Hamiltonian multiforms and multi-time Poisson brackets

Matteo Stoppato. Leeds

Lagrangians multiform were proposed by Nijhoff and Lobb in 2009 to capture multidimensional consistency for integrable hierarchies in a Lagrangian fashion. Using covariant Hamiltonian field theory techniques we investigate the Hamiltonian structure behind Lagrangian multiforms.  We introduce the notion of the symplectic multiform, which we use to define Hamiltonian multiforms  (via a suitable Legendre-like transform) and multi-time Poisson brackets that talk to every level of the integrable hierarchy. We will also mention that these Poisson brackets possess a classical r-matrix structure.

This is joint work with Vincent Caudrelier.