Lagrangian multiforms on coadjoint orbits

First introduced in 2009, Lagrangian multiforms provide a variational framework for describing integrable hierarchies using a generalised variational principle applied to an appropriate generalisation of a classical action. In this talk, I will give an overview of this framework and report some recent results based on joint works with V. Caudrelier, M. Dell’Atti, and B. Vicedo. In particular, I will explain how one can use the theory of Lie dialgebras to systematically construct Lagrangian multiforms living on coadjoint orbits for a large class of finite-dimensional integrable systems. Lie dialgebras are related to Lie bialgebras but are more flexible in that they incorporate the case of non-skew-symmetric r-matrices. After discussing some structural results for our multiforms, I will use the examples of the rational and the cyclotomic Gaudin models to illustrate the scope of our construction.