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We derive the first order phase space dependence of spin-orbit motion of aparticle in an accelerator by expanding the Thomas-BMT equation. Differentforms can be found in the literature and we show how these are related, andcare is taken to include fringe fields. The advantages of using certain formsis demonstrated by a detailed re-derivation of the spin matching conditions byV. Ptitsyn for the spin rotators in the Electron Storage Ring (ESR) of theElectron-Ion Collider (EIC) at Brookhaven National Laboratory.

Different forms of first order spin-orbit motion and their utility in spin matching in electron storage rings