Hydrodynamic Approach to the Free Electron Laser Instability
  Starting from the Vlasov-Maxwell system, an exact relativistic hydrodynamic closure for a special type water bag distributions satisfying the Vlasov equation has been derived. It has been shown that the hydrodynamic equations are fully equivalent to the original Vlasov-Maxwell equations. The linear stability of the exact hydrodynamic closure has been studied as well. It has been shown that all basic features of the small signal gain can be derived from the fluid dynamic description developed here. Satisfactory agreement with previously reported results has been also found. 
Stephan I. Tzenov and Kiril B. Marinov
physics.acc-ph physics.flu-dyn physics.plasm-ph