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

Abstract:
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).