Exact Maps to Second Order for Selected Elements in MAD-X Variables

Abstract

This report presents calculations for exact maps to second order about an arbitrary orbit for certain elements: drifts, solenoids, and rotations about a transverse axis. These expressions were used in recent updates to MAD-X, and thus the phase space variables used are those of MAD-X. Formulas are given here for the final orbit and its first and second derivatives with respect to the incoming phase space coordinates for a drift, solenoid, and coordinate system rotations about transverse axes. I do not claim these results to be new: these expressions are well-known and are presented here only for reference. This document does not collect all maps that could be expressed exactly, in particular dipoles are omitted.