Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems

Lagrangian multiforms provide a variational framework to describe integrablehierarchies. The case of Lagrangian $1$-forms covers finite-dimensionalintegrable systems. We use the theory of Lie dialgebras introduced bySemenov-Tian-Shansky to construct a Lagrangian $1$-form. Given a Lie dialgebraassociated with a Lie algebra $\mathfrak{g}$ and a collection $H_k$,$k=1,\dots,N$, of invariant functions on $\mathfrak{g}^*$, we give a formulafor a Lagrangian multiform describing the commuting flows for $H_k$ on acoadjoint orbit in $\mathfrak{g}^*$. We show that the Euler-Lagrange equationsfor our multiform produce the set of compatible equations in Lax formassociated with the underlying $r$-matrix of the Lie dialgebra. We establish astructural result which relates the closure relation for our multiform to thePoisson involutivity of the Hamiltonians $H_k$ and the so-called ``doublezero'' on the Euler-Lagrange equations. The construction is extended to ageneral coadjoint orbit by using reduction from the free motion of thecotangent bundle of a Lie group. We illustrate the dialgebra construction of aLagrangian multiform with the open Toda chain and the rational Gaudin model.The open Toda chain is built using two different Lie dialgebra structures on$\mathfrak{sl}(N+1)$. The first one possesses a non-skew-symmetric $r$-matrixand falls within the Adler-Kostant-Symes scheme. The second one possesses askew-symmetric $r$-matrix. In both cases, the connection with the well-knowndescriptions of the chain in Flaschka and canonical coordinates is provided.