The Study And Application Of Higher Order Lagrange Systems Of Modern Dynamics

The establishment of higher order Lagrange system deepen the depth and breadth of mechanics research. That Lagrange function is obtained and Lagrange equations is established make the study of higher order theory have a solid foundation. Under the given higher order system, we emphatically study the inverse problem of higher order system and Lie symmetry and other common mechanics method.In the beginning of the article, we reference and state previous research results that obtained the higher order Lagrange function and equations. The results can be as the basic function and equation for our research. Thus, it develop the article and study the corresponding problem.For the study of higher order system, we study the inverse problem of system firstly. Based on higher order equation of holonomic conservative system, the higher order Lagrange function is obtained by giving the self-adjoint conditions, generalizing the construction method of Lagrange function of second order system inverse problem and applied to higher order system. Moreover, a example is given to explain the application of the method.The mathematical graphics have the characteristics of the obvious. In the section of inverse problem, we introduce the MATLAB language and make the simple graphic. We use specific example to analyze graphic application. Generalized coordinates and Lagrange function are express as function of time t to make the graphics on time;. Then the relevant nature is analyzed specifically and compare with the classical results.Next, we study the Lie symmetry, Hojman theory and field method of higher order equation. In section of Lie symmetry, we introduce the k th order continuation formula of infinitesimal. According to the determining equation of Lie symmetry, the determining equation of higher order equation is obtained. The Noether-style conserved quantity of higher order equation is obtained by theory of Lie symmetry. In Hojman theory, it is changed as first order differential equations by transformed higher order Lagrange equation. Substituting into the classical Hojman theory, the Hojman conserved quantity of higher order equation is obtained. The field method is kind of effective and firsthand way. We apply the field method to higher order Lagrange equation and add diversity of solution of equation. We list part of examples as application of kinds of methods and intend to explain the using process. The article further research on inverse problem of higher order Lagrange mechanics and its graphic analysis after study of higher order Lagrange equation. The Lie symmetry, Hojman theory and field method are studied and get the corresponding conserved quantities. Moreover, the method is also suitable for low order situation and consistent with classical second order equation and universal.In the paper, the innovation is embodied in:The inverse problem of higher order system is obtained by using the method of comparison and expanding the classical method; To introduce the mathematical graphic analyze inverse problem and verify theoretical research; Apply the classical research method to higher order equation and obtain more general conclusions.