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 g and a collection H, k=1,⋯,N, of invariant functions on g, we give a formula for a Lagrangian multiform describing the commuting flows for H on a coadjoint orbit in g. 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 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 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.