Perturbation Methods Jeffrey Eldred 1 1 Perturbation Methods Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU 1 1 1 1 1 1

2 2 Direct Perturbation 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 2 2 2 2 2 2

3 Direct Perturbation A differential equation may be difficult (or impossible) to solve explicitly. We need to know the trajectory to calculate the force on the particle, but we need to know the force on the particle to calculate the trajectory! So one may be able to obtain a solution that is a series of simpler solutions. Make a severe approximation of the differential equation that admits a simple solution for the trajectory. Use that trajectory to calculate a simple force. Use that simple force to calculate a new trajectory. Iterate until the trajectory converges for a certain order of precision. 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 3 3 3

Example: Motion through a Sextupole 4 Example: Motion through a Sextupole Differential Equation: s2 = 0, S = 0: s5 =0, S2 = 0: S3 = 0: 4 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 4 4 4

Example: Motion through a Sextupole 5 Example: Motion through a Sextupole Differential Equation: s5 =0, S3 = 0: 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 5 5 5

Example: Motion through a Sextupole 6 Example: Motion through a Sextupole Differential Equation: s3 =0, S2 = 0: Imperfect Conservation of Energy: 6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 6 6 6

Lindstedt-Poincare Method 7 7 Lindstedt-Poincare Method 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 7 7 7 7 7 7

Lindstedt-Poincare Method 8 Lindstedt-Poincare Method Like the direct perturbation method, a few differences: Must be applied to a bounded system with oscillatory or periodic solutions. The frequency of the oscillation itself is perturbed. Energy is conserved. 8 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 8 8 8

Synchrotron Motion 9 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 9 9 9 9

Perturbation of Stable Motion Express the differential equations as a function of small angles: Write the small oscillation terms: Plug these terms into the differential equations: 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 10 10 10 10

Perturbation of Stable Motion Plug these terms into the differential equations: Match the oscillatory terms on the LHS and RHS: Solve for σ in the first equation and A3 in the second: 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 11 11 11 11

Slipping Motion 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 12 12 12 12

Perturbation of Slipping Motion Let’s determine what the differential equation is, and then approximate it for small perturbation (large slipping rate): Perturbation terms of the form: 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 13 13 13 13

Perturbation of Slipping Motion Perturbation terms of the form: Plug into the differential equation: Solve for the coefficients 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 14 14 14 14

Advanced Example: Time-Dependent System 15 Advanced Example: Time-Dependent System Lindstedt-Poincare Perturbation for a Time-Dependent System. https://journals.aps.org/prab/pdf/10.1103/PhysRevSTAB.17.094001 There is a 2D-series expansion that results. There will be many resonances, which complicates the convergence away from small perturbations. Extra credit: Spot the error in this paper! 15 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/12/2018 15 15 15