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For a harmonic oscillator with time varying coefficients a relation is obtained between the action integral and the first order adiabatic invariant. It is shown that a small resonant perturbation can modify the invariant. A canonical transformation generates a new action variable constant to first order in the perturbation with the old action-angle variables playing the role of oscillatory momentum and extension. There are two separate ways in which the new adiabatic constant can break down, leading either to a modified invariant or to a situation in which no invariant exists. A general theory is developed to use the procedure for multidimensional systems. It is shown that two distinct types of resonances are possible that lead to qualitatively different results. The general theory is applied to a coupled oscillator system in forms demonstrating both types of resonances. A canonical transformation is used to remove the more important type of resonance, and an average is performed over a time comparable to the slower period of the oscillator. Numerical integrations of both the averaged and exact Hamiltonian equations show the maximum value of the perturbation for which the averaging process is valued. The system is found to oscillate slowly about resonance but, as long as the averaging process is valid, this slow oscillation is adiabatically separated from the faster motion of the oscillator by the existence of an adiabatic invariant. When averaging is not valid, a higher order resonant coupling occurs between the slow drift about resonance and the motion of the oscillator. The canonical transformation as described in the general theory is made near a particular higher order resonance to obtain the oscillation in the adiabatic invariant. For oscillations about neighboring resonances which do not interact strongly a new approximate invariant exists. For strong interaction the breakup of the invariant curves is demonstrated and the strength of the perturbation necessary for breakup is compared with a simple analytic criterion.
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