Let’s assume that the reader has a basic knowledge of classical and relativistic mechanics.
Consier a particle of mass m and speed v. If c is the speed of light, it is customary to define a normalized velocity $latex \beta $ as $latex \beta =v/c$, and a relativistic mass factor $latex \gamma $, defined as $latex \gamma=1/\sqrt(1-\beta^2)$.
Some other important definitions include the relativistic momentum of a particle, $latex p=\gamma mv$, the kinetic energy, $latex W=(\gamma-1)mc^2 $, the rest energy, $latex mc^2$, and the total energy, $latex U=W+mc^2=\gamma mc^2$. The nonrelativistic limit applied when $latex \beta <<1$.
It is often convenient to convert between velocity, energy, and momentum, and the following relationship are helpful. The conversion from velocity $latex \beta $ to kinetic energy W is
$latex \gamma=1/\sqrt(1-\beta^2), W=(\gamma-1)mc^2$
The inverse conversion is
$latex \gamma =(W+mc^2)/mc^2, \beta=\sqrt(1-1/\gamma^2)$
The following relationships between small differences are sometimes useful:
$latex \delta\gamma=\gamma^3\beta\delta\beta, \delta\gamma=\beta\delta(\beta\gamma), \delta W=mc^2\delta\gamma,\delta p=mc\delta(\beta\gamma)$.
Particle dynamics is obtained from Newton’s law relating the force and the rate of change of momentum:
$latex F=dp/dt=md(\gamma v)/dt$
For a particle of charge q in an electromagnetic ﬁeld, the Lorentz force on particle with charge q and velocity v in an electric ﬁeld E and a magnetic ﬁeld B, is given by
From book of RF linear accelerators, second edition.