Non-Linear Canonical Equations of Coupled Synchro-Betatron Motion and Their Solution within the Framework of a Non-Linear 6-Dimensional (Symplectic) Tracking Program for Ultra-Relativistic Protons

摘要

Starting from the Lagrangian of a charged particle in an electromagnetic field, the Hamiltonian for non-linear coupled synchro-betatron oscillations of ultra-relativistic charged particles (protons) is derived. The canonical variables are x, psub(x), z, psub(z), sigma, eta which are well-known from the six dimensional linear theory (SLIM). Keeping only terms up to second order in the canonical momenta psub(x), psub(z), the equations of motion are then solved for various kinds of magnets (quadrupole, skew quadrupole, bending magnet, synchrotron-magnet, solenoid, sextupole, octupole, dipole kicker) and for cavities, taking into account the effect of energy deviation on the focusing strength. The equations so derived can serve to develop a non-linear, six dimensional (symplectic) tracking program for ultra-relativistic protons. (ERA citation 11:028064)