[MAP] Liouville's theorem and electromagnetic fields

[Tom Roberts]

On 3/10/11 3/10/11 - 6:33 PM, Robert D Ryne wrote:
>>>>> PPS Scott Berg notes that when one evaluates emittance at a fixed plane in
>>>>> space, rather than at a fixed time, it is better to use the “longitudinal”
>>>>> coordinates (E,t) rather than (P_z,z).
>>>>> Is there any written reference that explains this “well known” fact?

The ECALC9 program uses (E,t) at fixed z; described in Rick Fernow's document:
     http://nfmcc-docdb.fnal.gov/cgi-bin/ShowDocument?docid=280
It does not delve into the underlying theory, however.

> The above follows directly from whether we use the time t as the independent
> variable or the Cartesian coordinate z as the independent variable. When using
> the time, the longitudinal variables are (z,p_{z,canonical}). When using z, the
> longitudinal variables are (t, -E) where t is arrival time at location z, and
> where E is the total energy of a particle when it reaches location z, i.e.
> E=\gamma m c^2 + q \Phi.

Yes. But I must probe a bit more deeply.

I believe that Hamiltonian dynamics inherently uses t as the independent 
variable, but when one considers a beam, its uniform velocity can be used to 
change the independent variable to z. I suspect this includes the assumption of 
a paraxial beam. Note that the beams in our cooling channel are not paraxial, 
and dx/dz can be as large as 0.25 (implying significant path-length 
differences). Moreover, dp/p can be as large as 20%, and we are in a regime 
where v/c is ~ 0.8, so momentum differences imply speed differences. These are 
rather different from typical high-energy beams, and they each imply quite large 
time differences at fixed z.

Is the use of z as independent variable rigorously correct, or does it involve 
approximation(s) that are not valid for the beams in our cooling channels?


Tom Roberts