Mats Vermeeren - Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

ocnmp:7491 - Open Communications in Nonlinear Mathematical Physics, September 10, 2021, Volume 1 - https://doi.org/10.46298/ocnmp.7491
Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

Authors: Mats Vermeeren

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.


Volume: Volume 1
Published on: September 10, 2021
Submitted on: May 18, 2021
Keywords: Nonlinear Sciences - Exactly Solvable and Integrable Systems,Mathematical Physics,37K10, 37K06, 70S05


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