We elucidate the correspondence between Madey’s gain‐spread theorem for the free‐electron laser, and a similar theorem obtained for the Brownian motion of a stochastically driven oscillator. By using suitable changes of variables, these two different theorems can be shown to be special cases of a more general result valid for a mechanical system described by a Hamiltonian of the form H = H0(p,t)+λH1(q,p,t). For such a system, when terms up to second order in λ are kept, under certain specified conditions it follows that 〈Δp〉 = (1/2) (∂/∂pi) 〈(Δp)2〉, where Δp is the change in p in time Δt, and the average is over the initial value of the coordinate qi, for fixed time and initial momentum pi. In the case of the free‐electron laser, the canonical momentum p must be chosen as the total energy E of the electron, while for the driven oscillator, the necessary choice is the action variable J corresponding to the unperturbed periodic motion.

1.
J. M. J.
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64
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1979
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2.
N. M. Kroll, presented at the ONR Workshop on Free‐Electron Lasers at Sun Valley, Idaho, June 22–25, 1981 (unpublished).
3.
See, e.g.,
M. C.
Wang
and
G. E.
Uhlenbeck
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323
(
1945
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4.
See, e.g., H. Goldstein, Classical Mechanics (Addison‐Wesley, Mass., 1950).
5.
D. Boussard, G. Dome, and C. Graziani, Proceedings of the XIth International Conference on High‐Energy Accelerators, Geneva, 1980, edited by W. S. Newman (Birkhäuser Verlag Basel, 1980), p. 620.
6.
P. Luchini, C. H. Papas, and S. Solimeno (unpublished).
7.
A discussion of the inverse Hamiltonian formalism is also given in Appendix B of
E. D.
Courant
and
H. S.
Snyder
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Ann. Phys.
3
,
1
(
1958
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