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

[alex dragt]

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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