Use of Lie Transforms to Generalize Madey's Theorem for Computing the Gain in Microwave Devices

For near-integrable Hamiltonian systems with a nonintegrable piece of order ε≪1, we use Lie transforms to derive a generalized form of Madey’s theorem. Specifically, we find an expression for the average second-order (in ε) change of any function of momentum in terms of first-order quantities only. A formalism is given that makes gain calculations for devices like free-electron lasers and gyrotrons in complicated geometries tractable. An explicit expression is presented for the case where the nonintegrable part of the Hamiltonian is a harmonic function of the coordinates. As an example, the average change in particle kinetic energy is computed through second order in the field amplitude for gyrotrons in complicated geometries. The transform method is extended to non-Hamiltonian systems, and it is shown that there is a class of non-Hamiltonian differential equations to which Madey’s theorem applies.