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 √ w † 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.

Linear beam stability in periodic focusing systems: Krein signature and band structure

“…(40). Therefore, the algebraic meaning of the Krein signature is the sign of the action of an eigenmode [2,19]. Some studies interpret the Krein signature as the sign of mass [34] or the sign of energy [35].…”

“…The detailed proofs of the above properties are given in Refs. [1,2,35]. As the parameters of a stable periodic-coefficient linear Hamiltonian system vary, the multipliers move on the unit circle and may collide with each other.…”

“…The general question of how a beam becomes unstable remains the central theme for accelerator physics. Many fundamental stability problems in accelerator physics are described by (homogenous) linear differential equations with periodic coefficients [1,2]. The periodic coefficients are associated with the periodicity of the accelerators.…”

“…[9,10]. However, to introduce the concept of Krein signature and band structure in this study, we only focused on two basic cases where the state variables evolve according to dz ds = K(s)z, K(s) = K(s + L), (2) and the solution is provided in terms of a 4 × 4 transfer matrix M (s) as [11] z(s) = M (s)z 0 ,…”

“…[1]). For a simple system with one degree of freedom, a linear differential equation with a periodic coefficient takes the form of a harmonic oscillator with a periodic spring constant expressed as [2] d 2 x(s) ds 2 + κ(s)x(s) = 0,…”