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