A necessary and sufficient condition for the stability of linear Hamiltonian systems with periodic coefficients

TL;DR

This paper establishes a precise criterion for the stability of linear Hamiltonian systems with periodic coefficients, linking stability to the existence of periodic solutions of an envelope matrix equation, with broad physical implications.

Contribution

It introduces a novel stability condition based on the envelope matrix equation and employs advanced mathematical tools like time-dependent canonical transformations and symplectic matrix decompositions.

Findings

Stability iff the envelope matrix equation admits a periodic solution.

Envelope matrix w(t) has a clear physical interpretation.

Mathematical framework applicable to quantum algorithms for Hamiltonian systems.

Abstract

Linear Hamiltonian systems with time-dependent coefficients are of importance to nonlinear Hamiltonian systems, accelerator physics, plasma physics, and quantum physics. It is shown that the solution map of a linear Hamiltonian system with time-dependent coefficients can be parameterized by an envelope matrix $w(t)$, which has a clear physical meaning and satisfies a nonlinear envelope matrix equation. It is proved that a linear Hamiltonian system with periodic coefficients is stable iff the envelope matrix equation admits a solution with periodic $\sqrt{w^{\dagger}w}$ and a suitable initial condition. The mathematical devices utilized in this theoretical development with significant physical implications are time-dependent canonical transformations, normal forms for stable symplectic matrices, and horizontal polar decomposition of symplectic matrices. These tools systematically decompose the dynamics of linear Hamiltonian systems with time-dependent coefficients, and are expected to be effective in other studies as well, such as those on quantum algorithms for classical Hamiltonian systems.