Capture into resonance: An extension of the use of adiabatic invariants

The theory of the adiabatic invariant predicts the long term evolution of mechanical systems with slowly varying parameters.Unfortunately, it is not valid when the system goes across a critical trajectory. This case is important because it can lead to capture into resonance.Analysing the motion in the vincinity of the critical trajectory, we are able to give general formulae for the probability of capture and to show that in general, the adiabatic invariant is conserved (allowance being made for the geometrical discontinuity in its definition at the critical orbit).