Phase Advance

The concept of phase advance is best explained by the above picture. The one-turn map $T_i$ at some arbitrary position $s_i$ is normalized by the map $U_i$:

There are few points to make from which we deduce the phase advance:

1. The map at positions 1 and 2 are rotations in phase space and rotation in spin space around $e_2$.
2. The sizes of the circles at 1 and 2 are the same. This is automatically true if canonical transformations are used. If not, it could be enforced.

Item 1 is simply the assumption of the normal form. The reader may wonder how is this assumption justified? Is there any basis for it? Actually, thanks to the process of perturbation theory, a package like FPP is equipped with a normal form algorithm on the Taylor map. Therefore the formal existence of the normal form, i.e., its approximate existence closed to the origin is not in doubt.

Item 2 is a little more tricky. If the transformation $U$ is symplectic, then one can show that the circles at 1 and 2 are identical. If one allows a non-canonical transformation, it would not be the case. We disallow this for symplectic maps. In any event, the reader can see easily that one could scale the circles at position 2 to insure an identical size if non-canonical transformations were used.

Then it follows trivially that

$${R}_{12} = {U}_{2}^{-1}\circ {T}_{12}\circ {U}_{1}= \left({{r}_{12},{e}^{{\theta }_{12}{L}_{y}}}\right)$$ (11.20)

is a rotation in both phase space and spin space. By construction, it is of the same nature as the normal form. So the angle of the rotation in spin space is only a function of the phase space invariant radii.

The phase advance depends critically on the definition chosen for $U_s$. If $U_s$ is a functional of the one-turn map, then the phase advance is obviously unique between matched position, i.e., between point with identical one-turn map.