Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems
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| Format: | Preprint |
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2023
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| _version_ | 1866910917398102016 |
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| author | Caudrelier, Vincent Dell'Atti, Marta Singh, Anup Anand |
| author_facet | Caudrelier, Vincent Dell'Atti, Marta Singh, Anup Anand |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_07339 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems Caudrelier, Vincent Dell'Atti, Marta Singh, Anup Anand Mathematical Physics High Energy Physics - Theory Exactly Solvable and 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 $\mathfrak{g}$ 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. |
| title | Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems |
| topic | Mathematical Physics High Energy Physics - Theory Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2307.07339 |