Calculation of radiant energy in free electron lasers without inversion

The equations of motion of particles in free electron lasers without inversion (FELWI) were obtained using the Hamilton formalism. An one-dimensional decoupled phase equation of the type of mathematical pendulum equation was derived in the case of weak signal. For practical estimates, this equation was solved together with the equation of electron energy change within the frameworks of perturbation theory, and expressions for gain were obtained in FEL regime and for the output angle from the first undulator. Expressions for the gain coefficient, the phase and the energy change were found as functions of the parameters of beam and of the device.