  For multi-component superconductors, it is known that the presence of
symmetry breaking fields can lead to multiple superconducting phase
transitions. This phenomenon is best illustrated in UPt$_3$. Motivated by
recent small angle neutron scattering experiments on the vortex state of
UPt$_3$, the London theory in the vicinity of such phase transitions is
determined. It is found that the form of this London theory is in general quite
different than that for conventional superconductors. This is due to the
existence of a diverging correlation length associated with these phase
transitions. One striking consequence is that non-trivial vortex lattices exist
arbitrarily close to H$_{c1}$. It is also found that the penetration depth
develops a novel temperature dependence and anisotropy. Results of this theory
for UPt$_3$ are derived. Possible applications to CeIn$_3$,
U$_{1-x}$Th$_x$Be$_{13}$, electron doped cuprate superconductors,
Sr$_2$RuO$_4$, and MgCNi$_3$ are also discussed.
