Beam dynamics in #accelerators usually assumes your bunch has a finite temperature.  But what if your bunch has zero temperature?  This is the case for transporting ultra-low emittance bunches (Coulomb crystals) from ion traps.  The usual "beta functions" don't really work, so I developed equations for the 3D fully-coupled dynamics in the note below.  I'm hoping it can be used to do stability analysis (in 21 parameter space, so many ways it can go unstable!)   #physics 
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<br /><a href='https://stephenbrooks.org/ap/report/2025-9/model_order1_analytic.pdf'>https://stephenbrooks.org/ap/report/2025-9/model_order1_analytic.pdf</a>