  In this paper we calculate and visualize the dynamics of an ensemble of
electrons trapping in an electrostatic wave of slowly increasing amplitude,
illustrating that, despite disordering of particles in angle during the
trapping transition as they pass close to X-points, there is still an adiabatic
invariant for the great majority of particles that allows the long-time
distribution function to be predicted. Possible application of this approach to
recent work on the nonlinear frequency shift of a driven wave is briefly
discussed.
