Geodesic Flow On The Quadratic Surface Spectrum

The most part of this paper comes from the article of J. Moser.Here we give an explanation in details about this article. The main object of the article of Moser is to establish a connection of some classical integrable Hamiltonian systems with the elementary geometry of quadrics. The classical approach to finding the relevant integrals was baosed on solving the Hamilton-Jacobi equation by separation variables. This required the appropriate choice of variables and conmputatianal skill. In the recent studies of partial differential equation the integrals were found as eigenvalues of some linear operators which depend on the solution of the partial differential equation but have the feature that their spectrum is conserved for each solution of the partial differential equation considered. Thus under the time evoluation of this equation the linear opertor changes in such a way that its spectrum remains fixed, i.e., it undergoes an isospectral deformation. The eigenvalues, viewed as functionals, repesent the integrals.The question arises naturally whether all integrable Hamiltonian systems can be described by isospectral deformation. We will not attemp to answer this question but consider some classical examples, such as Jacobi's geodesic flow on the ellipsoid, and construct an isospectral deformation for them. This does not lead to new results for this old problem, but to an interesting geometrical interpretation of the eigenvalues and eigenvectors of these operators.We begin with the introduction by Moser about the relation of Hamiltonian system and many other fields such as PDE, spectrum of linear operator and KdV equation etc. In section 1, some concepts are given. There are quadrics, geodesics on ellipsoid, geodesic flow, sympletic manifold, and spectrum of linear operator etc. In section 2, we study the spectrum of a linear opertor and the symmetric functions of the eigenvalues of L . By x|. = H_y ,y = -H_x, we get the isospectral deformation of L . In section 3, we disscuss the confocal quadrics. Similarly, we get the isospectral deformtion and give an interesting geometrical interpretation for the eigenvalues and eigenvectors of these operators. In the last section, we give some examples of integrable flows.