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Polarized protons have never been accelerated to more than about $25$GeV. Toachieve polarized proton beams in RHIC (250GeV), HERA (820GeV), and theTEVATRON (900GeV), ideas and techniques new to accelerator physics are needed.In this publication we will stress an important aspect of very high energypolarized proton beams, namely the fact that the equilibrium polarizationdirection can vary substantially across the beam in the interaction region of ahigh energy experiment when no countermeasure is taken. Such a divergence ofthe polarization direction would not only diminish the average polarizationavailable to the particle physics experiment, but it would also make thepolarization involved in each collision analyzed in a detector stronglydependent on the phase space position of the interacting particle. In order toanalyze and compensate this effect, methods for computing the equilibriumpolarization direction are needed. In this paper we introduce the method ofstroboscopic averaging, which computes this direction in a very efficient way.Since only tracking data is needed, our method can be implemented easily inexisting spin tracking programs. Several examples demonstrate the importance ofthe spin divergence and the applicability of stroboscopic averaging.Comment: 39 page

Tracking algorithm for the stable spin polarization field in storage rings using stroboscopic averaging