The Maslov index and nondegenerate singularities of integrable systems

TL;DR

This paper investigates the Maslov index in integrable Hamiltonian systems, linking it to nondegenerate singularities and stability, and provides formulas for the index and Lyapunov exponents with applications to specific models.

Contribution

It establishes a new relationship between the Maslov index and nondegenerate singularities in integrable systems, extending classical index formulas to higher dimensions.

Findings

Maslov index sums contributions from enclosed singularities

Formula for Lyapunov exponent of invariant tori

Application to rotationally symmetric Hamiltonians and Toda chain

Abstract

We consider integrable Hamiltonian systems in R^{2n} with integrals of motion F = (F_1,...,F_n) in involution. Nondegenerate singularities are critical points of F where rank dF = n-1 and which have definite linear stability. The set of 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 is 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 Poincar\'e 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.