Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems

TL;DR

This paper develops a variational framework using Lagrangian multiforms and Lie dialgebras to describe finite-dimensional integrable systems, establishing links between multiform closure, Poisson involutivity, and integrability, with applications to Toda chains and Gaudin models.

Contribution

It introduces a novel construction of Lagrangian 1-forms for integrable systems via Lie dialgebras, connecting multiform closure to involutivity and extending to coadjoint orbits.

Findings

Constructed Lagrangian multiforms for integrable hierarchies on coadjoint orbits.

Linked multiform closure to Poisson involutivity of Hamiltonians.

Applied the framework to Toda chain and Gaudin model, illustrating different r-matrix structures.

Abstract

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 $\mathfrak{g}$ and a collection $H_k$, $k=1,\dots,N$, of invariant functions on $\mathfrak{g}^*$, we give a formula for a Lagrangian multiform describing the commuting flows for $H_k$ on a coadjoint orbit in $\mathfrak{g}^*$. 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 $H_k$ 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 $\mathfrak{sl}(N+1)$. 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.