  Let K be a compact Lie group and fix an invariant inner product on its Lie
algebra. Given a Hamiltonian action of K on a compact symplectic manifold X,
the normsquare of the moment map defines a Morse stratification of X by locally
closed symplectic submanifolds such that the stratum to which any x in X
belongs is determined by the limiting behaviour of its downwards trajectory
under the gradient flow with respect to a suitably compatible Riemannian metric
on X. The open stratum indexed by 0 retracts K-equivariantly via this gradient
flow to the minimum which is the zero-locus of the moment map (if this is not
empty). The usual 'symplectic quotient' for the action of K on any other
stratum is empty. Nonetheless, motivated by recent results in non-reductive
geometric invariant theory, we find that the symplectic quotient construction
can be modified to provide natural 'symplectic quotients' for the unstable
strata labelled by nonzero indices. There is an analogous infinite-dimensional
picture for the Yang--Mills functional over a Riemann surface with strata
determined by Harder-Narasimhan type.
