  For a field k, let G be a reductive k-group and V an affine k-variety on
which G acts. Using the notion of cocharacter-closed G(k)-orbits in V, we prove
a rational version of the celebrated Hilbert-Mumford Theorem from geometric
invariant theory. We initiate a study of applications stemming from this
rationality tool. A number of examples are discussed to illustrate the concept
of cocharacter-closure and to highlight how it differs from the usual
Zariski-closure. When k is perfect, we give a criterion in terms of closed
orbits for G to be k-anisotropic, answering a question of Borel.
