The theory of center manifold reduction is studied in thismonograph in the context of (infinite-dimensional) Hamil-tonian and Lagrangian systems. The aim is to establish a"natural reduction method" for Lagrangian systems to theircenter manifolds. Nonautonomous problems are considered aswell assystems invariant under the action of a Lie group (including the case of relative equilibria).The theory is applied to elliptic variational problemsoncylindrical domains. As a result, all bounded soluti ...