  We construct a density-functional formalism adapted to uniform external
magnetic fields that is intermediate between conventional Density Functional
Theory and Current-Density Functional Theory (CDFT). In the intermediate
theory, which we term LDFT, the basic variables are the density, the canonical
momentum, and the paramagnetic contribution to the magnetic moment. Both a
constrained-search formulation and a convex formulation in terms of
Legendre--Fenchel transformations are constructed. Many theoretical issues in
CDFT find simplified analogues in LDFT. We prove results concerning
$N$-representability, Hohenberg--Kohn-like mappings, existence of minimizers in
the constrained-search expression, and a restricted analogue to gauge
invariance. The issue of additivity of the energy over non-interacting
subsystems, which is qualitatively different in LDFT and CDFT, is also
discussed.
