Principles of Analytical Perturbation Theory

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”

Examples of 1 and 2

Example of 3

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)

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.

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”

quadrupole

Example: first order

1 2 Compute Perturbed Map

Canonically transform N 1 2 Canonically transform N

How can we choose F? Phasors to the rescue

Phase advancing xn using phasors

Choosing F to wipe out distortions

Resulting Map

1) Hamiltonian solved as well to first order in k

2) Proof

3) Continue…

4) Continue… Therefore Q.E.D.

Second Order Revisited

One turn map

Fs and map to first order

Fs and map to second order

Continue

Total Tune Shift Quadrupole

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

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

Application: Chromaticities and momentum compaction

Extract terms in J and d

Results

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