[MAP] Liouville's theorem and electromagnetic fields

[Kirk T McDonald]

Tom,

I agree with Alex (and Scott and Rob Ryne) on this.

The formal consequence is that when measuring emittances at fixed z, rather 
than fixed t, the "momenta" should include the vector potential in the case 
of p_x and p-y (or p_r and p_phi) and should include the scalar potential in 
the case of
p_t = E_mech + e V.

Of course, for DC magnets, there is no scalar potential (in a "sensible" 
gauge).  Only for transport through RF cavities do we need to worry about 
the scalar potential.

I don't believe that ECALC9 includes the scalar potential, so this may be 
the cause of the numerical problems shown in some talks at JLAB that 
launched this entire thread.

Also, it may be that ECALC9 also leaves out the vector potential for rf 
cavities, and only includes it for DC magnets.

This brings us to the debate as to whether it's really necessary to include 
A and V in the emittance calculation -- although the strict Hamiltonian 
formalism says we should do this.

--Kirk

PS Assuming that we do include A and V as per the Hamiltonian formalism, we 
are left with the question of how to calculate the scalar (and vector) 
potential of an rf cavity.

Juan Gallardo suggests that we do this in the Poincare gauge, which has the 
merit of providing the most straightforward calculation of A and V from E 
and B in any gauge.

In other gauges, only generally has to take derivatives of E and B (to 
calculate rho and J = charge and current densities before one can calculate 
A and V.

In any gauge, A and V are nonzero both inside AND outside the rf cavity, 
even if E and B are zero outside the cavity.

All this is consistent from the point of view of Hamiltonian dynamics, but 
it is disconcerting to some of us that the emittance calculation depends on 
nonzero A and V in regions where the E and B fields have NO effect on the 
beam!

Examples of calculations of A and V for rectangular and cylindrical pillbox 
cavities are at
http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf
http://puhep1.princeton.edu/~mcdonald/examples/cavity.pdf
In practice, we would use knowledge of E and B of our RF cavities to make 
numerical computations of A and V for use in ECALC9

Note how A and V are both nonzero outside the cavities, where E and B are 
zero!



-----Original Message----- 
From: alex dragt
Sent: Monday, March 14, 2011 3:18 PM
To: Tom Roberts
Cc: Robert D Ryne ; Yuri Alexahin ; Kirk T McDonald ; MAP List ; alex dragt
Subject: Re: [MAP] Liouville's theorem and electromagnetic fields

The use of z is rigorously correct.  See "Lie Methods for Nonlinear
Dynamics with Applications to Accelerator Physics", downloadable from

http://www.physics.umd.edu/dsat/

In particular, see Section 1.6 and its associated Exercises.  See also
Sections 1.5 and 1.7 and their associated Exercises.  Note that there
is no assumption of uniform velocity.

Best regards,

Alex Dragt


On Mar 14, 2011, at 7:15 AM, Tom Roberts wrote:

> On 3/10/11 3/10/11 - 6:33 PM, Robert D Ryne wrote
>>>>>> 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 
>>>>>> “longitudinal”
>>>>>> coordinates (E,t) rather than (P_z,z).
>>>>>> Is there any written reference that explains this “well known”  fact?
>
> The ECALC9 program uses (E,t) at fixed z; described in Rick Fernow's 
> document:
>    http://nfmcc-docdb.fnal.gov/cgi-bin/ShowDocument?docid=280
> It does not delve into the underlying theory, however.
>
>> 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.
>
> Yes. But I must probe a bit more deeply.
>
> I believe that Hamiltonian dynamics inherently uses t as the  independent 
> variable, but when one considers a beam, its uniform  velocity can be used 
> to change the independent variable to z. I  suspect this includes the 
> assumption of a paraxial beam. Note that  the beams in our cooling channel 
> are not paraxial, and dx/dz can be  as large as 0.25 (implying significant 
> path-length differences).  Moreover, dp/p can be as large as 20%, and we 
> are in a regime where  v/c is ~ 0.8, so momentum differences imply speed 
> differences. These  are rather different from typical high-energy beams, 
> and they each  imply quite large time differences at fixed z.
>
> Is the use of z as independent variable rigorously correct, or does  it 
> involve approximation(s) that are not valid for the beams in our  cooling 
> channels?
>
>
> Tom Roberts
>