TITLE:
Determination of an exact symplectomorphism 
from its primitive function

AUTHOR:
Alex Haro
Departament de Matematica Aplicada i Analisi
Universitat de Barcelona
Gran Via 585
Barcelona (SPAIN)

ABSTRACT:
To an exact symplectomorphism on an exact symplectic manifold we can
associate a primitive function, which is a primitive of a certain closed 
(in fact, exact) 1-form. This function is also known by many authors as 
generating function, but as we remark, this function does not generate 
our symplectomorphism. In fact, the primitive function generates a family 
of symplectomorphisms and we need some additional information in order 
to determine one of them.

We can relate this determination problem with the interpolation problem, 
that is to say, to get a time-dependent Hamiltonian whose 1-time flow be 
our symplectomorphism. As we shall see, these problems are related with 
a derivation on the Lie algebra of functions (endowed with the Poisson bracket).
We shall need to integrate respect to this derivative.

Our exact symplectic manifold will be the cotangent bundle of a manifold, 
endowed with the canonical symplectic form given by the differential of the 
Liouville form. We shall suppose that our exact symplectomorphism fixes the 
zero-section. We shall work in the analytic category.