Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems

TL;DR

This work develops a unified variational framework for finite-dimensional integrable hierarchies using Lagrangian 1-forms derived from Lie dialgebras. By encoding the r-matrix structure into a Lagrangian multiform on coadjoint orbits, the authors show that multi-time Euler–Lagrange equations yield compatible Lax flows and that the closure condition is equivalent to Hamiltonians in involution, with a precise double-zero term linking the two viewpoints. The reduction from cotangent bundles provides a bridge to the general coadjoint-orbit construction, and the framework is illustrated through explicit Lagrangian descriptions of the open Toda chain (both non-skew and skew r-matrix realizations) and the rational Gaudin model. The results illuminate the interplay between variational principles, r-matrix theory, and coadjoint geometry, offering a route to variational multiforms for Calogero–Moser–type systems and connections to 4D gauge-theoretic constructions. Overall, the paper provides concrete, computable Lagrangian coefficients that reproduce known Lax formulations while embedding them in a broad, structurally rich variational setting.

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.

Theorems & Definitions (16)

• Proposition 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4
• Remark 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4