Georgia Institute of Technology College of Sciences

The conventional point of view is that the Lagrangian is a scalar object, which through the Euler-Lagrange equations provides us with one single equation. However, there is a key integrability property of certain discrete systems called multidimensional consistency, which implies that we are dealing with infinite hierarchies of compatible equations. Wanting this property to be reflected in the Lagrangian formulation, we arrive naturally at the construction of Lagrangian multiforms, i.e., Lagrangians which are the components of a form and satisfy a closure relation. Then we can propose a new variational principle for discrete integrable systems which brings in the geometry of the space of independent variables, and from this principle derive any equation in the hierarchy.