1. Canonical Variables in MAD John Jowett J.M. Jowett, MAD-X meeting, 7/11/2013 1

2. Motivation Since the 1980s, MAD has used a set of canonical variables that are different from those found in accelerator physics textbooks Stems from a traditional separation between – transverse motion, treated by “optics” people, using s as independent variable and using normalised magnet strengths – longitudinal motion, treated by “RF” people, using t as independent variable J.M. Jowett, MAD-X meeting, 7/11/2013 2

3. Hamiltonian J.M. Jowett, MAD-X meeting, 7/11/2013 3

4. Coordinate system for cyclic accelerator Particles move in a neighbourhood of a reference trajectory (ideally a curve passing through centres of all magnets) defined by “bends”, etc J.M. Jowett, MAD-X meeting, 7/11/2013 4 Particle motion Radius of curvature Reference trajectory

5. Change of independent variable J.M. Jowett, MAD-X meeting, 7/11/2013 5 Azimuthal coordinate s can play role of time (independent variable) provided the particle never moves backwards (s is monotonic in t). Time t becomes the coordinate for the third degree of freedom (different particles pass s at different times).

6. Applied electromagnetic fields J.M. Jowett, MAD-X meeting, 7/11/2013 6

7. New Hamiltonian, the root of all evil in MAD J.M. Jowett, MAD-X meeting, 7/11/2013 7

8. Redemption by canonical transformation J.M. Jowett, MAD-X meeting, 7/11/2013 8

9. Equations of motion J.M. Jowett, MAD-X meeting, 7/11/2013 9 No mass terms in transverse motion! Slightly more complicated to compute the kicks imparted by cavities and these do involve the particle mass. Still no problem to implement, eg, in tracking.

10. Synchrotron motion J.M. Jowett, MAD-X meeting, 7/11/2013 10 For remainder of the development and how to introduce the optical and dispersion functions (and synchrotron radiation), see my old USPAS lectures, available as SLAC- PUB 4033 (1986) or US Particle Accelerator School, AIP Conference Proceedings, No. 153 (1985) which also include the synchrotron radiation terms. See also lectures by R. Ruth in same volum.

11. Conclusion One can have the benefits of working with “momentum deviations” in chromatic optics, without the messy terms containing mass, as usual (outside MAD) and treat synchrotron motion on the same footing – N.B. Transition energy emerges in a slightly different way Recommend: J.M. Jowett, MAD-X meeting, 7/11/2013 11