Paper 
Title 
Page 
WEPEA010 
Modeling Longitudinal Bunched Beam Dynamics in Hadron Synchrotrons using Scaled FourierHermite Expansions 
2513 

 K. Gross, D.E.M. Lens
TU Darmstadt, RTR, Darmstadt, Germany



To devise control strategies and to analyze the stability of systems with feedback, a set of few ODEs describing the underlying dynamics is required. Numerical FourierHermite solutions of the Vlasov equation have been studied for over fifty years. The idea to expand the distribution function in Fourier series in space and Hermite functions in velocity is transferred to the dynamics of bunched beams in hadron synchrotrons in this contribution. The Hermite basis is a natural choice for plasmas with Maxwellian velocity profile as well as for particle beams with Gaussian momentum spread. The Fourier basis used for spatially nearly uniform plasmas has to be adapted to bunched beams where the beam profile is not uniform in phase. This is achieved analogously to the deduction of the three term recursion relations to construct orthogonal polynomials, but applied to Fourier series with the weight function taken from the Hamiltonian. The resulting system of ODEs for the expansion coefficients of desired order  dependent on the number of functions retained  is checked against macro particle tracking simulations.


