Hamiltonian Evolutions of Twisted Polygons in Parabolic Manifolds: the Lagrangian Grassmannian

In this paper we show that the moduli space of twisted polygons in G/P , where G is semisimple and P parabolic, and where 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 GN defined in [23] to the moduli space GN/PN . We prove that any Hamiltonian evolution with respect to this bracket is induced on GN/PN by an invariant evolution of polygons. We describe in detail the Lagrangian Grassmannian case (G = Sp(2n)) 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.