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Thursday Week 2 Lecture

1. Thursday Week 2 Lecture. Jeff Eldred Review. 2. Overview. Lagrange, Hamilton, Poisson, Mechanics Accelerator Physics Electromagnetism Relativity Synchrotron Radiation. 3. Lagrange, Hamilton, Poisson Mechanics (see Lectures 1-5). Lagrangian Mechanics.

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Thursday Week 2 Lecture

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  1. 1 Thursday Week 2 Lecture Jeff Eldred Review

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

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

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

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

  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:

  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:

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

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

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

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

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

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

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

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

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

  17. Maxwell’s Equations

  18. Poynting Vector

  19. Plane Waves

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

  21. 21 Relativity (See Lecture 15, 18)

  22. Time & Length Dilation

  23. Lorentz Coordinate Transformation Relativistic Energy & Momentum

  24. Transformation of E & B fields

  25. Retarded Time Retarded-Time Potentials:

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

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

  28. Radiation Geometry

  29. Synchrotron Radiation Fields

  30. Radiated Power

  31. Radiation Spectrum If ψ= 0:

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