The Symmetry And Conserved Quantity And High-order Differential Motion Equations In The Mechanical System

The research of modern analytic mechanics mainly includes two aspects: First, the research of higher-order differential equations of motion; Second, the research of symmetry and conservation quantity of mechanical system.In view of the above two aspects content, we have done some meaningful fundamental research work and have made certain progress:1. By using Mathematica software to disclose the relationship between partial derivative and high-order derivative of different variables in the function r = r ( q ( t ),t), the symmetry of Yang Hui Triangle of the function r = r ( q ( t ),t) is found. Combining the symmetry of Yang Hui triangle with the Newtonian second law, the high-order differential equations of motion are deduced, which are satisfied by (n-1)-order force and n-1 order velocity energy. They describe (2n-m+1)-order differential equation of motion in mechanical system. With the further development of the ideal constraint concept, the high-order differential equation of motion under the holonomic ideal constraint system are discussed.2. The relationship between the high-order Lagrange function and generalized coordinates functional are found. Combining Lagrange equation, high-order differential equations of motion are deduced by using high-order Lagrange function. They described mechanical system's (n-m+ 3)-order differential equations of motion, and we take examples for their application.3. The vector of nonsimultaneous variation and differential and their n-order expansion under the infinitesimal transformation are abtained, The conserved quantity on mechanical system are deduced by using the relationship between differential and variation.4. The three-order Lagrange equations in different mechanical systems are discussed, and then the criteria of Noether symmetries and Noether conserved quantities for them are given. The criteria,structure equations and conserved quantities of Mei symmetry for three-order Lagrange equations are presented in both holonomic mechanic system and holonomic potential system. The relationship between Noether symmetry and Mei symmetry is also discussed. Finally, an exmaple is given to illustrate the application of the result.