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A necessary and sufficient condition for the stability of linear Hamiltonian systems with periodic coefficients. (arXiv:1810.03971v1 [math-ph])
来源于:arXiv
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
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