• H. Qin, R.C. Davidson
  PPPL, Princeton, New Jersey

Funding: Supported by the U.S. Department of Energy.

Courant-Snyder (CS) theory is generalized to the case of coupled transverse dynamics with two degree of freedom. The generalized theory has the same structure as the original CS theory for one degree of freedom. The four basic components of the original CS theory, i.e., the envelope equation, phase advance, transfer matrix, and the CS invariant, all have their counterparts, with remarkably similar formal expressions, in the generalized theory presented here. The unique feature of the generalized CS theory is the non-commutative nature of the theory. In the generalized theory, the envelope function is generalized into an envelope matrix, and the envelope equation becomes a matrix envelope equation with matrix operations that are not commutative. The generalized theory gives a new parameterization of the 4D symplectic transfer matrix that has the same structure as the parameterization of the 2D symplectic transfer matrix in the original CS theory.

• E. Startsev, R.C. Davidson, M. Dorf
  PPPL, Princeton, New Jersey

Funding: Research supported by the U. S. Department of Energy.

The transverse dynamics of an intense charged particle beam propagating through a periodic quadrupole focusing lattice is described by the nonlinear Vlasov-Maxwell system of equations where the propagating distance plays the role of time. To find matched-beam quasi-equilibrium distribution functions one need to determine a dynamical invariant for the beam particle moving in the combined external and self-fields. The standard approach for sufficiently small phase advance is to use the smooth focusing approximation, where particle dynamics is determined iteratively using the small parameter (vacuum phase advance)/(360 degrees) < 1 accurate to cubic order. In this paper, we present a perturbative Hamiltonian transformation method which is used to transform away the fast particle oscillations and obtain the average Hamiltonian accurate to 5th order in the expansion parameter. This average Hamiltonian, expressed in the original phase-space variables, is an approximate invariant of the original system, and can be used to determine self-consistent beam equilibria that are matched to the focusing channel.