Workshop — Lagrangian Multiform Theory and Pluri-Lagrangian Systems

In 2009 a new direction in variational calculus was proposed, called Lagrangian multiform theory (also referred to as pluri-Lagrangian systems), which breaks with the conventional approach in a fundamental way: in this new theory the Lagrangian is no longer a simple function of the fields, but an extended object (a differential or difference p-form). Furthermore, the Lagrangian is not put in by hand, i.e. chosen on the basis of secondary considerations (such as requiring certain symmetries), but emerges itself as a solution of the principal variational equations. These equations come from varying both the fields (as in the conventional theory) as well as the hyper-surfaces over which the action is integrated in the space of independent variables. The theory forms unique departure from the standard picture, in that it forms the first least-action principle that merges variational calculus with the key integrability notion of 'multidimensional consistency' (which is the key property of integrable systems, comprising the class of remarkable model equations which allow for the coexistence of infinite families of equations on one and the same mathematical functions). The fast development of the theory in the last decade, establishing the basic principles of the classical theory, demonstrating the applications to integrable systems and making the first steps towards a quantum theory, make it very timely to have a meeting on this new subject. The ambition of this novel viewpoint on the least-action principle, possibly as a candidate for a new foundational principle of physics, compel to branch out to researchers working in fundamental physics.