Busch´s theorem for mappings

For rotation-invariant Hamiltonian systems, canonical angular momentum is conserved. In beam optics, this statement is known as Busch´s theorem. This theorem can be generalized to symplectic mappings; two generalizations are presented in this paper. The first one states that a group of rotation-invariant mappings is identical to a group of the angular-momentum preserving mappings, assuming both of them symplectic and linear. The second generalization of Busch´s theorem claims that for any laminar beam which rotation symmetry happened to be preserved, an absolute value of the angular momentum of any particle from this beam is preserved as well; the linear symplectic mapping does not have to be rotation-invariant here