1. Lecture 2
How do we approach Perturbation theory with a Tracking Code equipped with a TPSA package?
Description of our goals
Concept of a Hierarchical theory and its advantages
Normal Form of an Accelerator
Computation of Approximate Invariants: Lattice Functions
Computation of Moments: Lattice Functions again
Tracking the Canonical Transformation A
Phase Advance: Map in the Normal Variables

2. Question
What was the quintessential (典型的な｛てんけいてき ) message of Lecture 1?

3. Exercise Of Lectures 1 Use Complex Taylor series!
Use the anti-derivative
Use an inverse DAMAP
Use an inverse DAMAP and compute it with a vector field!

4. What is our goal?
We want to be able to compute all the objects which normally enter into the analysis of a repetitive (periodic) system
For example: Twiss functions, linear and nonlinear resonance driving terms, tunes, etc…
We want a theory which relies on the objects computed by the tracking code
So, what can we compute? Remember 1st lecture?
only

5. So what is a Tracking Code?
A tracking code by definition is an algorithm to compute the ray, floating points numbers, for example, x,x’,y,y’,t,dE.
We have seen in Lecture 1, that the introduction of an overloaded TPSA package allows the extension of algorithm from floating to Truncated Power Series.
So we write our own toy Tracking code for this lecture!
Therefore, we decree (命令をする), that a tracking code is an algorithm to compute rays and Taylor series expansions of that ray in variables chosen by the user/programmer.

6. Goal: to understand the output of a Tracking Code
We will look at a system which is stable around some closed orbit.
A system which is Hamiltonian (canonical)
At first, we use a system in one degree of freedom, i.e. phase space is (x,p), in our example code.
Time for our weekly lesson of English: Canonical

7. English Lesson! Canonical : 教会法上に従った (したがった） Canon Law: 教会法律
Canonical　theory :  正準理論
Dogma:教会の教義
Hierarchical Abstraction: 階層抽象化｛かいそう ちゅうしょうか｝
My lectures will be canonical in the Church sense as well!
I will describe a hierarchical approach. Hierarchical methods are well suited for computer implementations because modern computers are highly hierarchical in structure.
In addition, in a hierarchical system full understanding is not necessary. One can stop at anywhere in the hierarchy without comprising the whole structure. For example, one may know how to use Microsoft Word without knowledge of programming or computer architecture!

8. Religious or Canonical Hierarchical Structure
Dogma
Level 0
Newton’s Laws: not subject to discussion at all!
Derived Laws
Level 1
Least Action, Lagrangian, Hamiltonian
Derived Mathematical Properties
of the Laws
Symplectic, determinant=1, Liouville,
Invariants, etc..
Level 2
Mathematical Tools Derived
from the Laws
Generating Functions, Lie Transformations,
Canonical Transformations, etc…
Level 3
Computer
Implementations
Canonical Integrators, TPSA, DA, TPSA vector
Fields, TPSA Poisson Brackets, etc…
Level 4
Read code manual, if do not understand,
it is OK! Just run the code!
Level 5
User Manual

9. My own library (FPP) is Hierarchical

10. Our Starting Point M11 M12 M21 M22 Tracking Rays using the code
Producing a Taylor map, here linear,
using the code equipped with TPSA.

11. Theory: a Level 2 Religious Result
For a stable accelerator there exists an infinite number Canonical Transformation A such that:
Ms=As◦R◦As-1
This result is exact for linear map and formal for nonlinear maps with irrational tunes.
R is called the Normal Form of a stable accelerator. It represents the fact that the accelerator produces pseudo-harmonic oscillations around its orbit.
Formal = Only True in a TPSA sense around the design orbit.
TPSA sense around the design orbit = DA sense.

12. How do we do this ? My_map = Code + TPSA
Code + TPSA + Analysis Library
my_a_inv =
As-1
my_a_inv = (My_normal%a_t)**(-1)
Type(damap) My_map
Type(normalform) My_normal
My_normal = My_map
**(-1) : Level 4 mystery
My_normal = My_map
Is a Level 3 religious mystery
My_normal contains the normal form and the transformation As in My_normal%a_t

13. Example Program available on WEB

14. Invariant εs Using As we can easily compute the invariant.
Obviously x2+p2=(x2+p2) ◦R
But, using the normal form Ms=As◦R◦As-1
x2+p2=(x2+p2)◦As-1◦Ms◦As
→ (x2+p2 )◦As-1◦Ms = (x2+p2 )◦As-1
→ εs= (x2+p2 )◦As-1 is invariant of Ms

15. Results for Linear Maps
ε= (x2+p2 )◦A-1
= {{A11-1}2 + {A21-1}2 } x2 + { {A12-1}2 + {A22-1}2 } p2
+ 2{ A11-1 A A21-1 A22-1 } xp γ β α

16. Code In FPP for εs
This is the algorithm to compute the Invariant
! Here we compute the invariant in the space of circles
! It is obviously a circle!
my_invariant=(1.d0.mono.1)**2+(1.d0.mono.2)**2
! Now we act with A**(-1) to compute the invariant in the normal space around the closed orbit
my_invariant=my_invariant*my_a_inv
注意 : Functions transform in an inverse manner from the rays!
my_invariant is of type TAYLOR and My_a_inv is of type DAMAP

17. Lattice Functions as Averages of Quadratic Moments
For example <xaxb> averaged for a single ray over many turns. Such averages can only depend on the initial conditions through the value of the invariant, and the coefficients of the invariant (i.e. (a,b,g) here)
Notice that the same results can be
derived using
This comes from the Det(A)=1 or more generally the symplectic condition.

18. Tracking the Map A: Bs=M0s◦A0 (Theory)
Suppose we have M0=A0◦R◦A0-1
From the tracking code with TPSA, we can also get M0s , the map from position 0 to position s.
Let us obtain a map which diagonalize the one-turn map Ms at s.
Ms=M0s◦M0◦M0s-1 → Ms=M0s◦A0◦R◦A0-1 ◦M0s-1
If Bs=M0s◦A then Ms=Bs◦R ◦Bs-1
Bs=M0s◦A0 is a fundamental result of the full nonlinear Normal form theory: the map A0 can be tracked.

19. Tracking the Map A: Bs=M0s◦A0 (FPP Coding)
!This is the algorithm to compute the lattice function
! Here we initialize the ray = Closed Orbit + A
my_y(1)=my_fix(1)+ my_normal%a_t%v(1)
my_y(2)=my_fix(2)+ my_normal%a_t%v(2)
my_y(3)=my_fix(3)
! Now let us do loop over lattice starting at position = pos
do k = pos , pos+lattice%n-1
! Beta function is computed here in front of magnet “k”
Beta = (my_y(1).sub.'1')**2+(my_y(1).sub.'01')**2
gamma = (my_y(2).sub.'1')**2+(my_y(2).sub.'01')**2
alpha = -((my_y(2).sub.'1')*(my_y(1).sub.'1')+(my_y(2).sub.'01')*(my_y(1).sub.'01'))
call track(my_y,lattice,k,k+1) ! Bk+1 = Mk,k+1 Bk
end do

20. Phase Advance: Difference between As and Bs
We know that a stable accelerator map M0 can be linearly exactly and nonlinearly formally put into a normal form i.e.
M0 = A0◦R◦A0-1
Of course we can also do this at some position s: Ms = As◦R◦As-1
Finally we can also use the map from 0 to s and achieve the same result:
Ms = Bs◦R◦Bs-1 where Bs=M0s◦A0
The difference between As and Bs is a rotation called the phase advance.

21. Code Example: Phase Advance = Map in Normal Coordinates

22. Theory deduced from pictures
Important Point:
As is really a function of the one-turn map at Ms and nothing else.
As = F[Ms]
Usual in one degree of freedom, people follow the Courant-Snyder recipe. So As = FC-S[Ms]
From above picture:
M12 = A2◦R12◦A1-1
thus
A2-1 ◦M12 ◦A1 = R12
A2-1 ◦ B2 = R12
Notice that F is periodic by construction:
F[Ms+C] = F[Ms]
But, Bs+C=Ms ◦ Bs is not !

23. Code for Phase Advance As = F[Ms] M2=M12◦M1◦M12-1 A2-1 ◦M12 ◦A1 = R12
my_y(1)=my_fix(1)+ (1.d0.mono.1)
my_y(2)=my_fix(2)+ (1.d0.mono.2)
my_y(3)=my_fix(3)
my_map1=my_map
! We track one turn with our favourite code
my_a1=my_normal%a_t
dphase=0.d0
k=pos4-1
if(pos4==pos) k=k+lattice%n
do i=pos,k
call track(my_y,lattice,i,i+1)
my_map12%v(1)=my_y(1); my_map12%v(2)=my_y(2);
my_map2=my_map12*my_map1*my_map12**(-1)
my_normal2=my_map2
my_a2=my_normal2%a_t
my_r12=my_a2**(-1)*my_map12*my_a1
onelie12=my_r12 ! This is a single exponent Vector Field Representation
h12=onelie12%vector ! H12 is a phasor representation
dphase=(h12%sin%v(2).k.2)/2.d0/pi +dphase
my_a1=my_a2
my_y(1)=(my_y(1).sub.'0')+(1.d0.mono.1)
my_y(2)=(my_y(2).sub.'0')+(1.d0.mono.2)
my_map1=my_map2
enddo
M2=M12◦M1◦M12-1
As = F[Ms]
A2-1 ◦M12 ◦A1 = R12
This code contains a two “vector” field objects.
It is time to describe in detail what these are
and what purpose they serve.