First-Order Averaging Principles for Maps with Applications to Beam Dynamics in Particle Accelerators

TL;DR

This paper develops first-order averaging principles for discrete-time systems, using Besjes' inequality, and applies these results to beam dynamics in particle accelerators and the Henon map.

Contribution

It introduces Besjes' inequality for perturbed identity maps and applies it to establish averaging principles in resonance and non-resonance cases.

Findings

Proves first-order averaging principles for discrete maps.

Extends mathematical results to accelerator beam dynamics.

Applies the theory to the Henon map.

Abstract

For slowly evolving, discrete-time-dependent systems of difference equations (iterated maps), we believe the simplest means of demonstrating the validity of the averaging method at first order is by way of a lemma that we call Besjes' inequality. In this paper, we develop the Besjes inequality for identity maps with perturbations that are (i) at low-order resonance (periodic with short period) and (ii) far from low-order resonance in the discrete time. We use these inequalities to prove corresponding first-order averaging principles, together with a principle of adiabatic invariance on extended timescales; and we generalize and apply these mathematical results to model problems in accelerator beam dynamics, and to the Henon map.