Abstract

I will present the relatively recent notion of Lagrangian multiforms whose aim is to capture integrability in a purely variational fashion. Lagrangian multiforms are ubiquitous in classical integrable models: they can be defined and used in continuous or discrete systems, finite or infinite dimensional (field theories). So far, the Hamiltonian formalism has been the overwhelming winner to define integrability, rooted in the Liouville-Arnold theorem. I will show how Lagrangian multiforms offer variational counterparts to the established Hamiltonian criteria for integrability, e.g. Poisson commuting Hamiltonians or the appearance of the classical Yang-Baxter equation. This continues the long interplay between Hamiltonian and Lagrangian formalisms, within the realm of integrability. Each one offers advantages and limitations. I will argue that, for field theories, Lagrangian multiforms offer a covariant and more natural criterion for integrability than the Hamiltonian counterpart. As a main example, I will use the AKNS hierarchy which contains several famous equations, e.g. nonlinear Schrödinger. I will comment on the open issue of quantization with hints that Feynman path integral techniques could perhaps be used. A long-term goal, but at this stage speculative, would be to offer a path integral alternative to the well-established, Hamiltonian driven, machinery of the Quantum Inverse Scattering Method.