Phase space structure of 4D symplectic mappings
                 and applications to beam optics
                          Ezio Todesco

                     University of Bologna,
                    INFN Sezione di Bologna,
                              Italy
                e-mail: ETODESCO@bologna.infn.it

    The  analytical  tools  of resonant  normal  forms  for  four 
dimensional symplectic mappings are outlined. The analysis of the 
perturbative  series allows one to determine a classification  of 
the  resonant  structures in  phase  space.  In  particular,  the 
existence   and  the  stability  of  fixed  points  for   generic 
resonances and mappings is worked out. The analytical tools prove 
to  be  useful for the optimization of  a  magnetic  lattice.  An 
application  to the sorting of the magnetic random errors of  the 
LHC is outlined.