[MAP] Liouville's theorem and electromagnetic fields

Chris Rogers chris.rogers at stfc.ac.uk
Mon Mar 14 13:20:15 EDT 2011


You can do either t or z as the independent variable. Indeed, you can
even use proper time and linearise your equations of motion if you
like... but you need 8d phase space then.

z as independent variable is standard for accelerators but has nasty
consequences...

Chris

Tom Roberts wrote:
> 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
> 
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