TITLE:
The primitive function of an exact symplectomorphism.

AUTHOR:
Alex Haro
Departament de Matematica Aplicada i Analisi
Universitat de Barcelona
Gran Via, 585
08007 Barcelona (SPAIN)
phone: +34 93 4021634
fax:   +34 93 4021601
E-mail: haro@mat.ub.es

Department of Mathematics
RLM 8100
University of Texas at Austin
Austin TX 78712 (USA)
phone: +1 (512) 2326187
fax:   +1 (512) 4719038
E-mail: haro@math.utexas.edu

ABSTRACT:
In this paper we propose a new way to describe an exact symplectomorphism, 
besides the classical methods using generating functions and Lie transforms.
We use the so-called primitive function of an exact symplectomorphism. 

Our method allows us to obtain 'all' the exact
symplectic dynamics around invariant exact Lagrangian manifolds. 
Accordingly, it can be used to study the symplectomorphism in a vicinity 
of such a manifold (for instance, to obtain normal forms).

To reconstruct the exact symplectomorphism from its primitive function,
we propose two methods, both of them implementable on a computer:
- a `generating-function-like' method: but, unlike the classical one,
our method does not involve the implicit function theorem (and non-degeneracy 
conditions are not needed) and compositions, and the coefficients of the 
symplectomorphism are computed by means of recurrences only;
- a `Hamiltonian like method': we demonstrate how to construct a 
time dependent Hamiltonian whose time-1 map is the symplectomorphism.

Specifically, let O be the zero-section of the cotangent bundle B of
a real analytic manifold M. Let F: (B,O) -> (B,O) be a real analytic local
diffeomorphism preserving the canonical symplectic form 'o= da' of N, where 
'a' denotes the Liouville form. Suppose that 'F^* a - a' is an exact 
form 'dS'. We prove that:
- We can reconstruct 'F' from 'S' and the restriction 'f= F_O'.
- If 'f' is included into a flow, then 'F' can be included into a 
Hamiltonian flow.