Generalizations of the neumann system a curve‐theoretical approach‐part I

Abstract

AbstractThe following observation, due to E. Trubowitz, illustrates an intimate relationship between spectral theory and Hamiltonian mechanics in the presence of constraints. Let q(s) be a real periodic function such that the Hill operator, magnified image has only a finite number gR of simple eigenvalues. There exist gR + 1 periodic eigenfunctions x1,…, x and corresponding eigenvalues a1,…, a of L such that magnified image where yr = dxr/ds. The equations Lxr = arxr, r = 1,…,gR+1, make up the classical Neumann system, a system of harmonic oscillators constrained to the unit sphere.H. Flaschka obtained similar results about the Neumann system from a more general point of view. His assumption, that there exists an operator of odd order that commutes with L, leads to algebraic curve theory by the method of Krichever and from there to the Neumann formulas above. The familiar Lax pairs, the constants of motion and the quadrics of the Neumann system emerge as consequences of the Riemann‐Roch theoreni. The existence of isospectral deformations of L, the Korteweg de‐Vries hierarchy of the soliton equations, underlies the complete integrability of the Neumann system. This paper extends Flaschka's techniques, replacing L by an operator of order n 2 2. Higher Neumann systems are defined in a way that leads naturally to interesting symplectic manifolds, Lax pairs and integrals of motion. C. Tomei, using scattering theory, obtained some of our n = 3 formulas.