Hamiltonian maps and normal forms for intense beams

The dynamics of a beam in a ring with a localized multipolar nonlinearity is described by a polynomial one turn map. The space charge forces act continuously along the ring, but their effect can be included by replacing the linear tune with the depressed tune which depends on the Courant Snyeder invariant. This approximation allows to use the normal forms to compute the nonlinear invariants, the nonlinear tune and the islands geometric parameters when a low order resonance is approached