Direct Vlasov solvers

In these proceedings we will describe the theory and practical steps required to build Vlasov solvers such as those commonly used to compute coherent instabilities in synchrotrons. Thanks to a Hamiltonian formalism, we will derive a compact and general form of the linearized Vlasov equation, written using Poisson brackets. This in turn will be the basis of a procedure to build Vlasov solvers, applied to the specific example of transverse instabilities arising from beam coupling impedance.