Chirped resonance dynamics in phase space

The dynamics of passage and capture into resonance of a distribution of particles driven by a chirped frequency perturbation is discussed. The resonant capture in this case involves crossing of the separatrix by individual particles and, therefore, the adiabatic theorem cannot be used in studying this problem no matter how slow the variation of the driving frequency is. It is shown that, if instead of analysing complicated single particle dynamics in passage through resonance, one considers the slow evolution of a whole distribution of initial conditions in phase space, the adiabaticity and phase space incompressibility arguments yield a solution to the resonant passage problem. This approach is illustrated in the case of an ensemble of electrons driven by a chirped frequency wave passing through Cherenkov resonances with the velocity distribution of electrons.