  This paper shall be concerned with three main results. After a brief
recollection of basic symplectic geometry, we prove using Moser's homotopy
method a special case of the Strong Darboux Theorem found, for instance, in
Theorem 21.1.6 of [Hor]. Next, we'll prove two conservative extension results
for a diffeomorphism on a circle. One uses Moser's homotopy method but loses a
degree of regularity. The other uses the method of generating functions as
found in [BCW] and [BGV]. Finally, we'll prove a conservative extension result
for a "diffeomorphism" defined on the boundary of $(0, 1)^2$ and use the
techniques developed there and by [M] to prove an ambient Dacarogna-Moser
Theorem.
