  We give an exponentially-accurate normal form for a Lagrangian particle
moving in a rotating shallow-water system in the semi-geostrophic limit, which
describes the motion in the region of an exponentially-accurate slow manifold
(a region of phase space for which dynamics on the fast scale are exponentially
small in the Rossby number). The result extends to numerical solutions of this
problem via backward error analysis, and extends to the Hamiltonian
Particle-Mesh (HPM) method for the shallow-water equations where the result
shows that HPM stays close to balance for exponentially-long times in the
semi-geostrophic limit. We show how this result is related to the variational
asymptotics approach of [Oliver, 2005]; the difference being that on the
Hamiltonian side it is possible to obtain strong bounds on the growth of fast
motion away from (but near to) the slow manifold.
