Liouville theorem and related forms

2 Liouville theorem and related forms The Helmholtz-Lagrange relation given in equ. (4.46) is related to many other forms which all state certain conservation laws (the Clausius theorem, Abbe s relation, the Liouville theorem). The most important one in the present context is the Liouville theorem [Lio38] which describes the invariance of the volume in phase space. The content of this theorem will be discussed and represented finally in a slightly different form which allows a new access to the luminosity introduced in equ. (4.14). 

The applications of the Helmholtz-Lagrange relation given in equ. (4.46) to an arbitrary conjugate object-image pair characterized by left (/) and right (r), respectively, leads to 

The product y a is called the emittance, and when divided by a factor of n it is termed the normalized emittance. Equ. (10.39) states that the (normalized) emittance remains constant, provided an ideal optical system, i.e., without intensity losses, is given. 

In the context of conservation relations another form will be added. If equ. (10.39b) is squared, one can introduce an area A to replace y2, a solid angle 2 to replace a2 and a so-called brightness B which is proportional to (see below). This gives 

In optical radiometry with plane solid surfaces the product 2, at the lens input represents the luminosity Lopt  

---