To devise control strategies and to analyze the stability of systems with feedback, a set of few ordinary differential equations (ODEs) describing the underlying dynamics is required. It is deduced by combining two approaches not used in that context before: (I) Numerical Fourier-Hermite solutions of the Vlasov equation have been studied for over fifty years [1, 2]. 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. (II) 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 roughly checked against macro particle tracking simulations.

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