[MAP] Liouville's theorem and electromagnetic fields

[alex dragt]

The use of z is rigorously correct.  See "Lie Methods for Nonlinear  
Dynamics with Applications to Accelerator Physics", downloadable from

http://www.physics.umd.edu/dsat/

In particular, see Section 1.6 and its associated Exercises.  See also  
Sections 1.5 and 1.7 and their associated Exercises.  Note that there  
is no assumption of uniform velocity.

Best regards,

Alex Dragt


On Mar 14, 2011, at 7:15 AM, 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
>