The Maslov index and nondegenerate singularities of integrable systems

Abstract

We consider integrable Hamiltonian systems in with integrals of motion F = (F1, …, Fn) in involution. Nondegenerate singularities of corank one are critical points of F where rank dF = n − 1 and which have definite linear stability. The set of corank-one nondegenerate singularities is a codimension-two symplectic submanifold invariant under the flow. We show that the Maslov index of a closed curve is a sum of contributions ± 2 from the nondegenerate singularities it encloses, the sign depending on the local orientation and stability at the singularities. For one-freedom systems this corresponds to the well-known formula for the Poincaré index of a closed curve as the oriented difference between the number of elliptic and hyperbolic fixed points enclosed. We also obtain a formula for the Liapunov exponent of invariant (n − 1)-dimensional tori in the nondegenerate singular set. Examples include rotationally symmetric n-freedom Hamiltonians, while an application to the periodic Toda chain is described in a companion paper (Foxman and Robbins 2005 Nonlinearity 18 2795–813).