[MAP] Liouville's theorem, kinematic invariants, and dynamic invariants

[Kirk T McDonald]

Alex,

Your latest email addresses mainly the important issue of whether we can/should be using some other form of approximation to emittance calculations than we are presently using.

I infer that the answer is NO, but that we should be aware of limitations of the scheme we do use.

Question: When one uses “rms emittance” calculations, are the results dependent on the choice of gauge, or whether or not we include the vector potential at all?

The vector potential is nonzero outside a closed fry cavity, but the cavity has no effect on the motion of particles when they are outside it.  If the “rms emittance” for the particles outside the cavity depends on the vector potential outside the cavity, that will continue to disconcert me.

Alexander argued, I believe, that in case the magnetic field is uniform and axial, the rms emittance is independent of whether or not the vector potential is included (which implies that this emittance is gauge invariant).

Does this happy result extend to arbitrary electromagnetic fields, as is the case of phase volume?

You say: “Moment invariants are currently known only for the case of linear maps, but there is some evidence that they should also exist in the  nonlinear case.”

Can I show my ignorance by asking what is a “linear map”?  I have the impression that transport through field-free vacuum is NOT an example of a “linear map”.

--Kirk

From: alex dragt 
Sent: Friday, March 11, 2011 4:24 PM
To: Sergei Nagaitsev ; Don Summers ; Yuri Alexahin 
Cc: Kirk T McDonald ; MAP List ; Robert D Ryne ; alex dragt ; srmane001 at gmail.com 
Subject: Liouville's theorem, kinematic invariants, and dynamic invariants

Dear All, 


     Confusion reigns:  As described in the papers


A. Dragt, F. Neri, et al., "LIE ALGEBRAIC TREATMENT OF LINEAR AND NONLINEAR BEAM DYNAMICS",
Annual Review of Nuclear and Particle Science 38, p. 455 (1988).


A. Dragt, R. Gluckstern, et al., "THEORY OF EMITTANCE INVARIANTS",  Lecture Notes in
Physics 343: Proceedings of the Joint US-CERN Capri School on Accelerator Physics,
Springer Verlag (1989).

A. Dragt, F. Neri, et al., "GENERAL MOMENT INVARIANTS FOR LINEAR HAMILTONIAN SYSTEMS", 
Physical Review A, 45, p. 2572 (1992).

there are two kinds of invariants, which I call "kinematic" and "dynamic".  It is important not to confound them.


     The Courant-Snyder invariants, generally treated in the linear approximation as described by Sergei below,  are examples of dynamic invariants.  Their use requires a knowledge of the equations of motion or, equivalently, a one-period or one-turn map.  (They generally apply to periodic systems for which the linear part of the map has all its eigenvalues on the unit circle and distinct, but in principle could also be constructed when the eigenvalues are off the unit circle.)  Contrary to the reservations that appear to be  expressed by Sergei, they are extendable to the nonlinear case using normal form methods.  See, for example, the first paper above and Sections 8.10 and 8.11 of the MaryLie manual available at the Web site


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


There are also papers in the literature that extend them, at least in the linear case, to non-periodic systems.


    Examples of kinematic invariants include what I call "moment invariants" and "eigen emitances".  They involve moments of particle distributions and, like Liouville's theorem, make no specific use of the equations of motion save for the symplectic condition.  It is these invariants that are the current focus of interest with regard to questions of "emittance partitioning".  See the draft paper


      arXiv:1010.1558v2 [physics.acc-ph] 


Their computation and use are described in the papers above and Section 8.37 of the MaryLie manual. 


     Moment invariants are currently known only for the case of linear maps, but there is some evidence that they should also exist in the  nonlinear case.  See Chapter 26 (only partially complete) and Section 6.8.2 of "Lie Methods ...", also available at the Web site above.  In the past I have tried to construct moment invariants for nonlinear maps by finding Casimir operators for the full Lie algebra of all symplectic maps.  The problem is difficult because the usual method for constructing Casimir operators requires that the Killing metric be invertible.  See Section 21.11 of "Lie Methods ...".  This metric is not invertible in the nonlinear case, and therefore the standard machinery for constructing Casimirs fails.  With considerable effort I found an alternate approach with some promising results, but was eventually overwhelmed by algebraic complexity.  Moment invariants for the nonlinear case, if they exist at all, will require many pages even to write them down.


     I hope these remarks provide some clarification.


Best,


Alex


On Mar 10, 2011, at 7:32 PM, Sergei Nagaitsev wrote:


  Dear Kirk,

  let me add my two cents to this discussion:

  1.  For the beam emittance to be a useful quantity, it needs to be conserved as the beam propagates along the beamline (consider non-interacting particles) through various external electromagnetic fields.  To this end, the definition of emittance that reflects such a property is "an ensemble-averaged action".  The particle motion is assumed to be integrable, i.e. there exists 3 functionally-independent constants of the motion in involution with the Hamiltonian.  Particle actions, expressed through such constants of motion, are also constants of motion.  Thus, the average actions (emittances) are conserved.  In a linear-focusing transport channel with linear rf focusing, such constants of motion exist and are called Courant-Snyder invariants (2 transverse and 1 longitudinal).  In a nonlinear (and generally time-dependent) focusing channel such constants of motion might not exist, therefore, the emittance (as a conserved quantity) is not defined.  We may use an approximate expression for the emittance by using linear-only focusing to define actions and then by treating non-linearities as a perturbation.  This leads to an effective emittance growth if a beamline has nonlinear elements even though the Liouville's theorem states that the phase-space density is conserved.  In some cases this emittance growth is not "real" (irreversible) but just a reflection that we are using an incorrect action definition.  Finally, my definition of the emittance (as the average action) is identical to the definition through eigen-values of a sigma matrix only in a case of a linear focusing channel.  However, where possible (like in case of a bunch occupying a large portion of an rf bucket) we should use exact actions, not approximate. 

  2. When averaging particle actions over the distribution function at a given time t it is useful to remember that a time slice t=const in one frame is not t'=const in another frame because of Lorentz transformations.

  Sergei  

  ----- Original Message -----
  From: Kirk T McDonald <kirkmcd at Princeton.EDU>
  Date: Thursday, March 10, 2011 8:49 pm
  Subject: Re: [MAP] Liouville's theorem and electromagnetic fields
  To: alex dragt <dragtnb at comcast.net>
  Cc: MAP List <map-l at lists.bnl.gov>



    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



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