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 solutionsbifurcating from a trivial state can be described by areduced finite-dimensional variational problem of Lagrangiantype. This provides a rigorous justification of rod theoryfrom fully nonlinear three-dimensional elasticity.The book will be of interest to researchers working inclassical mechanics, dynamical systems, elliptic variationalproblems, and continuum mechanics. It begins with theelements of Hamiltonian theory and center manifold reductionin order to make the methods accessible to non-specialists,from graduate student level.