# On emittance and optics calculation from the tracking data in periodic lattices

## TL;DR

This paper explores advanced mathematical methods to improve the calculation of beam emittance and optics in periodic lattices, linking established techniques with new decompositions for better analysis.

## Contribution

It introduces the use of Iwasawa decomposition for beam optics parameterization and connects it with sigma-matrix methods and Williamson's Theorem in a linear setting.

## Key findings

- Iwasawa decomposition offers a natural framework for beam optics analysis.
- The sigma-matrix method is connected to Williamson's Theorem.
- Application to realistic tracking examples demonstrates the approach's effectiveness.

## Abstract

In this work we examine the interplay between normal form and matched particle distributions in a linear setting. We first outline the connection between the established sigma-matrix method and Williamson's Theorem. Then we show that the Iwasawa decomposition provides a natural framework for a description of beam optics parameters. Along the way we will apply these methods to a realistic tracking example, as well as provide additional examples, including the connection to the parameterization of Courant-Synder.