Matrix theory of the motion of a charged-particle beam in curvilinear space-time Part II. Nonlinear theory. Formulae and algorithms

A general relativistic theory of the motion of a charged-particle beam motion along a curved optical axis, including the gravitational field, is of great significance in the design of optimal beam control systems. In a first paper, a new matrix approach was presented. In this second paper the expansion of the most general equations of motion and electromagnetic field equations in the Taylor series in powers of the derivation from the reference particle is presented. The nonlinear equations in phase space are reformulated as linear equations in phase moment space. A compact, conservative, recursive method of integrating the equations of motion is proposed. In some cases it is possible to optimize the control system by choosing a definite symmetry. This symmetry is considered in the paper.