  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.
