Abstract
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.
Highlights
Some of the most powerful descriptions of integrable systems use the Hamiltonian formalism
In the case of integrable PDEs, which have infinitely many degrees of freedom, integrability is often defined as having an infinite number of commuting Hamiltonian flows, where again each two Hamilton functions have a zero Poisson bracket
Many integrable PDEs have two compatible Poisson brackets. Such a bi-Hamiltonian structure can be used to obtain a recursion operator, which in turn is an effective way to construct an integrable hierarchy of PDEs
Summary
Some of the most powerful descriptions of integrable systems use the Hamiltonian formalism. In the case of integrable PDEs, which have infinitely many degrees of freedom, integrability is often defined as having an infinite number of commuting Hamiltonian flows, where again each two Hamilton functions have a zero Poisson bracket. Many integrable PDEs have two compatible Poisson brackets Such a bi-Hamiltonian structure can be used to obtain a recursion operator, which in turn is an effective way to construct an integrable hierarchy of PDEs. In many cases, especially in mechanics, Hamiltonian systems have an equivalent Lagrangian description. ]ocnmp[ Hamiltonian structures for hierarchies of Lagrangian PDEs equations (see for example [14], [3], [12, Chapter 12]). It is called the theory of Lagrangian multiform systems, or, of pluri-Lagrangian systems.
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