Forces in a thin cosine(n{theta}) helical wiggler

We commence with the derivation of the Lorentz force density on a surface of discontinuity based on the expressions of fields and currents previously derived (Appendix A). Applying such Lorentz body forces to the equilibrium condition of an infinitesimal surface area yields a set of differential equations for the local total force. In attempting to solve such differential equations it may prove to be useful and prudent to reduce their complexity by first transforming all fields, current densities and Lorentz forces to a coordinate system that is aligned with the direction of the current flow. A Frenet--Serret rotating unit vector coordinate system may serve such a purpose and will reduce the 3 components of the Lorentz force to 2. We proceed with obtaining such a conversion through the use of differential geometry, although a more straight forward approach may exist through the use of surface developability and coordinate transformation. Following a solution to the force equations we continue with and example of a nested set of a combined function dipole and quadrupole that employ an identical periodicity {omega}. The expressions for the self force and the mutual force on each magnet element are obtained. Finally, by reducing the periodicity {omega} to zero we obtain the force expressions for long (2D) multipole magnets including both the self and interactive forces