Moments and characteristic function of a nonstationary particle distribution after injection

This paper studies the injection process into a storage ring and presents an analytic model for the nonstationary particle distribution after mismatched or off-axis injection. The effects of nonlinear fields as well as the coupling to the longitudinal motion are described by analytic expressions for the first moments of the particle distribution. The result contains two distinct approximations: first, the Hamiltonian has been replaced by a Hamiltonian averaged over the phase variable; second, the characteristic function of the longitudinal distribution has been confined to the first two cumulants. Functions of moments that remain invariant for this averaged Hamiltonian are constructed.