Conservation Laws for Dynamical System Via Differential Variational principles

Abstract

The purpose of this thesis is to study the conservation law of a dynamical system which is an important research area in the investigation of analytical dynamics.The great impor- tance of the conservation law is to reduce the order of the di erential equation of motion. To nd the conservation we can used two methods, the rst one is based on the invariance of Hamilton's integral and the other is di erential variational principle. In this thesis we used the di erential variational principle to nd the conservation law of a dynamical system. To nd the conservation law of a dynamical system we will face two problems, either the motion of the dynamical system is under the ideal constraints or non-ideal constraints. For this purpose, we divides this thesis into four chapters. The rst three section of chapter one are devoted to the fundamental concepts of analytical dynamics including the notion of constraints and the generalized coordinates. Then the idea of variable mass is presented with some examples. After that the idea of fundamental and synchronous virtual variation (actual and virtual) is introduced and the exchange rule between actual and the virtual variation is established. In the chapter second, we calculated the conservation law of a non-conservative dy- namical system for Jourdian and Gauss principle under the ideal constraint, in which the virtual work is zero. Chapter third deals with the calculation of conservative law of a dynamical system for the D'Alembert principle under the non-ideal constraints. Besides the conservation law we also nd the magnitude of reaction force on the dynamical system. In this chapter we used ideas of supplementary virtual displacement and supplementary generalized coordinates. In the last chapter we developed the theory of conservation law for non conservative dynamical system of the Jourdian and Guass di erential variational principle under the non ideal constraints.