The Normal Form Itself

The map $a$ which brings us into normal form is far from being useless: not all maps are made of circles! It tells us the actual shape of the invariant quantities in real space. It allows us to transform any function into normal coordinates so that averages can be performed. Therefore all the lattice functions are computed from $a$ and/or its inverse.

The map $a$ can be factored as follows:

$$a= {a}_{1}\circ {a}_{\ell}\circ {a}_{n\ell}$$ (11.5)

The map $a_1$ brings the map $m$ to its parameter dependent fixed point. The parameters are $\vec{k}$ and additionally $z_5$ in the case of a coasting beam. Therefore the map

$${m}_{1}= {a}_{1}^{-1}\circ m\circ {a}_{1}$$ (11.6)

sends the origin of phase space into itself for all values of the parameters. The map ${a}_{\ell}$ normalizes the map $m_1$ linearly for all values of the parameters. Therefore ${a}_{\ell}$ provides us with the linear lattice functions as a function of the parameters. The end result,

$${r}_{\ell}= {a}_{\ell}^{-1}\circ {a}_{1}^{-1}\circ m\circ {a}_{1}\circ {a}_{\ell}$$ (11.7)

is a purely non-linear map around the parameter dependent fixed point. Finally ${a}_{n\ell}$ diagonalizes the map completely into the rotation $r$.

These maps in Eq. (11.5) can be obtained from $a$ using the PTC subroutines

 

CALL FACTOR(A,A_1,A_L,A_NL)   ! A=A_1 o A_L o A_NL

The idea is to extend this to spin maps.