Using Symbolic Algebra for the Generation of Orbit Simulation Codes from Hamiltonians

As a part of our new research activity aiming at a detailed understanding of space charge effects in ring cyclotrons and in the corresponding injection beam lines at the Paul Scherrer Institute we are currently developing a three dimensional space charge simulation code. The use of the Hamiltonian formalism, in combination with a symbolic algebra system (MAPLE), enables us to carry out the entire modelling on a high level. MAPLE easily handles the elaborate derivatives required to form the equations of motion and then casts them into FORTRAN. The subsequent embedding of the FORTRAN code into the MATLAB package is also handled automatically by the procedures of our framework. MATLAB is well suited for running small simulations followed by various post processing activities including graphics. In order to test this framework, two model cases have been chosen: the double focusing spectrometer and the motion of a single particle in a stable charged cloud. The successful processing of these and some further small model problems encourages us to apply this framework to produce code for the large scale simulation. The development steps from the Hamiltonian to the FORTRAN subroutine an the resulting simulations are shown for two model cases, as well as the general MAPLE and MATLAB procedures we have developed for this purpose.