Toroidal Polynomials Isomorphic to Ignition Scattering Functions in Fusion Conditions

Nuclear fusion is achieved through the self-sustaining chain reaction of particles in various geometries. One of the most interesting and novel structures developing fusion is the tokamak. Solving the Poisson equation and the Lagrangian for geodesics in the toroidal geometry requires the development of a new function called the toroidal elliptic function. As the coordinate system has problems with skewness, the components of the differential equation solution cannot be separated. Thus, the dot product is modified to account for overlapping coordinates. The importance of finding analytic solutions to fusion scattering and geodesic problems has implications in supersymmetry/standard model predictions and practical energy production in the H mode. This work is an effort to create polynomials of infinite order and recursion relations that define coefficients that completely describe a set of eigenfunctions in a toroidal shape. Denoted by a special letter, these toroidal elliptic functions are written in spherical, cylindrical, and Cartesian polynomials to simplify the difficult task of finding an isomorphic analytic and computational model for particles moving along geodesics in a tokamak.