This chapter gives an overview of the longitudinal dynamics of the particles
in an accelerator and, closely related to that, the issue of synchronization
between the particles and the accelerating field. Beginning with the trivial
case of electrostatic accelerators, the synchronization condition is explained
for a number of driven accelerators like Alvarez linacs, cyclotrons and finally
synchrotrons and storage rings, where it plays a crucial role. In the case of
the latter, the principle of phase focusing is motivated qualitatively as well
as on a mathematically more correct level and the problem of operation below
and above the transition energy is discussed. Throughout, the main emphasis is
more on physical understanding rather than on a mathematically rigorous
treatment.Comment: 15 pages, contribution to the CAS-CERN Accelerator School: Ion
Sources, Senec, Slovakia, 29 May - 8 June 2012, edited by R. Bailey,
CERN-2013-00