Longitudinal dynamics in storage rings

We derive the single-particle equations of motion for charged particles in a storage ring. Longitudinal space charge is included in the potential assuming an infinitely conducting circular beam pipe with a distributed inductance. The framework uses Hamilton's equations with the canonical variables Φ and W. The Twiss parameters for longitudinal motion are also defined for the small amplitude synchrotron oscillations. The space-charge Hamiltonian is calculated for both parabolic bunches and 'matched' bunches. We also give a brief analysis that includes second-harmonic rf contributions. In the final sections, we calculate dynamical quantities and particle simulations with the space-charge effects neglected. (orig.)