7 research outputs found
The Maslov index and nondegenerate singularities of integrable systems
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.Comment: 27 pages, 1 figure; published versio
Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain
The n-particle periodic Toda chain is a well known example of an integrable
but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold
singularities of the Toda chain, ie points where there exist k independent
linear relations amongst the gradients of the integrals of motion, coincide
with points where there are k (doubly) degenerate eigenvalues of
representatives L and Lbar of the two inequivalent classes of Lax matrices
(corresponding to degenerate periodic or antiperiodic solutions of the
associated second-order difference equation). The singularities are shown to be
nondegenerate, so that Sigma_k is a codimension-2k symplectic submanifold.
Sigma_k is shown to be of elliptic type, and the frequencies of transverse
oscillations under Hamiltonians which fix Sigma_k are computed in terms of
spectral data of the Lax matrices. If mu(C) is the (even) Maslov index of a
closed curve C in the regular component of R^{2n}, then (-1)^{\mu(C)/2} is
given by the product of the holonomies (equal to +/- 1) of the even- (or odd-)
indexed eigenvector bundles of L and Lmat.Comment: 25 pages; published versio
Maslov Indices and Monodromy
We prove that for a Hamiltonian system on a cotangent bundle that is
Liouville-integrable and has monodromy the vector of Maslov indices is an
eigenvector of the monodromy matrix with eigenvalue 1. As a corollary the
resulting restrictions on the monodromy matrix are derived.Comment: 6 page