Understanding Classical Mechanics and Electromagnetism at USPAS, June 2018

1 1 Review Lecture Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU

First Monday 1B Oscillators Oscillators from Linearized Potential Nonlinear Damped Oscillator Oscillators Driven with an arbitrary force 1C Perturbation Direct Perturbation Poincare Lindstedt Pertubation 1D Special Relativity Length Contraction, Time-Dilation, Lorentz Boosts Relativistic Momentum & Energy Plane Waves & Doppler Shifts Causality, Minkowsky Diagrams & Paradoxes 2 2 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

First Monday 1E Lagrangians Nonrelativistic & Relativistic Lagrangian Equations of Motion Least Action Principal Change of coordinates for a Lagrangian Wave Equation for a String 1E Hamiltonians Canonical Momentum & Obtaining H from L Obtaining H from equations of motion Hamiltonian & Conservation Laws 3 3 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

First Tuesday 2A Phase-Space Fixed Points & Stability of Points Eigenvalues & Eigenvectors Equipotential Surfaces 2B Canonical Transformations Properties of Poisson Brackets Poisson Brackets to Find Invariants Canonical Transformations & Generating Functions Jacobian Matrix & Checking for preservation of PhSp area. Liouville s Theorem for Volume, Number, Density Foiling Liouville s Theorem Action-Angle Coordinates, what they mean Action-Angle Coordinates, 1D system Frequency as a function of Action 4 4 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

First Tuesday 2A Phase-Space Fixed Points & Stability of Points Eigenvalues & Eigenvectors Equipotential Surfaces 2B Canonical Transformations Properties of Poisson Brackets Poisson Brackets to Find Invariants Canonical Transformations & Generating Functions Jacobian Matrix & Checking for preservation of PhSp area. Liouville s Theorem for Volume, Number, Density Foiling Liouville s Theorem Action-Angle Coordinates, what they mean Action-Angle Coordinates, 1D system Frequency as a function of Action 5 5 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

First Tuesday 2C Accelerator Hamiltonian Electromagnetic Relativistic Hamiltonians EM Canonical Momentum Making Canonical Transformations Assumptions for an Accelerator Hamiltonian Action-Angle Variables for an Accelerator Linear Accelerator Motion 2D Accelerator Physics Overview Transfer Matrices & Finding TWISS Parameters Particle Motion in terms of TWISS, Action Phase-Advance Particle Emittance & Adiabatic Damping Dispersion & Chromaticity 6 6 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

First Tuesday First Wednesday 2E Longitudinal Dynamics Synchrotron Focusing Equations Phase-slip Factors & Transition 3A Magnetic Multipoles Vector Potential & Scalar Potential Basic Magnetic Multipoles Taylor Expansion for Multipole Components 3B Magnetostatics Calculating Magnetic Fields from Sources Amperian Loops Concept of Magnetization & Hysteresis 7 7 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

First Wednesday 3C Nonlinear Accelerator Resonance Action-Angle Hamiltonian Integrated Hamiltonian Identifying Resonances from Multipoles Calculating & Correcting Resonances Equations of Motion near Resonance Tune Diagrams 3D Numerical Methods Poincare Section FMA Analysis 8 8 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

First Thursday 4A Electrostatics Calculating Electric Fields from Sources Gaussian Surfaces Electrostatic Multipoles 4A Transformation of EM Fields Relativistic Charges & Currents Lorentz Boosts of EM Fields 4B Space-charge Electric Field from Beams KV Distributions & Linear Focusing Beam Envelope Equation Space-charge Tune-shift 9 9 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

First Thursday First Friday 4C Plane Waves Orientation & Polarization Poynting Vector Retarded Time Fields from a Point Charge Larmor Formula / Synchrotron Radiation 5A Symplectic Maps Symplectic vs Nonsympletic Tracking Meaning of a Lie Operator BCH Formula for a Corrected Invariant 10 10 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

Second Monday 6A Cavities & Waveguides Standing Waves Boundary Conditions for E & B Separation of Variables TM vs TE modes Rectangular & Cylindrical Geometry Waveguides & Cutoff Frequency Phase Velocity & Group Velocity RF Cavities & Eigenmodes Reflection of EM Waves 11 11 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

Second Monday Second Tuesday 6B Transmission Lines & Resonators Complex Impedance, Capacitors, Inductors Load-matching & Reflections Propagation of V & I over z RLC Resonator Model Loaded Transmission Lines as Resonators 7A Radiation Synchrotron Radiated Power Synchrotron Angular Distribution Synchrotron Spectrum & Critical Frequency Velocity vs. Radiative Fields Formation Length Coherent Radiation 12 12 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU

Second Tuesday Second Thursday 7B Space-charge Debye Length, Collisional vs. Space-charge Pancaking of E&B-fields Longitudinal Space-charge Image Charges 8A Wakefields Panofsky Wenzel with application Beam moments & Cylindrical Decomposition Wake potentials & Wake functions Impedances as Fourier Transform Collectivity Instabilities from Wakes Landau Damping Concepts 13 13 3/21/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU