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It should therefore be the goal to generalize this concept to non-linear dynamics.
The computation of a reliable one-turn-map and the analysis of its properties will
provide all relevant information.
Given that the non-linear maps can be rather complex objects, the analysis of the
one-turn-map should be separated from the calculation of the map itself.
3.5 Linear Normal Forms
3.5.1 Sequence of Maps
Starting from a position s 0 and combining all matrices to get the matrix to position
s 0 + L (shown for 1D only):
x
x
s 0 + L
= M N ◦ M N−1 ◦ . . . ◦ M 1
M(s 0 ,L)
◦
x
x
s 0
(3.11)
For a ring with circumference C one obtains the One-Turn-Matrix (OTM) at s 0
x
x
s 0 + C
=
m 11 m 12
m 21 m 22
M OT M
◦
x
x
s 0
(3.12)
Without proof, the scalar product:
x
x
s 0
· M OT M
x
x
s 0
= const. = J
(3.13)
is a constant of the motion: invariant of the One Turn Map.
With this approach we have a strong argument that the construction of the One
Turn Map is based on the properties of each element in the machine. It is entirely
independent of the purpose of the machine and their global properties. It is not
restricted to rings or in general to circular machine.
Once the One Turn Map is constructed, it can be analysed, but this analysis does
not depend on how it was constructed.
As a paradigm: the construction of a map (being for a circular machine or not)
and its analysis are conceptual and computational separated undertakings.