Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems

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 ${\mathfrak{g}}^{\ast }$, we give a formula for a Lagrangian multiform describing the commuting flows for $H_k$ on a coadjoint orbit in ${\mathfrak{g}}^{\ast }$. 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 $H_k$ 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.

arXiv commentAuthors' accepted version, to appear in Lett. Math. Phys. New section 4 added about reduction, corrected typos and references added. 42 pages

References in corpus (19)

• Lax pair for the Adler (lattice Krichever-Novikov) System
• Lagrangian multiforms and multidimensional consistency
• Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms
• Discrete-time Calogero-Moser system and Lagrangian 1-form structure
• On the Zakharov-Mikhailov action: $4$d Chern-Simons origin and covariant Poisson algebra of the Lax connection
• A Variational Approach to Lax Representations
• On the Lagrangian structure of integrable hierarchies
• Variational symmetries and Lagrangian multiforms
• Variational symmetries and pluri-Lagrangian systems in classical mechanics
• Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs
• Continuum limits of pluri-Lagrangian systems
• Lagrangian 3-form structure for the Darboux system and the KP hierarchy
• Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies
• Multiform description of the AKNS hierarchy and classical r-matrix
• Hamiltonian multiform description of an integrable hierarchy
• Lagrangian multiforms on Lie groups and non-commuting flows
• Adler-Kostant-Symes systems as Lagrangian gauge theories
• Hamiltonian structures for integrable hierarchies of Lagrangian PDEs
• 3-dimensional mixed BF theory and Hitchin's integrable system

Cited by in corpus (3)

• Lagrangian multiforms and dispersionless integrable systems
• On the geometry of Lagrangian one-forms
• Lagrangian Multiform for Cyclotomic Gaudin Models