conserved quantity can bedirectly denoted as Appell function in this study. But this method has not got effectiveextension and application.Lagrange system is also an important one of the three mechanical systems. In order toimprove the theory of symmetry and conserved quantity in Lagrange system, we also need todo a lot of researches.The main purpose on the research of this article is further improved the theory ofsymmetry and conserved quantity in Nielsen system, made up shortages in Appell equations,obtained some important results, developed the theory of symmetry of Appell system, andperfected the theory of symmetry and conserved quantity for Lagrange system. The chaptersare arranged as following:In chapter1, there are three parts include: firstly, the cognitive process of symmetrytheory and its relationship with conserved quantity; secondly, the significance of symmetryand conserved quantity and their current research at home and abroad; thirdly, the keyproblems need to be solved and innovative achievements of this paper.In chapter2, some basic concept, the important symmetry methods and theory involvedin the research of this paper have been introduced.In chapter3, mainly study a type of new conserved quantity deduced from Meisymmetry for Nielsen equations in a holonomic system with unilateral constraints. Thedifferential equations of motion in Nielsen systems are established. The definitions and thecriterions of Mei symmetry under corresponding constraint of Nielsen equations are given.And the expressions of the structural equation and new conserved quantity of Mei symmetryare achieved.In chapter4, firstly study Lie symmetry and approximate Hojman conserved quantity ofAppell equations for a weakly nonholonomic system, and special Lie symmetry and Hojmanconserved quantity of Appell equations for a Chetaev nonholonomic system. Appell equations and the differential equations of motion are set up. The definitions and the criterions of Liesymmetry of Appell equations are given. And the expression of Hojman conserved quantitydeduced from Lie symmetry is obtained. Then we study conformal invariance and conservedquantity of Mei symmetry for Appell equations in holonomic system. We define Meisymmetry and conformal invariance for holonomic system. The determining equation of Meisymmetry and conformal invariance are obtained. And taking advantage of a structureequation that gauge function satisfies, the Mei conserved quantity is derived.In chapter5, mainly study Lie symmetry and approximate Hojman conserved quantity ofLagrange equations for a weakly nonholonomic system, and a type of the new exact andapproximate conserved quantity deduced from Mei symmetry for a weakly nonholonomicsystem. The differential equations of motion for system are established, and the definitions ofsymmetry for weakly nonholonomic systems and its first-degree approximate holonomicsystem are given. And then, the exact and the approximate conserved quantities deduced fromsymmetry are obtained.Finally, we summarize the innovation points, the method and the main work of this paper,and raise some future application and work that can continue to do in this field.