Combinatorics of linear stability for Hamiltonian systems in arbitrary dimension: Preliminaries

Table of Links

• Abstract
• Introduction
• Preliminaries
• The B-signature
• GIT sequence: low dimensions
• GIT sequence: arbitrary dimension
• Appendix A. Stability, the Krein–Moser theorem, and refinements and References

2. Preliminaries

In order to recall the definition of the GIT sequence, we need the following notion.

Definition 2.1 (GIT quotient). Let G be a group acting on a topological space X by homeomorphisms. The GIT quotient is the quotient space X//G defined by the equivalence relation x ∼ y if the closures of the G-orbits of x and y intersect, endowed with the quotient topology.

In particular, half of the symmetric periodic orbit is a Hamiltonian chord (i.e. trajectory) from Fix(ρ) to itself. Hence we can think of a symmetric periodic orbit in two ways, either as a closed string, or as an open string from the Lagrangian Fix(ρ) to itself.

The monodromy matrix of a symmetric orbit at a symmetric point is a Wonenburger matrix, i.e. it satisfies

where

equations which ensure that M is symplectic. The eigenvalues of M are determined by those of the first block A (see [FM]):

Theorem 1 (Wonenburger). Every symplectic matrix M ∈ Sp(2n) is symplectically conjugated to a Wonenburger matrix.

In other words, the natural map

is surjective.

In the presence of a symmetric periodic orbit, the above algebraic fact has a geometric interpretation: the monodromy matrix at each point of the orbit (a symplectic matrix) is symplectically conjugated via the linearized flow to the monodromy matrix at any of the symmetric points of the orbit (a Wonenburger matrix).