Hamiltonian Formulation of Beam Dynamics

Abstract

Particles in electromagnetic fields behave like oscillators and are conservative systems as long as we ignore statistical effects like the emission of quantized photons in the form of synchrotron radiation. Even in such cases particles can be treated as conservative systems on average. In particular, particle motion in beam transport systems and circular accelerators is oscillatory and can be described in terms of perturbed oscillators. The Hamiltonian formalism provides a powerful tool to analyze particle motion and define conditions of stability and onset of instability. In this chapter we will derive the Lagrangian and Hamiltonian formalism with special consideration to particle beam dynamics.