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

[Kirk T McDonald]

Alex,

Thanks for that useful clarification.  I now agree that Rob's slide 16 is correct.

This leads to several other comments.

If you find this tedious, but are willing to consider some of the argument, skip to item 7.

1.  With H = E_mech + q V,
the partial derivative on the diagonal of the Jacobian of the transformation from 
canonical coordinates
(x, y, t, p_mech_x+qA_x/c, p_mech_y_qA_y/c, H=E_mech+qV)
to (noncanonical) coordinates
(x, y, t, p_mech,x, p_mech,y, E_mech)
is
partial H / partial E_mech = 1.

Hence, the determinant of the Jacobian is 1, which means that the phase volume is the same whether or not one includes the potentials A and V in the calculation, using either t or z as the independent variable.   This a the generalization of Swann's "theorem".

So, in principle, we are free to leave out all consideration of the potentials in calculation of phase volume, and its approximate representation by emittance!

There remains the vexing issue of which potentials, I.e., which gauge, to use such that the numerical value of the calculated emittance is the best representation of the underlying phase volume.

2.  If we choose to include the potentials A and V in the calculation, we must do so consistently.

However, we can give a corollary to Swann's theorem which indicates that in principle we have a lot of flexibility.

For example, if we decide to change from the canonical coordinates
(x, y, t, p_mech_x+qA_x/c, p_mech_y+qA_y/c, H=E_mech+qV)
to
(x, y, t, p_mech_x+qA_x/c, p_mech_y+qA_y/c, E_mech)
or to
(x, y, t, p_mech_x+qA_x/c, p_mech_y, H=E_mech+qV)
or to
(x, y, t, p_mech_x, p_mech_y+qA_y/c, H=E_mech+qV)
or to
(x, y, t, p_mech_x+qA_x/c, p_mech_y, E_mech)
or to
(x, y, t, p_mech_x, p_mech_y+qA_y/c, E_mech)
or to
(x, y, t, p_mech_x, p_mech_y, H=E_mech+qV),
in every case the determinant of the Jacobian is 1, so phase volume is invariant under all 6 of these transformations [as well as under the original transformation to coordinates
(x, y, t, p_mech,x, p_mech,y, E_mech) ].

[Similarly, if we use t as the independent variable, we obtain 7 versions of Swann's theorem, for a total of 14 versions so far.]

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)

3.   If we have potentials due to several different charge/current sources, I believe that it is OK to use different gauges for the different sources, if this proves to be computationally convenient.

It is less clear to me that we have the freedom to include the potentials from some sources, but to leave then out for others.

4.  There is a special gauge called the Hamiltonian gauge in which the scalar potential is everywhere zero.

See sec. 8 of 
http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_917_02.pdf

user: archive
pass alpha137

However, for time-dependent fields it is not so easy to calculate the vector potential in this gauge.   So, I doubt that we will use it much.

[For static electric fields, V = 0 and A = - E t.    For static magnetic fields the vector potential is the same as the Coulomb gauge potential (which is the same as the Lorenz gauge potential for this case).]

5.  If we include the potentials in our emittance calculations, we will have to learn to live with the fact that the potentials in most gauges can be nonzero in regions where the fields E and B are zero.   In these regions, where the particle motion is unaffected by E and B, the emittance calculation will nonetheless depend on the potentials.   In principle, this should make no difference -- as we should get the same value for the emittance even if we neglect the potentials altogether.

6.  Juan Gallardo points out that if the E and B fields are known, the simplest calculation of corresponding potentials is in the so-called Poincare gauge.

See sec. 9 of Jackson's paper linked above.

In this calculation, the potentials depend on the E and B fields only along a line from the origin to the point in question.

This means that the value of the potentials depends on the choice of the origin.

If the origin is in a region of nonzero E or B field, we immediately see that the potentials in the Poincare gauge are nonzero throughout all space.

Juan and I have been making a few toy analytic calculations of potentials in the Poincare gauge to get some experience as to how it works:
http://puhep1.princeton.edu/~mcdonald/examples/cavity.pdf
http://puhep1.princeton.edu/~mcdonald/examples/solpot.pdf
http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf
http://puhep1.princeton.edu/~mcdonald/examples/axial.pdf
[This is a work in process.  These notes are still being updated.

A price of using the Poincare gauge to simplify the calculation of the potentials will be that they must be calculated over the entire volume where our beams go.

The emittance calculation in the final cooling section must include the vector potential from the magnets back at the target station, etc.

I believe this complexity can be mitigated by choosing different origins to calculate the potentials from different field regions.

7.  All this discussion emphasizes how it would be simpler if we just omit the potentials when calculating the emittance.

Swann's "theorem" shows that this is OK for phase volume, but it may not be OK for calculations of rms emittance.

Challenge:  Give versions of Swann's theorem for rms emittance calculations.  Clarify whether the theorem holds only for 6D emittance, or for 4D and 2D subemittances as well.

Alexander Shemyakin claims that this has been done for the 4D transverse rms emittance for beams inside an ideal DC solenoids, but perhaps not for general electromagnetic fields.   Apparently, Alexey Burov does not believe the theorem holds even for this special case.

However, I get the impression from comments by Alex Dragt that rms emittances, are not, in general, invariants with respect to the independent variable of the Hamiltonian, be that either t or z.   So it may well be that rms emittances are not invariant under gauge transformations, or under transformations from canonical to noncanonical variables (I.e., neglecting the potentials).

Worry:  If rms emittances are not gauge invariant, then we have a whole new class of issues to deal with.

--Kirk 



From: alex dragt 
Sent: Saturday, March 12, 2011 11:52 AM
To: Kirk T McDonald 
Cc: MAP List ; Robert D Ryne ; alex dragt 
Subject: Re: [MAP] Use of x,y,t as Hamiltonian coordinates.


Dear Kirk,   


     If you employ (1.5.29) and (1.5.30) in my book, you will find that


H=\gamma mc^2+q\psi.


Then. from (1.6.5), it follows that


p_t=-H=-\gamma mc^2-q\psi,


in agreement with Rob's slide 16.


Best,


Alex


If you 


On Mar 11, 2011, at 8:10 PM, Kirk T McDonald wrote:


  Rob,

  Thanks for this comment.

  Your slide 16 seems to imply that the canonical momentum associated with coordinate t, when using (x,y,t) as coordinates, is
  p_t = - (E_mech + q V).

  This does not quite match what I infer from Alex Dragt that

  p_t = - H
  = - { sqrt[ m^2 c^4 + (p_mech - q A / c)^2 ] + q V }

  How did you arrive at your simplification?

  Your result matches Dragt's if the vector potential is zero.....

  Are you saying that we can ignore the vector potential, but not the scalar potential?

  --Kirk


  From: Robert D Ryne 
  Sent: Saturday, March 12, 2011 12:07 AM
  To: Kirk T McDonald 
  Cc: alex dragt ; MAP List 
  Subject: Re: [MAP] Use of x,y,t as Hamiltonian coordinates.


  Kirk, 


  The 6-vector of canonical variables is shown on slide 16 of my presentation at the MAP meeting. These are the variables that should be used for eigen-emittance calculations. Of course this happens "for free" in a code that uses canonical variables. When I calculate eigen-emittances from a non-canonical code, the diagnostic subroutine does the conversion to canonical variables.


  Rob





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