Hamiltonians will be expanded to fourth order in the variables.

In sections 3, 4, and 5, special forms of the Hamiltonian will be derived for transverse horizontal and vertical motion, for longitudinal motion, and for synchrobetatron coupling of horizontal and longitudinal motions. In section 2, formulas will be derived for the vector potentials that describe the various lattice components. He switches to a form more convenient for most purposes using the distance s along the reference orbit as independent variable. He then starts from the general Hamiltonian for a particle in an electromagnetic field, using the accelerator coordinate system, with time t as independent variable. The author will use the usual accelerator coordinate system, which will be introduced first, along with a list of handy formulas. The generic term accelerator will be used to refer to any of these devices. Except where noted, the treatment will apply generally to linear and circular accelerators, storage rings, and beamlines. In this report various forms of the Hamiltonian for particle motion in an accelerator will be derived. Applications include easy calculation of the transfer functions of clusters of beam guiding elements and of generalized field gradients for any multipole contribution up to any order. Methods for incorporating the obtained field in both analytical and numerical computation of transfer functions are outlined. Relative errors in the field approximation do not exceed the maximal relative errors in the input data.

A boundary element method is presented to fit the fields to a discrete set of field data, obtained, for instance, from field measurements, on the two-dimensional surface. Magnetic field data on a two-dimensional surface, e.g., a cylindrical surface or median plane, serve as input for the calculation of the three-dimensional magnetic field. In the field description, pseudodifferential operators described by Bessel functions are used to obtain the various multipole contributions. It is also both divergence free and curl free, and takes fringe field effects up to any desired order into account. The presented description of the magnetic field differs from the classical approach in that it does not rely on power series approximations. It can be applied to single beam guiding elements as well as to clusters of elements. The method relates the calculation to measurement data of the magnetic field in a direct way.

A new method for the calculation of the magnetic field of beam guiding elements is presented.

Fringe field effect