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We establish a Lagrangian variational framework for general relativisticcontinuum theories that permits the development of the process of Lagrangianreduction by symmetry in the relativistic context. Starting with a continuumversion of the Hamilton principle for the relativistic particle, we deduce twoclasses of reduced variational principles that are associated to eitherspacetime covariance, which is an axiom of the continuum theory, or materialcovariance, which is related to particular properties of the system such asisotropy. The covariance hypotheses and the Lagrangian reduction process areefficiently formulated by making explicit the dependence of the theory on givenmaterial and spacetime tensor fields that are transported by the world-tube ofthe continuum via the push-forward and pull-back operations. It is shown thatthe variational formulation, when augmented with the Gibbons-Hawking-York (GHY)boundary terms, also yields the Israel-Darmois junction conditions between thesolution at the interior of the relativistic continua and the solutiondescribing the gravity field produced outside from it. The expression of thefirst variation of the GHY term with respect to the hypersurface involves someextensions of previous results that we also derive in the paper. We consider indetails the application of the variational framework to relativistic fluids andrelativistic elasticity. For the latter case, our setting also allows toclarify the relation between formulations of relativistic elasticity based onthe relativistic right Cauchy-Green tensor or on the relativistic Cauchydeformation tensor. The setting developed here will be further exploited formodelling purpose in subsequent parts of the paper.

General relativistic Lagrangian continuum theories -- Part I: reduced  variational principles and junction conditions for hydrodynamics and  elasticity