Using methods of symmetrization, geometric nomography and asimptotic approximation for facilitating the solving of classical differential equations of charged particle motion in magnetiic and electric fields
The aim of this report is to demonstrate, using some examples of solving differential equations of motion of charged particles in magnetic and electric fields, that symmetrietization-simplification, nomographic methods and the methods of asymptotic approach are a very efficient tool in solving and investigating linear and non-linear differential equations. The new compact and symmetric form of the first and second order coefficients for transformation of charged particle trajectory in dipole and quadrupole magnets has been obtained. The geometric nomograms are give for determination of principal ion-optical parameters and optical coefficients. The asymptotic approximation method developed by N.V. Krylov and N.N. Bogolyubov has allowed to analyse the so-called secular solutions to equations of motion of charged particles in axial symmetric magnetic and electric fields of the Penning trap. In addition it has enabled us to analytically construct an electrostatic field with strong quadrupole focusing and an accelerating components for direct acceleration of charged particles (patent of RF).