Hill Formula And Its Application In Mathieu Equation And Elliptic Lagrangian Equation

This paper studies mainly the eigenvalues of the monodromy matrix of the periodic orbits of Mathieu equation and Lagrangian equation with Hill formula for the S-periodic orbits in Hamiltonian systems, subsequently infers the condition of the existence of the periodic solutions and the linear stability of the solutions.There are five chapters in the paper. In Chapter1, we introduce the historical background,development of Hill formula,some recent results of Mathieu equation and Lagrangian equation and our main work in this paper. In Chapter2, we introduce some definitions and basic knowledge.In Chapter3,we elaborate on the research method,and preliminary calculate the Hill formula which will be used in the latter. In Chapter4, we obtain the condition of the existence of the periodic solutions of Mathieu equation by using Hill formula. In Chapter5, we obtain the spectrum of the monodromy matrix of the periodic orbits of Lagrangian equation with equal mass with Hill formula.The main idea of this paper is to calculate the eigenvalues of the monodromy matrix of the periodic orbits with Hill formula of the S-periodic orbits in the2dimen-sionality Hamiltonian systems, and to get the property of the solutions through an analysis of the eigenvalues. Because Hill formula relates a conditional Fredholm de-terminant, the corresponding differential operator and the characteristic polynomial of the monodromy matrix for the periodic orbits.The key technology is to calculate the corresponding conditional Fredholm determinant which would be written to be a determinant of a infinite matrix, and we will get the value by MATLAB. What more, some diagrams will be shown in this paper. The condition of the existence of the periodic solutions of Mathieu equation is consistent with before but it is a different formula. And the eigenvalues of the monodromy matrix of the periodic orbits of Lagrange equation is turn out to be a positive real number, which is consistent with theoretical result.