# Non Linear Dynamics - Methods and Tools II

- Institution: CERN
- Department: CAS - CERN Accelerator School
- Speaker: Yannis Papaphilippou
- Year: 2024
- Level: Graduate
- Language: English
- Subjects: Mathematics, Physics
- Source: [CERN](https://videos.cern.ch/record/3022735)
- Licence: [CERN](https://copyright.web.cern)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

The goal of this lecture is to present the contemporary methods and numerical tools of non-linear dynamical systems in order to analyse the motion in particle accelerators. After a short introduction to non-linear effects and their impact to beam performance, the lecture will briefly review elements of classical mechanics, essential for the study of non-linear dynamics, including the Lagrangian and Hamiltonian formalism, canonical transformation and simplicity. Starting from the relativistic Hamiltonian for E/M fields, elements of canonical perturbation theory will be presented, showing its limitation, for particle accelerators. In this respect, concepts of linear and non-linear beam transport will be introduced, represented by matrices or more generally maps. The Lie formalism will be employed in order to elaborate and analyse these maps through non-linear normal form construction. An introduction to Truncated Power Series through differential Algebra will be also given, as it is essential for constructing 1-term maps. Finally, elements of symplectic integration will be reviewed.

0:00:00 Slide 1

0:28:38 Slide 2

0:31:10 Slide 3

0:32:55 Slide 4

0:34:22 Slide 5

0:36:12 Slide 6

0:38:07 Slide 7

0:40:53 Slide 8

0:42:18 Slide 9

0:44:26 Slide 10

0:45:27 Slide 11

## Lectures

1. [Non Linear Dynamics - Methods and Tools II](https://invisiblenetwork.ai/learn/cern/3022735/4cd83aafd9ed), 50 min (CERN's player)