# Coordinate System and Hamiltonian for a Circular Accelerator

## Abstract

In this chapter, we derive the Hamiltonian for a particle moving in a circular accelerator. Our derivation uses several simplifying assumptions. First, we assume that there is no electrostatic field, \(\phi = 0\), and the magnetic field does not vary with time. Second, the magnetic field is arranged in such a way that there is a *closed reference* (or *nominal*) *orbit* for a particle with a nominal momentum \(p_0\)—this is achieved by a proper design of the magnetic lattice of the ring. We will also assume that this reference orbit is a *plane curve* lying in the horizontal plane. Our goal is to describe the motion in the vicinity of this reference orbit of particles having energies (or, equivalently, momenta) that can slightly deviate from the nominal one.

## References

- 1.J.M. Jowett, Introductory statistical mechanics for electron storage rings. AIP Conf Proc
**153**, 864–970 (1987)ADSCrossRefGoogle Scholar - 2.K.R. Symon, Derivation of Hamiltonians for accelerators. Report ANL/APS/TB-28, Argonne National Laboratory, 1997Google Scholar
- 3.A.W. Chao, M. Tigner,
*Handbook of Accelerator Physics and Engineering*, 3rd edn. (World Scientific Publishing, Singapore, 2006)Google Scholar - 4.E. Kreyszig,
*Differential Geometry*(Dover Publications, 1991)Google Scholar