Matrix methods in periodic focusing systems

A linear differential equation with periodic driving matrix P in the n-dimensional phase space, the matrizant R(P) of this equation and an envelope matrix σ, representing the simultaneous transmission of an ensemble of trajectories, are considered. Three new n×n matrices are introduced: the oscillating antisymmetric matrix, the amplitude matrix and the phase orthogonal matrix, elements of which are derived as functions of the envelope and the driving matrices. The Courant–Snyder parametrization for n=2 in periodic systems is generalized to an arbitrary n. The generalized multiplicative representation of the matrizant R(P) is derived via the amplitude and phase matrices. For the particular case n=2 the Courant–Snyder representation is obtained.