Averaged dynamics of ultra-relativisitc charged particles beams

TL;DR

This thesis demonstrates that for ultra-relativistic charged particle beams, an averaged Lorentz force model effectively approximates the traditional model, simplifying analysis while maintaining accuracy for narrow distribution functions.

Contribution

The work introduces an averaged Lorentz force equation derived via differential geometry, justifying its use for ultra-relativistic beams and simplifying kinetic models in accelerator physics.

Findings

Averaged Lorentz force equation closely matches the original for narrow distributions.

The averaged Vlasov equation is simpler and yields similar solutions to the original.

Beam dynamics can be derived from the Jacobi equation of the averaged model.

Abstract

In this thesis, we consider the suitability of using the charged cold fluid model in the description of ultra-relativistic beams. The method that we have used is the following. Firstly, the necessary notions of kinetic theory and differential geometry of second order differential equations are explained. Then an averaging procedure is applied to a connection associated with the Lorentz force equation. The result of this averaging is an affine connection on the space-time manifold. The corresponding geodesic equation defines the averaged Lorentz force equation. We prove that for ultra-relativistic beams described by narrow distribution functions, the solutions of both equations are similar. This fact justifies the replacement of the Lorentz force equation by the simpler {\it averaged Lorentz force equation}. After this, for each of these models we associate the corresponding kinetic model, which are based on the Vlasov equation and {\it averaged Vlasov equation} respectively. The averaged Vlasov equation is simpler than the original Vlasov equation. This fact allows us to prove that the differential operation defining the averaged charged cold fluid equation is controlled by the {\it diameter of the distribution function}, by powers of the {\it energy of the beam} and by the time of evolution $t$. We show that the Vlasov equation and the averaged Vlasov equation have similar solutions, when the initial conditions are the same. Finally, as an application of the {\it averaged Lorentz force equation} we re-derive the beam dynamics formalism used in accelerator physics from the Jacobi equation of the averaged Lorentz force equation.