  The use of Liouvillian forms to obtain symplectic maps for constructing
numerical integrators is a natural alternative to the method of generating
functions, and provides a deeper understanding of the geometry of this
procedure. Using Liouvillian forms we study the generating function introduced
by Poincar\'e (1899) and its associated symplectic map. We show that in this
framework, Poincar\'e's generating function does not correspond to the
symplectic mid-point rule, but to the identity map. We give an interpretation
of this result based on the original framework constructed by Poincar\'e.
