Geodesic
Simple proof that an adiabatic invariant is conserved to exponential accuracy over a~complete interval of development
Teoretičeskaâ i matematičeskaâ fizika, Tome 66 (1986) no. 1, pp. 146-149

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The Bogolyubov–Zubarev fast-phase method [1] is used to give a simple proof of the following well-known proposition [2]: when a perturbation is switched on and off smoothly, the total increment of the adiabatic invariant (the action) for one-dimensional periodic motion in a slowly varying potential is a quantity less than any power of $\alpha$, where $\alpha$ is a parameter that characterizes the rate of change of the potential. 
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     author = {A. N. Vasil'ev and M. A. Guzev},
     title = {Simple proof that an adiabatic invariant is conserved to exponential accuracy over a~complete interval of development},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {146--149},
     publisher = {mathdoc},
     volume = {66},
     number = {1},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1986_66_1_a10/}
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A. N. Vasil'ev; M. A. Guzev. Simple proof that an adiabatic invariant is conserved to exponential accuracy over a~complete interval of development. Teoretičeskaâ i matematičeskaâ fizika, Tome 66 (1986) no. 1, pp. 146-149. http://geodesic.mathdoc.fr/item/TMF_1986_66_1_a10/