Lie algebraic treatment of space charge

The purpose of the paper is to discuss the application of Lie algebraic methods to beam transport calculations with space charge. Specifically, the authors present a method for finding approximate solutions of the Vlasov-Poisson equations, treated as an initial value problem. First, they discuss the formal solution of the Vlasov equation using Hamiltonian methods. The formal solution is shown to depend on a (generally nonlinear) transfer map, which is just the inverse of the single particle transfer map; this reduces the problem of beam transport with space charge to the self-consistent computation of a transfer map. To obtain the evolution equations for the transfer map, they formally represent it (through third order) as a product of Lie transformations. This representation is then used to obtain an approximate solution of Poisson's equation for the scalar potential. The scalar potential is substituted back into the evolution equations for the transfer map, making the calculation self-consistent. Since the resulting equations cannot be solved analytically, they obtain solutions by numerical integration. The authors' results are compared with other methods, including particle simulation programs