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The Hamiltonian structure of spacetimes with two commuting Killing vectorfields is analyzed for the purpose of addressing the various problems of timethat arise in canonical gravity. Two specific models are considered: (i)cylindrically symmetric spacetimes, and (ii) toroidally symmetric spacetimes,which respectively involve open and closed universe boundary conditions. Foreach model canonical variables which can be used to identify points of spaceand instants of time, {\it i.e.}, internally defined spacetime coordinates, areidentified. To do this it is necessary to extend the usual ADM phase space by afinite number of degrees of freedom. Canonical transformations are exhibitedthat identify each of these models with harmonic maps in the parametrized fieldtheory formalism. The identifications made between the gravitational models andharmonic map field theories are completely gauge invariant, that is, nocoordinate conditions are needed. The degree to which the problems of time areresolved in these models is discussed.Comment: 36 pages, Te

Internal time formalism for spacetimes with two Killing vectors