Title: Lagrangian multiforms and conservation laws

Abstract:

Lagrangian multiform theory is a variational principle for hierarchies of commuting differential equations. Lagrangian 1-forms typically describe systems of commuting ODEs, Lagrangian 2-forms typically describe hierarchies of (1+1)-dimensional PDEs, and so on. Typical examples include soliton hierarchies such as the KdV. In these examples, there is a close relation between the existence of a Lagrangian multiform and the fact that the flows are variational symmetries of each other.

In this talk, we give a number of examples of Lagrangian multiforms that break this pattern. We will present Lagrangian 2-forms that can be interpreted as conservation laws for 3-dimensional PDEs. In these examples, the conservation Law does not arise via Noether's theorem from a known Lagrangian. Almost the opposite is true: the construction of a Lagrangian multiform relies on the existence of a suitable conservation Law for an equation that is not variational by itself.

We also give additional examples of Lagrangian multiforms for dispersionless integrable systems, including some systems that arise via hydrodynamic reductions of Plebanski's second heavenly equation.