Two-dimensional variational systems on the root lattice $Q(A_{N})$

We study certain two-dimensional variational systems, namely pluri-Lagrangian systems on the root lattice $Q(A_{N})$. Here, we follow the scheme which was already used to define two-dimensional pluri-Lagrangian systems on the lattice $\mathbb{Z}^{N}$ and three-dimensional pluri-Lagrangian systems on the lattice $\mathbb{Z}^{N}$ as well as on $Q(A_{N})$. We will show that the two-dimensional pluri-Lagragian systems on $Q(A_{N})$ are more general than the ones on $\mathbb{Z}^{N}$, in the sense that they can encode several different pluri-Lagrangian systems on $\mathbb{Z}^{N}$. This also means that the variational formulation of several systems of certain hyperbolic equations, so-called quad-equations, can be obtained from one and the same pluri-Lagrangian system on $Q(A_{N})$.