Equations of motion and Hamiltonian for synchrotron oscillations and synchro-betatron coupling

摘要

We discuss the equations of motion and Hamiltonian for synchrotron oscillations and synchro-betatron coupling while taking into account the localized nature of RF cavities. We start with differential and difference equations for the phase and energy equations, respectively. We derive approximate differential equations as well as difference equations for mapping from the basic equations. The Hamiltonian formalism developed in this note will help readers to understand the equations of motion from a more formal point of view. It also helps to derive the equations of motion for synchro-betatron coupling. The effect of betatron acceleration is also taken into account in the formalism. Synchrotron radiation effects are only briefly mentioned. Wake-field effects are not treated. This note is based on lectures given at the Accelerator Department of the Graduate University for Advance Studies attached to KEK, but has been extended to include the Hamiltonian formalism. (author)