1. Principles of Analytical Perturbation Theory

2. Ideas Assume a simple Hamiltonian H0 for the unperturbed system
Assume that this H0 can be put in Normal Form
Add the perturbation which is to be studied
Expand this perturbation in a simple basis in the time-like variable “s”

3. Examples of 1 and 2

4. Example of 3

5. Expansion in “s” The trick is to expand the perturbation
in terms of “solvable” functions
Two ways:
Delta-functions in “s” : equivalent to a map based theory. So let us look at the map based analytical theory.
Fourier modes in “s” (standard accelerator physics)

6. Rules of Analytical Perturbation Theory
Choose a perturbation potential V (quadrupole, sextupole, Beam-Beam, etc…)
Choose the order of the calculation, i.e., kth order in V
Introduce k “kick” maps exp(-:ds Vi:) i=1,k at arbitrary locations
Compute the quantities of interest
Sum/Integrate over the variable i: the index i serves as a time ordering label.

7. Index Summation
To get the continuous result we first reject any term where the same index appears twice
We then interpret the index “i” as a time ordering label
We then integrate over each variable “i”

8. quadrupole

9. Example: first order

10. 1 2
Compute Perturbed Map

11. Canonically transform N1 2
Canonically transform N

12. How can we choose F?
Phasors to the rescue

13. Phase advancing xn using phasors

14. Choosing F to wipe out distortions

15. Resulting Map

16. 1) Hamiltonian solved as well to first order in k

17. 2) Proof

18. 3) Continue…

19. 4) Continue…
Therefore
Q.E.D.

20. Second Order Revisited

21. One turn map

22. Fs and map to first order

23. Fs and map to second order

24. Continue

25. Total Tune Shift
Quadrupole

26. Continue (Sextupole) All the other terms can be computed similarly!

27. Other Normal Forms
No cavity: the longitudinal plane is a drift-like map z=(x,px,y,py,d,t)
M=ARA-1

28. Application: Chromaticities and momentum compaction

29. Extract terms in J and d

30. Results

31. Other Normal Forms (Continue)
Radiation
M=ANA-1