1 Accelerator Hamiltonian Jeffrey Eldred Classical Mechanics and
- Slides: 12
1 Accelerator Hamiltonian Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU
2 Relativistic Accelerator Hamiltonian 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 10/21/2021
Relativistic Electromagnetic Hamiltonian Lorentz Force: Vector Potential: Lagrangian: Canonical Momentum: Hamiltonian: 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 10/21/2021
The Accelerator Hamiltonian (Frenet-Serret) Cartesian Coordinates for a circular system: Frenet-Serret Coordinates for varying bend radius: 4 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 10/21/2021
The Accelerator Hamiltonian (Frenet-Serret) Hamiltonian: Change to Frenet-Serret Coordinates: 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 10/21/2021
The Accelerator Hamiltonian (Frenet-Serret) Note: The new coordinates are curvilinear, grad, div, and curl must change to reflect that in the new coordinates: 6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 10/21/2021
The Accelerator Hamiltonian (s as the time coordinate) Beam diagnostics and changes in beam focusing occur at specific locations in the accelerator, along the variable s. But Hamilton’s equations of motion describe how the coordinates change over time, in this case the variable t. So can we change the “time” variable to the location s? Ideal for beams that are already collinear and relativistic. Use -Πs as the new Hamiltonian, h and t become new coordinates. 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 10/21/2021
The Accelerator Hamiltonian (s as the time coordinate) This only works because the new derivatives match the old ones: 8 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 10/21/2021
The Accelerator Hamiltonian (Transverse Magnetic Fields) Only magnetic focusing, no electrostatic elements: Only transverse magnetic fields, no solenoidal fields: 9 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 10/21/2021
The Accelerator Hamiltonian (Paraxial Approximation) Rescale the Hamiltonian by p 0 -1 (reference momentum): Small angle-approximation: 10 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 10/21/2021
The Accelerator Hamiltonian (Paraxial Approximation) 11 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 10/21/2021
Linear Transverse Focusing See Lecture 7 and Lecture 8. 12 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 10/21/2021
