1. 1
Thursday Week 2 Lecture
Jeff Eldred
Review 1 1 1 1 1

2. Overview Lagrange, Hamilton, Poisson, Mechanics Accelerator Physics2
Overview
Lagrange, Hamilton, Poisson, Mechanics
Accelerator Physics
Electromagnetism
Relativity
Synchrotron Radiation 2 2 2 2 2

3. Lagrange, Hamilton, Poisson Mechanics (see Lectures 1-5)3
Lagrange, Hamilton, Poisson
Mechanics (see Lectures 1-5) 3 3 3 3 3

4. Lagrangian Mechanics The Lagrangian is defined by:
Lagrange’s Equations:
Every independent set of phase-space coordinate has its own Lagrange equation. 4 4 4 4

5. Electromagnetic Lagrangian
The Electromagnetic Lagrangian is:
With the conjugate momentum: 5 5 5 5

6. Hamiltonian Mechanics
The conjugate momentum is defined:
The Hamiltonian is defined by:
When there is no explicit time dependence:
The equations of motion are given by: 6 6 6 6

7. Poisson Brackets Poisson Brackets are defined:
Where pk is the conjugate momentum.
With Poisson Brackets we can consider the time dependence of any function of the coordinates:
A set of coordinates is canonical iff: 7 7 7 7

8. Oscillator Examples Harmonic Oscillator with Damping:
Driven Harmonic Oscillator without Damping: 8 8 8

9. Generating Functions q, Q independent q, P independent
p, P independent 9 9 9

10. Finding Action-Angle Coordinates
Method 1 (Position - Momentum):
Method 2 (Time - Energy): 10 10 10 10

11. 11
Accelerator Physics
(see Lectures 6-10) 11 11 11 11 11

12. Longitudinal Dynamics
Longitudinal Equations of Motion:
Synchrotron Motion: 12 12 12

13. Linear Betatron Motion
Betatron Tune, Phase-Advance:
Courant-Snyder Parameters: 13 13 13

14. Nonlinear Resonance Accelerator Hamiltonian:
Then take Fourier series near a resonance.
Sextupole example: 14 14 14

15. 15
Electromagnetism
(see Lecture 13-17, Suppl) 15 15 15 15 15

16. Vector Identities Dot Product: Vector Product:
Gauss Theorem: Stokes Theorem: 16 16 16 16

17. Maxwell’s Equations 17 17 17 17

18. Poynting Vector 18 18 18 18

19. Plane Waves 19 19 19

20. Eikonal Approximation
General E&M Wave:
Eikonal Approximation:
Longitudinal-Transverse Independent:
Cylindrically Symmetric: 20 20 20

21. 21
Relativity
(See Lecture 15, 18) 21 21 21 21 21

22. Time & Length Dilation 22 22 22

23. Lorentz Coordinate Transformation
Relativistic Energy & Momentum 23 23 23

24. Transformation of E & B fields24 24 24

25. Retarded Time
Retarded-Time Potentials: 25 25 25

26. Fields from a Point Charge
Lienard-Wiechert Potentials: 26 26 26 26

27. Synchrotron Radiation27
Synchrotron Radiation
(see Lecture 20, 24) 27 27 27 27 27

28. Radiation Geometry 28 28 28 28

29. Synchrotron Radiation Fields29 29 29 29

30. Radiated Power 30 30 30 30

31. Radiation Spectrum
If ψ= 0: 31 31 31 31