Type DAMAP is a collection of 2*ND Taylor series. The size of ND is determined by the appropriate call to the function INIT. For example:

CALL INIT(NO1=2,ND1=1,NP1=0,NDPT1 =0) 

This call will produce a package dealing with second degree (NO1=2) polynomial maps in one degree of freedom  (ND1=1).

 

Example:   A rotation of angle 30o  followed by a sextupole kick. Click here for source code.

Usually the DAMAP represents an expansion around some special orbit such as the closed orbit. Therefore, in DAMAP operation, it is usual to ignore the constant part.

Other Forms for the DAMAP

Damaps can be transformed into convenient vector fields representations and also into a characteristic function representation. These things are explained the Big Table page (here).

One goes from DAMAP to another type of map representation using the FORTRAN90 assignment (=). For example, if DF is of type DRAGTFINN and M is of type DAMAP, one can get the Dragt-Finn representation by simply typing DF=M.

However we list here the kinds of Maps FPP provides. (L=Linear part, a DAMAP, T constant part, ID=identity map put here out of mathematical rigor,   Ok=Fk · grad ).

  1. DAMAPs (of course)

  2. Dragt-Finn : M = exp(ONo)...exp(O2) L T Id

  3. Reverse Dragt-Finn : M = Lexp(O2)...exp(ONo) T Id   

  4. One Lie Exponent : M = exp(F·grad)Id    F = Σ Fk

  5. Normal Form:  M = AoRoA-1 where R is very special, often a rotation.

  6. Generating function: used if you want an exact symplectic map for tracking. (limited use)

Representations 2,3 and 4 rely on vector fields (on general vector fields although the Poisson Bracket field is also available for free). The One-Lie-Exponent  representation is possible for maps near the identity. It is computed by an iterative procedure which does not necessarily converge on maps too far from identity. The Normal form is also a representation which is often ill behaved, near resonances for example. The generating function representation can also be ill-behaved since it requires a partial inversion of the p-variables (momenta).

We do not discuss these maps any further: A lot of  examples are found in the Big Table page by following the links. Click here.