[MAP] Liouville's theorem and electromagnetic fields

Sergei Nagaitsev nsergei at fnal.gov
Fri Mar 11 00:32:13 EST 2011


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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