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Fundamentals of Lagrange, Hamilton, and Poisson Mechanics Overview

Dive into Lagrangian and Hamiltonian mechanics, Poisson brackets, accelerator physics, electromagnetism, and relativity with this detailed review. Explore concepts like conjugate momentum, canonical coordinates, and synchrotron radiation. Learn essential equations and principles in theoretical physics.

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Fundamentals of Lagrange, Hamilton, and Poisson Mechanics Overview

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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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