Asymptotic Behaviours of Adiabatic Invariants

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Abstract
The asymptotic expansion of Liouville's distribution in a one-dimensional system is investigated. The expansion leads to the adiabatic theorem with respect to the action integral. The first order invariance with respect to the slowness parameter ε involved in the external distortion is explained in terms of the canonical mapping between two energy curves in the phase plane but at two different instants, which is possible to be found in the one-dimensional system between those two curves with a common area. The canonical mapping determines the mechanism of the external distortion, and also yields the second order invariance of mean action integral when the action integral is averaged over all possible phases at a fixed initial energy, if the distortion is imposed on, or released from, the system in a sufficiently smooth manner.

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