Transient and sustained resonance in very slowly varying oscillatory Hamiltonian systems with application to free-electron lasers

Slowly varying oscillatory systems occur frequently in both natural and man made applications, including the free electron laser. The long time solutions to these systems often prove difficult as numerical integration becomes too time consuming and inaccurate, and standard asymptotic techniques require large expansions not easily calculable by hand. This is especially true for the very slow time considered, which requires higher order expansions than the usual slow time. By formulating these systems into Hamiltonian standard form, the technique of averaging through a series of canonical transformations can be performed automatically by symbolic manipulation programs. When resonance is exhibited in these systems, the averaging technique results in N - 1 adiabatic invariants which are constants of the motion, and reduces the original system of 2N first order differential equations to two differential equations which embody the resonance behavior. For the very slow time considered, transient resonance (where the system makes a slow passage through the resonance) is characterized by an O(1) transfer in energy, which is extremely sensitive to initial conditions. Sustained resonance, also referred to as phase locking, is a strictly nonlinear phenomenon. A frequently occurring class of reduced problems is examined and a highly accurate asymptotic solution is found. The free electron laser is a device which operates in the resonance state; therefore the results for sustained resonance can be directly applied. A solution is found for the energy and phase of the relativistic electrons in the electron beam, which indicates significant design advantages for this case of very slow variation of the parameters.