  Polarized protons have never been accelerated to more than about $25$GeV. To
achieve polarized proton beams in RHIC (250GeV), HERA (820GeV), and the
TEVATRON (900GeV), ideas and techniques new to accelerator physics are needed.
In this publication we will stress an important aspect of very high energy
polarized proton beams, namely the fact that the equilibrium polarization
direction can vary substantially across the beam in the interaction region of a
high energy experiment when no countermeasure is taken. Such a divergence of
the polarization direction would not only diminish the average polarization
available to the particle physics experiment, but it would also make the
polarization involved in each collision analyzed in a detector strongly
dependent on the phase space position of the interacting particle. In order to
analyze and compensate this effect, methods for computing the equilibrium
polarization direction are needed. In this paper we introduce the method of
stroboscopic averaging, which computes this direction in a very efficient way.
Since only tracking data is needed, our method can be implemented easily in
existing spin tracking programs. Several examples demonstrate the importance of
the spin divergence and the applicability of stroboscopic averaging.
