  Beam matching is a common technique that is routinely employed in accelerator
design with the aim of minimizing beam losses and preservation of beam
brightness. Despite being widely used, a full theoretical understanding of beam
matching in 6D remains elusive. Here, we present an analytical treatment of 6D
beam matching of a high-intensity beam onto an RF structure. We begin our
analysis within the framework of a linear model, and apply the averaging method
to a set of 3D beam envelope equations. Accordingly, we obtain a matched
solution that is comprised of smoothed envelopes and periodic terms, describing
envelope oscillations with the period of the focusing structure. We then
consider the nonlinear regime, where the beam size is comparable with the
separatrix size. Stating with a Hamiltonian analysis in 6D phase space, we
attain a self-consistent beam profile and show that it is significantly
different from the commonly used ellipsoidal shape. Subsequently, we analyze
the special case of an equilibrium with equal space charge depression between
all degrees of freedom. Comparison of beam dynamics for equipartitioned, equal
space charge depression, and equal emittances beams is given. Finally, we
present experimental results on beam matching in the LANSCE linac.
