# Longitudinal beam dynamics recap

- Institution: CERN
- Department: CAS - CERN Accelerator School
- Speaker: Frank Tecker
- Year: 2025
- Level: Graduate
- Language: English
- Subjects: Earth and environment
- Source: [CERN](https://videos.cern.ch/record/3027010)
- 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)

This talk gives a short review of longitudinal beam dynamics for both linear and circular accelerators. It covers the basics of acceleration in a linac. For radio-frequency (RF) acceleration, it introduces the concepts of synchronicity and the oscillation around the synchronous particle, including the criterium for stability of the oscillations. For circular accelerators, the lecture focuses on synchrotrons, their operation principle and their longitudinal beam dynamics. It reviews the synchrotron oscillations, their stability conditions and their representation in longitudinal phase space. Then it explains the concept of longitudinal phase space matching and its importance when you transfer the beam from one accelerator to another. Finally, it presents the Hamiltonian for the longitudinal motion.

00:00:00 Slide 1

00:01:57 Slide 2

00:03:36 Slide 3

00:05:20 Slide 4

00:07:36 Slide 5

00:10:19 Slide 6

00:10:32 Slide 7

00:11:44 Slide 8

00:15:12 Slide 9

00:15:24 Slide 10

00:17:49 Slide 11

00:18:58 Slide 12

00:20:41 Slide 13

00:22:28 Slide 14

00:23:34 Slide 15

00:25:30 Slide 16

00:27:25 Slide 17

00:28:44 Slide 18

00:31:29 Slide 19

00:33:18 Slide 20

00:34:07 Slide 21

00:35:19 Slide 22

00:36:30 Slide 23

00:38:38 Slide 24

00:39:30 Slide 25

00:39:58 Slide 26

00:40:59 Slide 27

00:41:55 Slide 28

00:42:26 Slide 29

00:43:08 Slide 30

00:44:19 Slide 31

00:45:42 Slide 32

00:46:20 Slide 33

00:47:07 Slide 34

00:48:28 Slide 35

00:49:22 Slide 36

00:50:12 Slide 37

00:50:51 Slide 38

00:51:34 Slide 39

00:52:58 Slide 40

00:53:35 Slide 41

00:54:28 Slide 42

00:55:09 Slide 43

00:55:43 Slide 44

00:56:39 Slide 45

## Lectures

1. [Longitudinal beam dynamics recap](https://invisiblenetwork.ai/learn/cern/3027010/d7efd63dc302), 57 min (CERN's player)