Approximate matched solution of an intense charged particle beam propagating through periodic quadrupole focusing lattice

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 needs to determine a dynamical invariant for the beam particles moving in the combined applied and self-generated fields. In this paper, we present a perturbative Hamiltonian transformation method which is an expansion in the particle's vacuum phase advance $\epsilon=\sigma_v/2\pi$, treated as a small parameter, which is used to transform away the fast particle oscillations and obtain the average Hamiltonian accurate to order $\epsilon^3$. The average Hamiltonian 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. In the third-order, the average self-field acquires a octupole component which results in the average motion of some beam particles being non-integrable and their trajectories chaotic. This chaotic behavior of the beam particles may significantly change the nature of the Landau damping (or growth) of collective excitations supported by the beam.