Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems

Lagrangian multiforms provide a variational framework to describe integrable hierarchies. The case of Lagrangian 1-forms covers finite-dimensional integrable systems. We use the theory of Lie dialgebras introduced by Semenov-Tian-Shansky to construct a Lagrangian 1-form. Given a Lie dialgebra associated with a Lie algebra \(\mathfrak {g}\) $\mathfrak{g}$ and a collection \(H_k\) $H_k$ , \(k=1,\dots ,N\) $k=1,\cdots ,N$ , of invariant functions on \(\mathfrak {g}^*\) ${\mathfrak{g}}^{\ast }$ , we give a formula for a Lagrangian multiform describing the commuting flows for \(H_k\) $H_k$ on a coadjoint orbit in \(\mathfrak {g}^*\) ${\mathfrak{g}}^{\ast }$ . We show that the Euler–Lagrange equations for our multiform produce the set of compatible equations in Lax form associated with the underlying r-matrix of the Lie dialgebra. We establish a structural result which relates the closure relation for our multiform to the Poisson involutivity of the Hamiltonians \(H_k\) $H_k$ and the so-called double zero on the Euler–Lagrange equations. The construction is extended to a general coadjoint orbit by using reduction from the free motion of the cotangent bundle of a Lie group. We illustrate the dialgebra construction of a Lagrangian multiform with the open Toda chain and the rational Gaudin model. The open Toda chain is built using two different Lie dialgebra structures on \(\mathfrak {sl}(N+1)\) $\mathfrak{sl}(N+1)$ . The first one possesses a non-skew-symmetric r-matrix and falls within the Adler–Kostant–Symes scheme. The second one possesses a skew-symmetric r-matrix. In both cases, the connection with the well-known descriptions of the chain in Flaschka and canonical coordinates is provided.