The interactions of energetic protons with whistlers propagating near the quasi-electro-static limit, are investigated using a test particle, Hamiltonian formalism. We assume that wave packets exist with finite bandwidths of frequencies, which are close to the equatorial electron gyrofrequency and propagate obliquely with respect to the geomagnetic field. Near the equator the protons interact with the waves which appear Doppler shifted to some harmonic of their cyclotron frequency. In an inhomogeneous geomagnetic field the spacing between cyclotron harmonic resonances is very small. The Hamiltonian equations of motion are solved including multiple, independent harmonics for each resonance. The wave frequency varies as a function of the distance along the field line, with only one frequency being resonant at a given point. Thus the inhomogeneity of the magnetic field is compensated by the frequency variation. The proton whistler interactions satisfy the conditions for second-order resonances for all the harmonics. The resonances may also overlap in phase space, leading to significant changes in the protons energies and pitch angles. The combined contributions of positive and negative harmonics allow protons to diffuse toward smaller pitch angles. Numerical calculations applying this formalism to parameters relevant to the plasmasphere and controlled VLF transmission experiments are presented. ¿ American Geophysical Union 1993