Hamiltonian evolutions of twisted polygons in parabolic manifolds: The Lagrangian Grassmannian

Gloria Marí Beffa

Vol. 270 (2014), No. 2, 287–317

DOI: 10.2140/pjm.2014.270.287

Abstract

We show that the moduli space of twisted polygons in $G∕P$, where $G$ is semisimple and $P$ parabolic, and where $\mathfrak{g}$ has two coordinated gradations has a natural Poisson bracket that is directly linked to $G$-invariant evolutions of polygons. This structure is obtained by reducing the quotient twisted bracket on $G^N$ (as defined by M. Semenov-Tian-Shansky) to the moduli space $G^N∕P^N$. We prove that any Hamiltonian evolution with respect to this bracket is induced on $G^N∕P^N$ by an invariant evolution of polygons. We describe in detail the Lagrangian Grassmannian case ($G=Sp\left(2n\right)$) and we describe a submanifold of Lagrangian subspaces where the reduced bracket becomes a decoupled system of Volterra Hamiltonian structures. We also describe a very simple evolution of polygons whose invariants evolve following a decoupled system of Volterra equations.

Keywords

discrete Hamiltonian systems, discrete Lagrangian Grassmannian, Hamiltonian evolutions of polygons in parabolic manifolds, discrete Poisson reduction

Mathematical Subject Classification 2010

Primary: 39AXX

Milestones

Received: 20 May 2013

Revised: 10 April 2014

Accepted: 11 April 2014

Published: 22 August 2014