No Access Published Online: 14 August 1998
Journal of Applied Physics 53, 5453 (1982);
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  • S. Krinsky
  • J. M. Wang
  • P. Luchini
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.
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  1. © 1982 American Institute of Physics.