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 as , and a relativistic mass factor , defined as .
Some other important definitions include the relativistic momentum of a particle, , the kinetic energy, , the rest energy, , and the total energy, . The nonrelativistic limit applied when .
It is often convenient to convert between velocity, energy, and momentum, and the following relationship are helpful. The conversion from velocity to kinetic energy W is
The inverse conversion is
The following relationships between small differences are sometimes useful:
Particle dynamics is obtained from Newton’s law relating the force and the rate of change of momentum:
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.