[MAP] Liouville's theorem and electromagnetic fields

[Kirk T McDonald]

Sergei,

Thanks for your thoughtful comments.

One theme of them is a different, and important issue from the ones that I 
have been raising.

Namely that depending one how one calculates "emittance" there can be 
apparent "emittance" growth in "nonlinear" transport systems.

Apparently a drift space in (field free) vacuum is an example of a 
"nonlinear" transport system.

To me, this "fact" seems ludicrous, and suggests to me that better working 
definitions of "emittance" can/should be found.

Do the folks in the "symplectic" world have any such better definitions, 
such that "emittance" is invariant for transport through field-free vacuum?

-----------------------
There remains the issue of whether or not its better to include the vector 
potential in the calculation of "emittance" when electromagnetic fields are 
present.  If the answer is YES, then which gauge should we use?

Note that the vector potential in many gauges is nonzero even in field-free 
regions!!!!

Since phase volume is gauge invariant, and also the same if we ignore the 
vector potential altogether, we have the option of changing gauges, or 
ignoring the vector potential altogether, in any particular calculation of 
emittance along a transport system.

If at one plane in the transport system there are electromagnetic fields 
from two different sources, I believe that we could use the vector potential 
for one source in one gauge, and use another gauge for the other source --  
as well as ignoring one or both the the vector potentials.

This is "gauge freedom".

In practice, what should we do to get the best numerical estimates of the 
invariant phase volume by our so-called "emittance" calculations?

--Kirk

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From: "Sergei Nagaitsev" <nsergei at fnal.gov>
Sent: Friday, March 11, 2011 12:32 AM
To: "Kirk T McDonald" <kirkmcd at Princeton.EDU>
Cc: "alex dragt" <dragtnb at comcast.net>; "MAP List" <map-l at lists.bnl.gov>
Subject: Re: [MAP] Liouville's theorem and electromagnetic fields

> Dear Kirk,
>
> let me add my two cents to this discussion:
>
> 1.  For the beam emittance to be a useful quantity, it needs to be 
> conserved as the beam propagates along the beamline (consider 
> non-interacting particles) through various external electromagnetic 
> fields.  To this end, the definition of emittance that reflects such a 
> property is "an ensemble-averaged action".  The particle motion is assumed 
> to be integrable, i.e. there exists 3 functionally-independent constants 
> of the motion in involution with the Hamiltonian.  Particle actions, 
> expressed through such constants of motion, are also constants of motion. 
> Thus, the average actions (emittances) are conserved.  In a 
> linear-focusing transport channel with linear rf focusing, such constants 
> of motion exist and are called Courant-Snyder invariants (2 transverse and 
> 1 longitudinal).  In a nonlinear (and generally time-dependent) focusing 
> channel such constants of motion might not exist, therefore, the emittance 
> (as a conserved quantity) is not defined.  We may use an approximate 
> expression for the emittance by using linear-only focusing to define 
> actions and then by treating non-linearities as a perturbation.  This 
> leads to an effective emittance growth if a beamline has nonlinear 
> elements even though the Liouville's theorem states that the phase-space 
> density is conserved.  In some cases this emittance growth is not "real" 
> (irreversible) but just a reflection that we are using an incorrect action 
> definition.  Finally, my definition of the emittance (as the average 
> action) is identical to the definition through eigen-values of a sigma 
> matrix only in a case of a linear focusing channel.  However, where 
> possible (like in case of a bunch occupying a large portion of an rf 
> bucket) we should use exact actions, not approximate.
>
> 2. When averaging particle actions over the distribution function at a 
> given time t it is useful to remember that a time slice t=const in one 
> frame is not t'=const in another frame because of Lorentz transformations.
>
> Sergei
>
> ----- Original Message -----
> From: Kirk T McDonald <kirkmcd at Princeton.EDU>
> Date: Thursday, March 10, 2011 8:49 pm
> Subject: Re: [MAP] Liouville's theorem and electromagnetic fields
> To: alex dragt <dragtnb at comcast.net>
> Cc: MAP List <map-l at lists.bnl.gov>
>
>
>> Folks,
>>
>> I have added Alex' paper to DocDB 560.  See Appendix A.
>>
>> It is gratifying to see that the fact that Liouville's theorem holds
>> for
>> both mechanical and canonical phase space is "well known to those who
>> know".
>>
>> The challenge now is to learn how best to use the "freedom" offered to
>> us by
>> this apparently nonintuitive result.
>>
>> --Kirk
>>
>> --------------------------------------------------
>
>