On a class of ideal focusing systems for energy analysis

Every plane electrostatic system produces pair of orthogonal curve families -- field lines and equipotentials, which can be associated with trajectories and lines of constant action of some mechanical system. The pair of opposite parallel line charges has a two-dimensional representation on a plane perpendicular to charge lines. Such electrostatic system is used as trajectories/action generator. Both action and its arbitrary function are orthogonal to particle trajectories simultaneously. This fact can be used to regularize result two-dimensional potential function and to optimize three-dimensional charged particle motion. The method is applied to create charged particle energy analyzer with an ideal focusing in the symmetry plane. The system useful property is that its energy dispersion tends to infinity when the angle between direction to detector and initial velocity tends to π. Also, for some simple examples it was shown that together with the ideal focusing in a plane of symmetry the system has transversal first order focusing for some initial angles.