Following the Lagrangian formalism, the conjugate momenta of the spatial coordinates x, are defined as

with у, = dx,/dt. The electromagnetism Lagrangian reads [187]

L = m0)c2(1 — (1 — ф2)1/2) — дф + qv ■ A

where ф is the electric scalar potential and A the magnetic vector potential. From equations (III.43) and (III.44) the conjugate momentum vector is obtained:

(III.45)

The six-dimensional space (x, p) is the phase space. The Liouville theorem states that, in an energy conservative system, the density in the phase space is invariant. In other words, particles initially enclosed in a small phase volume dV = dx • dp remain in an equal volume throughout their motion. If A can be considered to be constant over the phase volume, then dx • dP is also conserved. It follows that the normalized emittances defined in equations (III.42) are such that

ex. ey. ez = constant. (III.46)

If the coupling between the transverse and longitudinal motions can be neglected, then

ex. ey = constant. (III.47)

In many instances,[70] in particular when fields are locally linear, coupling between the two transverse coordinates can also be neglected, and thus both transverse emittances are constant:

axagx = constant; (III.48)

emittances are usually expressed in mm-mrad.