Application of Lie-algebraic maps to the orbit dynamics of cyclotrons

A simplified model of the magnetic field of a cyclotron is used to analyse the orbit dynamics in a Lie-algebraic framework. The basic Lie-algebraic methods are outlined, including the powerful normal-form method in the context of a perturbation expansion. Analytic results are obtained for the motion of the central trajectory as well as the radial and vertical tunes. form methods are used to investigate both non-resonant and resonant behaviour. It is demonstrated that for vx = 1, all radial non-linearities vanish, leaving only the chromaticities and the axial aberrations. The one third integer resonance is investigated, including the iterated sextupole term to octupole order. y, analytic results were obtained for the decentering resonance, which although generally pathological, is universally used in compact cyclotrons as an aid to extracting the beam.