Transforming Lie Operators Directly

 

Consider the following sequence of Lie maps using Vector Fields:

    M=A Exp(F·grad) A-1 =  Exp( A F·grad A-1 ) = Exp( Ft·grad )

or in the language of ordinary DAMAPS,

    M=A-1 o {Exp(F·grad)Id} o A = Exp( Ft·grad )Id

FPP can compute Ft directly with the syntax:

 Ft =A*F

of course the same thing applies to Poisson Bracket operators.

 

Click here for Example of with Vector Field Operators

Click here for Example of with Poisson Bracket Operators

 

Warning: It makes sense to change a Poisson Bracket Operator by a non-symplectic map and obtain a general vector field. For example:

Ft·grad = A :h: A-1

It is definitely a meaningful equation. However, the following WILL NOT WORK in FPP if A is not symplectic:

Ft= A h              (1)

The reasons are due to programming complexity. To achieve the intended result of equation (1), first turn h into a vector field. The following will work:

F= h              

Ft= A F