Positive paths in the linear symplectic group

TL;DR

This paper extends Krein's theory on positive paths in the symplectic group, analyzing eigenvalue behavior, path extensions, and their relation to stability and geometry, with implications for Hamiltonian systems and symplectic topology.

Contribution

It generalizes Krein's results by studying positive paths avoiding eigenvalue 1, and relates path index to group regions, especially in four dimensions.

Findings

Positive paths can be extended to have eigenvalues on the unit circle.

Eigenvalues of positive paths can move off the unit circle without encountering 1.

In 4D, the path index relates to regions crossed in the symplectic group.

Abstract

A positive path in the linear symplectic group $\Sp(2n)$ is a smooth path which is everywhere tangent to the positive cone. These paths are generated by negative definite (time-dependent) quadratic Hamiltonian functions on Euclidean space. A special case are autonomous positive paths, which are generated by time-independent Hamiltonians, and which all lie in the set $\Uu$ of diagonalizable matrices with eigenvalues on the unit circle. However, as was shown by Krein, the eigenvalues of a general positive path can move off the unit circle. In this paper, we extend Krein's theory: we investigate the general behavior of positive paths which do not encounter the eigenvalue 1, showing, for example, that any such path can be extended to have endpoint with all eigenvalues on the circle. We also show that in the case $2n=4$ there is a close relation between the index of a positive path and the regions of the symplectic group that such a path can cross. Our motivation for studying these paths came from a geometric squeezing problem in symplectic topology. However, they are also of interest in relation to the stability of periodic Hamiltonian systems and in the theory of geodesics in Riemannian geometry.