  This paper provides a precise sense in which the time t map for the Euler
equations of an ideal fluid in a region in R^n (or a smooth compact n-manifold
with boundary) is a Poisson map relative to the Lie-Poisson bracket associated
with the group of volume preserving diffeomorphism group. This is interesting
and nontrivial because in Eulerian representation, the time t maps need not be
C^1 from the Sobolev class H^s to itself (where s > (n/2) + 1). The idea of how
this difficulty is overcome is to exploit the fact that one does have
smoothness in the Lagrangian representation and then carefully perform a
Lie-Poisson reduction procedure.
