Longitudinal dynamics in storage rings

The single-particle equations of motion are derived 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 phi 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. A brief analysis including second-harmonic rf contributions is also given. The final sections supply calculations of dynamical quantities and particle simulations with the space-charge effects neglected