Lie Factorization of DAMAPS

DAMAPS can factorized in many different ways. FPP provided a few built-in ways. Of course users can (and generally must) produce new ways depending on the problems they face.

In all these factorization, the general vector field is computed. It is the primary Lie representation in FPP. In addition, a Poisson bracket field is also computed for convenience. Notice that this often provides a way to "symplectify" a map.

Here are the ways:

1) One Lie Exponent

 This factorization was used in the example introducing vector fields. Please refer to the page by clicking here.

We must add that it is  not  always possible to express a map as a single Lie exponent. FPP uses an iterative scheme to compute the vector F. If the map is far from the identity map, this scheme may not converge even if F exists!

    M=Exp(F·grad)Id

2) Dragt-Finn Factorization

The Dragt-Finn factorization and its alter ego the reverse Dragt-Finn appear naturally in perturbation theory on map. (Normal form). It is also the kind of representation used to store maps in the code Marylie.

    M = exp(O_No)...exp(O₂) L T Id         L is the linear map and T contains the constant part if any.

3) Reverse Dragt-Finn Factorization

    M = Lexp(O₂)...exp(O_No) T Id                O_k=F_k · grad

4) Are there other useful factorization? Answer: yes, but I cannot provide them all! However, using the tools of FPP on can program anything.

ALL THESE LIE REPRESENTATION HANDLE THE CONSTANT PART CORRECTLY IF PRESENT

Click here for example