Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

Many integrable hierarchies of differential equations allow a variational description, called Lagrangian multiform or pluri-Lagrangian structure. The fundamental object in this theory is not Lagrange function but $d$-form that integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for field to solution the problem. In paper we present procedure obtain Hamiltonian structures from formulation an hierarchy PDEs. As prelude, review similar ODEs. We show exterior derivative closely related Poisson brackets between corresponding Hamilton functions. ODE (Lagrangian 1-form) case discuss as examples Toda and Kepler PDE 2-form) potential Schwarzian Korteweg-de Vries hierarchies, well Boussinesq hierarchy.