The Orbital Normal Form Itself

The content of this section are documented at

 http://mad.web.cern.ch/mad/PTC_proper/

It suffices to say that FPP allows for the normalization of a type damap.

 
  TYPE DAMAP
     TYPE (TAYLOR) V(NDIM2)    ! NDIM2=6 BUT ALLOCATED TO ND2=2,4,6 
  END TYPE DAMAP

The normal form is central to all the self-consistent algorithms of PTC. So what is a normal form? A normal form is a reduction of the one-turn map into a form so simple that anyone can discover the basic property of the map. For example, in the case of a complete orbital normalization, we have:

$$m = a\circ r\circ {a}^{-1}$$ (11.1)

The map $r$ has the following Lie representation:

$$r = \exp\left({:h({J}_{1},{J}_{2},{J}_{3};\vec{k}):}\right)I~$$ (11.2)

$$\vphantom{{\rm o}{\rm r} {\rm o}{\rm r} {\rm o}{\rm r} {\rm o}{\rm r}}{\rm o}{\rm r}$$

$$= ~\exp\left({:h({J}_{1},{J}_{2},{x}_{5};\vec{k}):}\right)I~~~{\rm ... ...{\rm f}~{\rm p}{\rm h}{\rm a}{\rm s}{\rm e}~{\rm s}{\rm p}{\rm a}{\rm c}{\rm e}$$ (11.3)

In Eq. (11.2), the map is fully normalized into 3 amplitude dependent rotations. In Eq. (11.3), the transverse part of the map is normalized into 2 rotations. The longitudinal part is normalized into a drift-like map; here $x_5$ is an energy like variable (either energy or momentum). It is canonically conjugate to either time or path length. This last normal form, a Jordan normal form, applies to the case of a coasting beam. In PTC one can choose either:

$${z}_{5} = \delta = \ {p-{p}_{0} \over {p}_{0}}~{\rm w}{\rm i}{\rm... ...rm E}~{\rm s}{\rm t}{\rm a}{\rm t}{\rm e}~{\rm o}{\rm f}~{\rm P}{\rm T}{\rm C})$$ (11.4)

If we accept the potential or formal existence of this normal form, then many things become trivial. If one instead is obstinate, then even the simplest of properties become mysterious even to the smartest of accelerator physicists. For example, it has been stated by some that the amplitude dependent path length is the chromaticity produced by the quadrupoles only. In the normal form of Eq. (11.2), this can be immediately computed as the derivative of $h$ with respect to $x_5$. Also the tunes are the derivative with respect to $J_1$ and $J_2$. It is clear that the term in $h$ which of the form $x_5 (a_1 J_1 + a_2 J_2 )$ will produce the chromaticies and the amplitude dependent path length.

So what went wrong? The original treatment had no concept of normal form what so ever and thus this trivial mistake crept in due to an incorrect averaging process.

There exists other normal forms. For example we can leave one resonance in the map: this could describe appropriately an isolated resonance disrupting phase space. We can also normalize the map in the presence of radiation: this is useful in the theory of synchrotron radiation.