The interaction of auroral electrons with kinetic Alfv¿n waves is complicated by the fact that the Alfv¿n speed above the ionosphere is strongly inhomogeneous, leading to a region often referred to as the ionospheric Alfv¿n resonator (IAR). For these waves, the wave-particle interaction must be treated with a nonlocal kinetic approach. Linear damping of these waves due to the wave-particle interaction has been calculated based on a quasi-dipolar field line model; however, particle orbits in the wave field are not well described by perturbation theory due to the development of new turning points in the particle trajectories, especially for low-energy particles. Full orbit calculations of such particles have been performed in order to assess the validity of the linear theory and to determine how much wave energy is converted into electron energy flux that is precipitated into the ionosphere. In addition, the precipitating electrons show a phase shift with respect to the field-aligned current in the waves, leading to a modification of theories of ionospheric feedback in the ionospheric Alfv¿n resonator.