[MAP] Correction (Re: Hamiltonian-gauge potentials

[Pavel Snopok]

Rob,

This is exactly what I tried to convey both at the MAP meeting, but 
maybe not so successfully, and in my later message you might not have 
received, since it was only sent to the MAP mailing list. Let me forward 
it to you next. I also performed a nonlinear drift exercise for 
eigen-emittances, and sure enough they grow too in the nonlinear case.

Pavel

On 3/17/2011 21:38, Robert D Ryne wrote:
> What I wrote below is not correct. As Kirk just pointed out to me, and
> Alex wrote earlier, it is true only in the linear approximation.
> As Alex wrote earlier,
>> But the eigen emittances are invariant under linear symplectic
>> transformations, and therefore also under gauge transformations in the
>> linear approximation.
>
> Rob
>
>
> On Mar 17, 2011, at 4:29 PM, Robert D Ryne wrote:
>
>> Kirk,
>>
>>> On Mar 17, 2011, at 3:24 PM, Kirk T McDonald wrote:
>>> It remains that rms emittance (and eigen-emittance, I believe) is not
>>> gauge invariant. But we have to choose some gauge to proceed.
>>
>> The eigen-emittances /are/ gauge invariant.
>>
>> More precisely, consider a numerical distribution of particles in the
>> presence of electromagnetic fields. These can be defined, e.g., using
>> Cartesian coordinates and velocities at some time, or using transverse
>> coordinates and velocities along with energy and arrival time at some
>> location z, etc. Next, express the numerical distribution using
>> /canonical/ variables. Some elements of the 6-vector describing each
>> particle /will/ depend on your choice of gauge (e.g. p_{x,canonical}
>> and p_{y,canonical}). Next, compute the beam second moment matrix
>> \Sigma. Some of the matrix elements /will/ depend on your choice of
>> gauge. Next, compute the moduli of the eigenvalues of J \Sigma (these
>> are the eigen-emittances). Though the numerical distribution (the
>> collection of 6-vectors) and the \Sigma matrix will depend on the
>> gauge, the eigen-emittances will be /independent/ of the choice of gauge.
>>
>> Rob
>>
>>
>> On Mar 17, 2011, at 3:24 PM, Kirk T McDonald wrote:
>>
>>> Folks
>>> I believe that what Alex Dragt has been trying to tell us is that for
>>> applications involving Hamiltonian dynamics it is most favored to use
>>> the so-called Hamiltonian gauge, in which the scalar potential V is
>>> zero everywhere. Further, this ties in nicely with the use of
>>> coordinates (x,y,t) rather than (x,y,z).
>>> For a general expression for the vector potential A in this gauge,
>>> see sec. 8 of
>>> http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_917_02.pdf
>>> For fields with time dependence e^{- i omega t), with nonzero wave
>>> number k = omega / c, then the vector potential is simply
>>> A = – i E / k. (Gaussian units).
>>> Note that A = 0 wherever E = 0, in which regions B can only be static.
>>> For a static magnetic field, the vector potential is the same as that
>>> in the Coulomb gauge (and in the Lorenz gauge). [In some static
>>> cases, such a toroidal magnets, the static vector potential will be
>>> nonzero in regions where B is zero; but in regions where a Fourier
>>> component B_omega is zero for nonzero omega, A_omega will be zero also.]
>>> A static electric field has the vector potential
>>> A = – c(t – t_0) E.
>>> This seems a bit odd, but will not bother us.
>>> An unusual feature of the Hamiltonian gauge vector potential is that
>>> is is not continuous at a perfectly conducting surface – which did
>>> bother me for quite a while.
>>> However, in using A to compute B, the (ill-defined) normal derivative
>>> at a perfectly conducting surface is not needed to deduce the
>>> tangential B.
>>> Of course, the Hamiltonian gauge vector potential makes the canonical
>>> momentum p = p_mech + q A / c
>>> discontinuous at the surface of the rf cavity.
>>> This is troublesome if we use coordinates (x,y,z) and have an
>>> accelerating cavity with E_z, such that p_z takes a step on entering
>>> or leaving the cavity.
>>> However, if we switch to coordinates (x,y,t) the t-canonical momentum is
>>> p_t = – E_mech – q V = – E_mech
>>> in the Hamiltonian gauge, which is continuous at the cavity wall.
>>> Also, canonical momenta p_x and p_y are continuous if the particles
>>> enter and leave the cavity through faces at constant z (of a good
>>> conducting material).
>>> All this is serendipitous for the cooling sections of a muon
>>> collider, where we are almost certain to use cavities with flat (Be)
>>> faces where the particles enter and exit.
>>> It remains that rms emittance (and eigen-emittance, I believe) is not
>>> gauge invariant. But we have to choose some gauge to proceed.
>>> It now looks like the Hamiltonian gauge is the one to use, along with
>>> coordinates (x,y,t) rather than (x,y,z).
>>> 1. Time-dependent A_omega follows immediately from knowledge of
>>> E_omega, and is zero where E_omega and B_omega are zero.
>>> 2. Static A is same as in the Coulomb gauge (which is what we almost
>>> always use now).
>>> 3. No steps in canonical momenta so long as all rf cavities have flat
>>> faces where particles enter/exit.
>>> 4. No scalar potential to worry about, so p_t = – E_mech, as in our
>>> present software. (It doesn’t hurt anything to define p_t = + E_mech,
>>> as we actually do.)
>>> ---------------------------------------------------------
>>> 5. Although phase volume is the same where or not we include the
>>> potentials in the momenta, this is not true for rms emittance (or
>>> eigen-emittance). So we should stop using ECAL9, and switch to
>>> emitcalc (with a better approximation to the vector potential) and/or
>>> equivalent programs developed by Tom Roberts, Chris Rogers, et al.
>>> --Kirk
>>> _______________________________________________
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>>
>
>
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