Abstract

During the DIS programme of 2009 Lobb and Nijhoff introduced a novel point of view on the Lagrangian theory of systems integrable in the sense of multidimensional consistency. The key observation was that suitably chosen Lagrangians obey a “closure” relation when embedded in multidimensional, discrete or continuous, space-time and subject to the equations of the motion. The apparent universality of this phenomenon has now been confirmed for many integrable systems, both continuous and discrete with defining equations in one, two and three dimensions. From a physics point of view this could set a new paradigm for least-action principles in physics where the Lagrangian itself is a solution of a system of generalized Euler-Lagrange equations, and where the geometry in the embedding space is a variational variable together with the field variables.