[MAP] Use of x,y,t as Hamiltonian coordinates.

[Gregory Penn]

This has been a very interesting discussion.

It's easier for me to talk about the calculations in "emitcalc.for"
because that was written more recently.  I think the 4D and 6D
calculations are very similar to those in ECALC9.

Because all linear correlations are automatically removed in the 6D
calculation, and all linear transverse correlations are removed in the
4D calculation, any errors arise from nonlinear corrections to the
scalar potential (assumed to be zero) and vector potential (assumed to
be static and usually from a solenoid field, but can include high
order nonlinear terms).  There are certainly errors calculated within
an RF cavity because of the dynamic fields, and the nonlinearity will
be strong because typical beams have a time spread comparable to the
RF period.  However, what's important are the nonlinear terms in the
transverse vector potential, and I think those terms will usually lead
to weak corrections, with terms like A_r~r sin(omega t) and r^3
sin(omega t).  It would certainly be better to correct the transverse
momentum if the vector potential as a function of time is known, and
that could be added to the code.

Courant's invariant emittances (what Rob calls eigen emittances) are
calculated similarly and should have the same accuracy.  Removing the
nonlinear correlations of energy and transverse amplitude changes the
emittances, of course, but that feature can be turned off.

The emittance growth in a drift is more fundamental, there is always
going to be a nonlinear time-energy (and z-pz) correlation in a drift
when the gamma factor is only moderately bigger than 1.  See attached
PDF for a calculation.  For a long enough drift, the longitudinal
phase space will be boomerang-shaped rather than an ellipse.  One
could correct the energy as a function of time to remove this
correlation, but it does have real consequences especially for
transport using RF.  To do this automatically would mean also
removing, for example, the quadratic droop in energy resulting from
accelerating a beam on crest.

-Gregg

On Mon, Mar 14, 2011 at 12:55 PM, Kirk T McDonald <kirkmcd at princeton.edu> wrote:
> Rick,
>
> Thanks for this.
>
> It sounds to me like ECALC9 only includes a vector potential for DC magnets.
>
> For rf cavities, it sounds like neither the vector nor scalar potential are
> considered.
>
> This is inconsistent with the nominal rules of Hamiltonian dynamics.
>
> I am still trying to clarify whether we can get away with such apparent
> inconsistencies.
>
> --Kirk
>
> From: Fernow, Richard C
> Sent: Monday, March 14, 2011 1:51 PM
> To: MAP-l at lists.bnl.gov
> Subject: Re: [MAP] Use of x,y,t as Hamiltonian coordinates.
>
>
> Kirk,
>
>
>
> First let me say that the physics content of ECALC9 is entirely the work of
> Gregg Penn. I made a modified version ECALC9F that took its input from an
> input file instead of the keyboard and have tried to maintain it as problems
> arose over the years. I also wrote the Mucool note 280 to help me understand
> what Gregg was doing. Sometimes this isn’t very clear in the code. I don’t
> think FORTRAN was Gregg’s native language. Anyway looking over the code
> again this morning
>
>
>
> 1.      Greg calculates em6, emT, and emL using three different matrices.
>
> 2.      The variables for em6 are {x, y, t, px, py, E}. The momenta appear
> to be mechanical, i.e. the correction for Bz is not applied here.
>
> 3.      The variables for emT are {x, y, px, py}. The momenta also appear to
> be mechanical in this matrix. The statement I made about this at JLAB was
> apparently wrong.
>
> 4.      The variables for emL are {t, E, AT2}. The third variable is the
> square of the transverse amplitude.
>
> 5.      The only place where the on-axis field appears to be used is in
> calculating the canonical angular momentum and the transverse amplitude.
>
>
>
> Rick
>
>
>
>
>
> It is my understanding the ICOOL/ECALC presently uses the first of these
> transformations, I.e., it uses coordinates
>
> (x, y, t, p_mech_x+qA_x/c, p_mech_y+qA_y/c, E_mech)
>
>
>
> [Rick Fernow: Can you comment on this?]
>
>
>
> It now seems to me that this is actually OK in principle (although perhaps a
> somewhat "ugly" transformation)
>
>
>
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