43
2.3. Conservation laws
This is called Liouville’s theorem. It means that ρ = const, and
the density of trajectory points in phase space is conserved.
Applying this to a beam, the geometry is shown schematically
in Figure 2.5. We imagine particles emitted from an infinitesimal
Figure 2.5: Geometry for brightness conservation.
area element dA into an infinitesimal solid angle element dΩ centered around the kinetic momentum vector p. The phase space
density is given locally by
d
6 N
ρ (x, P) =
= const,
(2.91)
d 3 x d 3 P
where
d
3 x = v cos θ dt dA,
d
3 P = p
2 dp dΩ,
(2.92)
and the scalar kinetic momentum p is related to the velocity v
by (2.18). Passing to the limit of an infinitesimally thin volume
element in the z-axis, the density ρ is a delta function in z. Integrating over all z, the result is unity, by the property of the delta
function. It follows that
1
dN
= const.
(2.93)
p 2 dp dA dΩ