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

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.