FACTORIZATION OF A

and

Phase Advance Computation 

M denotes a DAMAP and DS denotes a DAMAPSPIN:    DS=(M,S)  where S is a 3 by 3 matrix.

A symplectic map M can be put in normal form

                                        R=A^-1 * M * A

FPP  does choose an A very carefully. If one insists on a "canonical" form for A, then the following routine can be called.

call factor(a_t,a_f,a_l,a_nl,  DR=dr,  r_te=r_te,cs_te=cs_te,cosLIKE=cosLIKE)

1)  a_t=a_f * a_l * a_nl : These are compulsory DAMAPS.

A_F=Parameter dependent fixed point map; may include time in coasting beam case ndpt/=0.

A_L= Parameter dependent linear map around the fixed point!

A_NL = the nonlinear part of A around the fixed point!.

2)  Difference between the input A_T and its value on exit, i.e., puts A_T in "canonical" form.

A_T_entrance = A_T_exit * DR ; DR is the phase advance (nonlinear terms included). Notice that A_L and A_NL are changed by DR.

Definition of the canonical form of A: 

• Linear part: A₁₂=A₃₄(=A₅₆)=0  → that is a Langrangian definition of A consistent with Courant-Snyder. It insures that the phase advance between monitors is the phase difference of the position oscillations. This concept has no easy extension to the nonlinear case unfortunately.
• In the nonlinear case, FPP offers two choices:

1. The reverse Dragt-Finn representation has no tune shift phasors    →   onelie=false
2. The one-lie exponent representation has no tune shift phasors        →   onelie=true

DR contains the phase difference between the input map and its canonical form

3) Some people like this parameterization of Teng and Edwards for a 4x4 oscillatory matrix. (No RF-Cavity).

                        a_l= R_TE * cs_te

The parameter  cosLIKE is

• TRUE if the component c and s of R_TE   are  c=cos(j)  and s=sin(j)
• FALSE if the component c and s of R_TE   are  c=cosh(j)  and s=sinh(j) with -s on the second row also equal to sinh(j).

Example routine from PTC Click here

BS Report:

Apparently it is unknown, according to Frank, that the coefficient "c" of the matrix R_TE can be greater than 1! This is actually implicit in the paper of Ohmi, Hirata and Oide of 1994. However Frank may well be correct. According to Ohmi, the reviewer of a recent paper by Y. Seimiya, K. Ohmi, D. Zhou, J. W. Flanagan and Y. Ohnishi needed to be reminded that "c" can be greater than one. It is in reference [7] of that paper. Nothing new to competent people apparently!