Average amplitude and phase detuning near driven 3rd integer resonance

For resonant slow extraction of accelerated particle beams from synchrotrons, a 3rd order resonance driven by sextupole magnets is commonly used. This note reviews the non-linear particle dynamics under these conditions as described by the Kobayashi Hamiltonian, focussing on the amplitude and phase dependent detuning. Using the Kobayashi Hamiltonian as the invariant, an expression for the action and the average detuning as a function of the Hamiltonian is derived and compared to numerically obtained results.