The normal form of the Hamiltonian 1:3:4 resonance, which exhibits two simultaneous resonances of differing orders, is studied asymptotically. Since the two resonances have different strengths, the exact solution of the primary single resonance system may be used to construct an action-angle transformation. The resulting standard form system is solved asymptotically by canonical near-identity averaging transformations. In addition to the Hamiltonian itself and its unperturbed part, which are two exact constants of the motion, a third independent adiabatic invariant of the original Hamiltonian system is constructed. The results apply directly to the problem of a free electron laser with weak self-fields. A specific model problem is studied numerically to verify the asymptotic validity of the results over long times.
