Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems: the open Toda chain

Marta Dell'Atti, (University of Portsmouth)

Tuesday 7th November, 2023 16:00-17:00 Maths 311B

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 general Lagrangian 1-form. Given a Lie dialgebra associated with a Lie algebra g and a collection of invariant functions Hk ( k = 1, ..., N ) on g*, we give a formula for a Lagrangian multiform describing the commuting flows for Hk on a coadjoint orbit in 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 illustrate these points and our construction with the open Toda chain.

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