On the Hamiltonian and Lagrangian structures of time-dependent reductions of evolutionary PDEs

In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of time--dependent symmetries. In particular we describe how that the finite dimensional Hamiltonian structure of the reduced system is obtained from the Hamiltonian structure of the initial PDE and we construct the time--dependent Hamiltonian function. We also present a very general Lagrangian formulation of the procedure of reduction. As an application we consider the case of the Painlev\'e equations PI, PII, PIII, PVI and also certain higher order systems appeared in the theory of Frobenius manifolds.