1. Perturbation Methods Jeffrey Eldred1 1
Perturbation Methods
Jeffrey Eldred
Classical Mechanics and Electromagnetism
June 2018 USPAS at MSU 1 1 1 1 1 1

2. 2 2
Direct Perturbation 2
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3. 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
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4. Example: Motion through a Sextupole4
Example: Motion through a Sextupole
Differential Equation:
s2 = 0,
S = 0:
s5 =0,
S2 = 0:
S3 = 0: 4
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5. Example: Motion through a Sextupole5
Example: Motion through a Sextupole
Differential Equation:
s5 =0,
S3 = 0: 5
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6. Example: Motion through a Sextupole6
Example: Motion through a Sextupole
Differential Equation:
s3 =0,
S2 = 0:
Imperfect Conservation of Energy: 6
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7. Lindstedt-Poincare Method7 7
Lindstedt-Poincare Method 7
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8. Lindstedt-Poincare Method8
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
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9. Synchrotron Motion 9
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10. 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
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11. 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
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12. Slipping Motion 12
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13. 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
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14. Perturbation of Slipping Motion
Perturbation terms of the form:
Plug into the differential equation:
Solve for the coefficients 14
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15. Advanced Example: Time-Dependent System15
Advanced Example: Time-Dependent System
Lindstedt-Poincare Perturbation for a Time-Dependent System.
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
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