| Focusing on the integral theory of symmetry and conserved quantity for constraint dynamical systems, mainly study the Noether symmetry and Noether conserved quantity,Lie symmetry and Hojman conserved quantity and Mei symmetry and the Mei conserved quantity in Lagrange systems,Nielsen systems and Appell systems.Now, about the symmetry and conserved quantity in Nielsen systems, the study in a dynamical system of the relative motion involves little.In 1899, Appell P, a famous French scholar, got Appell equations. Hereafter, Appell equations have become one of the three mechanical systems of analytical mechanics, and play an important role in the theories of analytical mechanics. In recent decades, though Chinese scholars have received important results in studying of Appell systems and their symmetries and conserved quantities, they have a broad developing space.Moveover, the study of the special symmetry and the Noether-Lie in Lagrange systems is not sound.This paper made up the questions of the symmetry and conserved quantity in Nielsen systems, developed the study of the symmetry and conserved quantity in Appell systems, and improved the shortages of the symmetry and conserved quantity in Lagrange systems.In chapter 1, a general discussion of the historical development, as well as the current research of the symmetries and conserved quantities at home and abroad. The significance, purpose and content of the research are also in this paper.In chapter 2, the author introduces the basic concept and basic theory of the research in this article.In chapter 3, mainly study some questions about the symmetry and conserved quantity in Nielsen systems.In chapter 4, mainly study some questions about the symmetry and conserved quantity in Appell systems.In chapter 5, mainly study some questions about the symmetry and conserved quantity in Lagrange systems.In chapter6, we summarize the main results of our research and envision the future research directions.
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