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The Adiabatic Invariant of the Linear or Nonlinear Oscillator
1.H. R. Lewis, Jr., J. Math. Phys. 9, 1976 (1968);
1.H. R. Lewis, Jr., Phys. Rev. Letters 18, 510 (1967);
1.also an erratum: H. R. Lewis, Jr., Phys. Rev. Letters 18, 636 (1967).
2.E. D. Courant and H. S. Snyder, Ann. Phys. (N.Y.) 3, 1 (1958).
3.G. D. Birkhoff, Dynamical Systems (Am. Math. Soc. Colloquium Publ., New York, 1927), Chap. 3.
4.J. Moser, Nachr. Akad. Wiss. Goettingen, Math. Physik. Kl. IIa 6, 87 (1955).
5.In order to avoid a profusion of π’s we have chosen to define the action variable J so that it is the phase area of the corresponding ellipse divided by 2π. Our J, therefore, differs by a factor of 2π from the more customary definition. Likewise, the angle variable y is a true angle, increasing by 2π in one revolution.
6.A case of this type is discussed by R. W. B. Best, Physica 40, 182 (1968).
7.R. M. Kulsrud, Phys. Rev. 106, 205 (1957).
8.F. Hertweck and A. Schlüter, Z. Naturforsch. 12a, 844 (1957).
9.G. Backus, A. Lenard, and R. Kulsrud, Z. Naturforsch. 15a, 1007 (1960).
10.P. O. Vandervoort, Ann. Phys. (N.Y.) 13, 436 (1961).
11.J. E. Howard, “The Non Adiabatic Harmonic Oscillator,” submitted to Phys. Fluids.
12.The quantities A and B in Eq. (A4) differ by a factor from those in Eq. (25).
13.I am indebted to J. B. Taylor for pointing out this flaw in previous derivations of including my own first draft of this paper.
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