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.