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The structure of a diffeomorphism invariant Lagrangians for an extendedobject W embedded in a bulk space M is discussed by following a close analogywith the relativistic particle in electromagnetic field as a system that isreparametrization-invariant. The current construction naturally contains,relativistic point particle, string theory, and Dirac--Nambu--Goto Lagrangianswith Wess--Zumino terms. For Lorentzian metric field, the non-relativistictheory of an integrally submerged W-brane is well defined provided that thebrane does not alter the background interaction fields. A natural time gauge isfixed by the integral submergence (sub-manifold structure) within a Lorentziansignature structure. A generally covariant relativistic theory for thediscussed brane Lagrangians is also discussed. The mass-shell constraint andthe Klein--Gordon equation are shown to be universal when gravity-likeinteraction is present. A construction of the Dirac equation for the W-branethat circumvents some of the problems associated with diffeomorphism invarianceof such Lagrangians by promoting the velocity coordinates into a non-commutinggamma variables is presented.Comment: added references and minor format changes, 5 pages revtex4 style, no  figures, talk presented at the 3rd International Symposium on Quantum Theory  and Symmetries, and the Argonne Brane Dynamics Worksho

ASPECTS OF DIFFEOMORPHISM INVARIANT THEORY OF EXTENDED OBJECTS