  The structure of a diffeomorphism invariant Lagrangians for an extended
object W embedded in a bulk space M is discussed by following a close analogy
with the relativistic particle in electromagnetic field as a system that is
reparametrization-invariant. The current construction naturally contains,
relativistic point particle, string theory, and Dirac--Nambu--Goto Lagrangians
with Wess--Zumino terms. For Lorentzian metric field, the non-relativistic
theory of an integrally submerged W-brane is well defined provided that the
brane does not alter the background interaction fields. A natural time gauge is
fixed by the integral submergence (sub-manifold structure) within a Lorentzian
signature structure. A generally covariant relativistic theory for the
discussed brane Lagrangians is also discussed. The mass-shell constraint and
the Klein--Gordon equation are shown to be universal when gravity-like
interaction is present. A construction of the Dirac equation for the W-brane
that circumvents some of the problems associated with diffeomorphism invariance
of such Lagrangians by promoting the velocity coordinates into a non-commuting
gamma variables is presented.
