[MAP] Liouville's theorem and electromagnetic fields

[Kirk T McDonald]

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

--------------------------------------------------
From: "alex dragt" <dragtnb at comcast.net>
Sent: Thursday, March 10, 2011 9:23 PM
To: "Don Summers" <summers at phy.olemiss.edu>
Cc: "Robert D Ryne" <rdryne at lbl.gov>; "Yuri Alexahin" <alexahin at fnal.gov>; 
"Kirk T McDonald" <kirkmcd at Princeton.EDU>; "Alex Dragt" 
<dragtg5 at comcast.net>; "MAP List" <map-l at lists.bnl.gov>; "Alex Dragt" 
<dragt at physics.umd.edu>
Subject: Re: [MAP] Liouville's theorem and electromagnetic fields

> Dear all,
>
> The fact that Liouville's theorem holds in both mechanical and  canonical 
> phase space is also proved in
>
> A. Dragt
> SOLAR CYCLE MODULATION OF THE RADIATION BELT PROTON FLUX, J.
> Geophys. Res. 76: 2312-2344 (1971)
>
> also done in the context of the Van Allen Radiation, and hence for  motion 
> in the Earth's Magnetic Field.
>
> But we are interested in more than Liouville's theorem.  Also note  that 
> gauge transformations are symplectic maps, and hence do not  affect  the 
> eigen emitances.  See the book Lie Methods ... available  at the Web site
>
> http://www.physics.umd.edu/dsat/
>
> Best,
>
> Alex
>
>
> On Mar 10, 2011, at 3:55 PM, Don Summers wrote:
>
>> The exact reference for Swann's paper is
>>
>> W. F. G. Swann,  Application of Liouville's Theorem to Electron  Orbits 
>> in the Earth's Magnetic Field,
>> Phys. Rev. 44, 224–227 (1933)
>> http://prola.aps.org/abstract/PR/v44/i3/p224_1
>>
>> Best,
>> Don
>>
>>
>> On Thu, 10 Mar 2011 16:33:10 -0800, Robert D Ryne wrote
>>> I have not yet read the papers mentioned. But here are some brief
>>> comments. Alex Dragt and I (cc to Alex) have been thinking about
>>> this  a lot in the past months.
>>>
>>> The natural quantities to be computed are called "eigen-emittances."
>>> To compute them properly they need to be derived from a beam 2nd
>>> moment matrix, Sigma, formed using canonical variables.
>>> The eigen-emittances are invariant under linear symplectic
>>> transformations.
>>>
>>> The eigen-emittances can be computed in various ways, but the
>>> simplest  is to compute the eigen-values of J Sigma, where J is the
>>> fundamental  symplectic 2-form; the eigen-emittances are the modulii
>>> of the eigen- values of J Sigma (which are pure imaginary and in +/-
>>> pairs). If one  is interested in calculating the symplectic matrix
>>> that transforms  Sigma to Williamson normal form, Alex Dragt has an
>>> algorithm to do  this and has implemented it in the MaryLie code.
>>>
>>> Though the entries of Sigma will depend on the choice of gauge, the
>>> eigen-emittances themselves are gauge invariant. We can't just set
>>> the  vector potential to zero inside elements where it is nonzero,
>>> and  expect to calculate the correct eigen-emittances (as was
>>> suggested  below).
>>>
>>>>>>> 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 [WINDOWS-1252?]“longitudinal” coordinates (E,t)  rather 
>>>>>>> than (P_z,z).
>>>>>>>
>>>>>>> Is there any written reference that explains this  [WINDOWS-1252?]“well 
>>>>>>> [WINDOWS-
>> 1252?]known”
>>>>>>> fact?
>>>>>>>
>>>
>>> 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.
>>>
>>> Rob
>>>
>>> On Mar 10, 2011, at 4:29 PM, Yuri Alexahin wrote:
>>>
>>>> Hi Kirk,
>>>>
>>>> Thank you for digging out these interesting papers.
>>>> Of course the Poincare invariants remain the same no matter what
>>>> momenta are used.
>>>> But this is not what we calculate from tracking or measurement data
>>>> using standard definition.
>>>> So a clarification is still needed of what and how we should
>>>> calculate.
>>>>
>>>> Yuri
>>>>
>>>> ----- Original Message -----
>>>> From: Kirk T McDonald <kirkmcd at Princeton.EDU>
>>>> Date: Thursday, March 10, 2011 4:09 pm
>>>> Subject: [MAP] Liouville's theorem and electromagnetic fields
>>>> To: MAP List <map-l at lists.bnl.gov>
>>>> Cc: Kirk McDonald <kirkmcd at Princeton.EDU>
>>>>
>>>>
>>>>> Folks,
>>>>>
>>>>> There is a technical question as to how we should be calculating
>>>>> emittance for beams in electromagnetic fields.
>>>>>
>>>>> The formal theory of [WINDOWS-1252?]Liouville’s theorem is clear  that 
>>>>> the invariant
>>>>> volume in phase space is to be calculated with the canonical  momentum
>>>>> gamma m v + e A / c
>>>>> and not the mechanical momentum m v.
>>>>>
>>>>> This is awkward in two ways:
>>>>> 1.   We [WINDOWS-1252?]don’t always know the vector potential of  our 
>>>>> fields
>>>>> 2.   The vector potential is subject to gauge transformations, so
>>>>> canonical momentum is not gauge invariant.
>>>>>
>>>>> The second issue is disconcerting in that it suggests that phase-
>>>>> space
>>>>> volume, and emittance, are not actually invariant  -- with  respect to
>>>>> gauge transformations.
>>>>>
>>>>> Hence, it is useful to note a very old paper,
>>>>> W.F.G. Swann, Phys. Rev. 44, 233 (1933)
>>>>> which shows that the phase-space volume for a set of noninteracting
>>>>> particles is the same whether or not the term e A / c is included  in
>>>>> the [WINDOWS-1252?]“momentum”.
>>>>>
>>>>> This result has the consequence that phase-space volume (and
>>>>> emittance) is actually gauge invariant [WINDOWS-1252?]– although  the 
>>>>> location of a
>>>>> volume element in space space is gauge dependent.
>>>>>
>>>>> ---------------
>>>>> This suggests that we could simply calculate emittances based  only on
>>>>> the mechanical momentum, and avoid having to worry about the  accuracy
>>>>> of our model for the vector potential.
>>>>>
>>>>> Of course, our calculations are actually of rms emittance, which  is a
>>>>> better representation of the [WINDOWS-1252?]“ideal” emittance if  the 
>>>>> phase-space
>>>>> volume is more [WINDOWS-1252?]“spherical”, and not elongated/ twisted.
>>>>>
>>>>> It could be that the shape of the phase-space volume is better for
>>>>> rms
>>>>> emittance calculation if the vector potential, in some favored  gauge,
>>>>> is included in the calculation.....
>>>>>
>>>>> --Kirk
>>>>>
>>>>> PS  I have placed [WINDOWS-1252?]Swann’s paper as DocDB 560
>>>>> http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase
>>>>> user = ionization pass = mucollider1
>>>>>
>>>>> See also the paper by Lemaitre that used  [WINDOWS-1252?]Liouville’s 
>>>>> theorem for
>>>>> cosmic rays in the [WINDOWS-1252?]Earth’s atmosphere (using 
>>>>> mechanical momentum).
>>>>> This may well be the earliest paper about particle beams and
>>>>> [WINDOWS-1252?]Liouville’s theorem.
>>>>>
>>>>> 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
>>>>> [WINDOWS-1252?]“longitudinal” coordinates (E,t) rather than  (P_z,z).
>>>>>
>>>>> Is there any written reference that explains this  [WINDOWS-1252?]“well 
>>>>> [WINDOWS-
>> 1252?]known” fact?
>>>>>
>>>>> How is this prescription affected by electromagnetic fields?
>>>>>
>>>>> The vector potential of even a simple rf accelerating cavity has an
>>>>> A_z component (which is zero on axis, but nonzero off it).
>>>>> http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf
>>>>> Note that the vector potential is nonzero outside the cavity, even
>>>>> though the E and B fields are zero there!
>>>>>
>>>>> Do we know how to include A_z in our longitudinal emittance
>>>>> calculations?
>>>>> _______________________________________________
>>>>> MAP-l mailing list
>>>>> MAP-l at lists.bnl.gov
>>>>> https://lists.bnl.gov/mailman/listinfo/map-l
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>>
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>
>