Semi-discrete Lagrangian 2-forms and the Toda hierarchy

We present a variational theory of integrable differential-difference equations (semi-discrete systems). This is natural extension the ideas known by names "Lagrangian multiforms" and "Pluri-Lagrangian systems", which have previously been established in both fully discrete continuous cases. The main feature these to capture hierarchy commuting single principle. Our example illustrate new semi-discrete Lagrangian multiforms Toda lattice. ODE describes evolution time 1-dimensional lattice particles with nearest-neighbour interaction. It part an ODEs, each involves derivative respect variable number shifts. will use multiform derive PDEs variables hierarchy, do not involve any As second example, we briefly discuss potential KdV equation, related Volterra