  It is well known that a covariant Lagrangian for relativistic charged
particles can lead to a vanishing Hamiltonian. Alternatively, it is shown that
using a "space+time" Lagrangian leads to a new canonical momentum and minimal
coupling rule that describes the coupling of both electromagnetic and
gravitational fields to a relativistic charged particle. Discrepancies between
Hamiltonians obtained by various authors are resolved. The canonical momentum
leads to a new form of the London equations and London gauge. Using the
linearized Einstein field equation in harmonic coordinates, and a
non-relativistic ideal fluid, leads to gravito-electromagnetic field equations.
These are used to obtain new penetration depths for both the magnetic and
gravito-magnetic fields. A key result is that the gravito-magnetic field is
expelled from a superconductor only when a magnetic field is also present. The
flux quantum in the body of a superconductor, and the quantized supercurrent in
a superconducting ring are derived. Lastly, the case of a superconducting ring
in the presence of a charged rotating mass cylinder is used as an example of
applying the formalism developed.
