ADIABATIC INVARIANCE AND TRANSIENT RESONANCE IN VERY SLOWLY VARYING OSCILLATORY HAMILTONIAN-SYSTEMS

Canonical averaging techniques are applied to very slowly varying oscillatory systems in Hamiltonian standard form to very high orders, which are required for uniformly valid solutions. When resonance is exhibited in these systems N - 1 adiabatic invariants are found, reducing the original system of 2N first-order differential equations to two differential equations that embody the resonance behavior. Three examples exhibiting transient resonance are examined. Transient resonance occurs when the leading-order frequency of the reduced system makes a slow passage through zero. Depending on the rate of this slow passage, three distinguished cases are identified: the subcritical, the critical, and the supercritical. For each case, asymptotic solutions are found illustrating the nature of the resonance for certain classes of problems. In both the critical and supercritical cases, the action (and correspondingly the energy) can undergo changes of O(1) or greater. Specific examples are used to illustrate and numerically verify all results.