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 g and a collection Hₖ H k , k=1, ,N k = 1 , ⋯ , N , of invariant functions on g^* g ∗ , we give a formula for a Lagrangian multiform describing the commuting flows for Hₖ H k on a coadjoint orbit in g^* 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 Hₖ 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 sl(N+1) 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.