Orbit-averaged quantities, the classical Hellmann-Feynman theorem, and the magnetic flux enclosed by gyro-motion
Action integrals are often used to average a system over fast oscillations and obtain reduced dynamics. It is not surprising, then, that action integrals play a central role in the Hellmann-Feynman theorem of classical mechanics, which furnishes the values of certain quantities averaged over one period of rapid oscillation. This paper revisits the classical Hellmann-Feynman theorem, rederiving it in connection to an analogous theorem involving the time-averaged evolution of canonical coordinates. We then apply a modified version of the Hellmann-Feynman theorem to obtain a new result: the magnetic flux enclosed by one period of gyro-motion of a charged particle in a non-uniform magnetic field. These results further demonstrate the utility of the action integral in regards to obtaining orbit-averaged quantities and the usefulness of this formalism in characterizing charged particle motion.
© 2015 AIP Publishing LLC. Received 14 August 2014; accepted 23 December 2014; published online 3 February 2015. This material was based upon work supported by the U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences under Award Nos. DE-FG02-04ER54755 and DE-SC0010471, by the National Science Foundation under Award No. 1059519, and by the Air Force Office of Scientific Research under Award No. FA9550-11-1-0184. We thank John Preskill for pointing out the connection to the Aharonov-Bohm effect.
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