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The dynamics of charged particles in general linear focusing lattices withquadrupole, skew-quadrupole, dipole, and solenoidal components, as well astorsion of the fiducial orbit and variation of beam energy is parameterizedusing a generalized Courant-Snyder (CS) theory, which extends the original CStheory for one degree of freedom to higher dimensions. The envelope function isgeneralized into an envelope matrix, and the phase advance is generalized intoa 4D symplectic rotation, or an U(2) element. The 1D envelope equation, alsoknown as the Ermakov-Milne-Pinney equation in quantum mechanics, is generalizedto an envelope matrix equation in higher dimensions. Other components of theoriginal CS theory, such as the transfer matrix, Twiss functions, and CSinvariant (also known as the Lewis invariant) all have their counterparts, withremarkably similar expressions, in the generalized theory. The gauge groupstructure of the generalized theory is analyzed. By fixing the gauge freedomwith a desired symmetry, the generalized CS parameterization assumes the formof the modified Iwasawa decomposition, whose importance in phase space opticsand phase space quantum mechanics has been recently realized. This gauge fixingalso symmetrizes the generalized envelope equation and express the theory usingonly the generalized Twiss function \beta.The generalized phase advancecompletely determines the spectral and structural stability properties of ageneral focusing lattice. For structural stability, the generalized CS theoryenables application of the Krein-Moser theory to greatly simplify the stabilityanalysis. The generalized CS theory provides an effective tool to study coupleddynamics and to discover more optimized lattice design in the larger parameterspace of general focusing lattices.

Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory