Matrix nonlinear beam dynamics in curvilinear space-time

A general relativistic matrix theory of charged particle beam motion along an arbitrary curved optical axis in 4-space-time has been developed. This theory uses three basic matrix functions: the reference frame matrix, the curvature matrix and the electromagnetic matrix. The Cartan method of the moving 3-vector is generalized as the method of the moving 4×4 reference matrix. The curvature matrix function consists of the normal curvature, the geodesic curvature and torsion and three components of the gravitational force acting on the reference particle. The matrix equations of the beam motion and of the electromagnetic field are written. The nonlinear equations in phase space are reformulated as linear equations in phase moment space. A new compact recursive method is proposed for integrating these linear equations. Using this method the phase volume of the beam will be strictly conserved in each step of the numerical integration