Marco Pedroni

In this paper we review a recently introduced method for solving the Hamilton–Jacobi equations by the method of Separation of Variables. This method is based on the notion of pencil of Poisson brackets and on the bihamiltonian approach to integrable systems. We discuss how separability conditions can be intrinsically characterized within such a geometrical set-up, the definition of the separation coordinates being encompassed in the bihamiltonian structure itself. We finally discuss these constructions studying in details a particular example, based on a generalization of the classical Toda Lattice.

We present a fairly new and comprehensive approach to the study of stationary flows of the Korteweg–de Vries hierarchy. They are obtained by means of a double restriction process from a dynamical system in an infinite number of variables. This process naturally provides us with a Lax representation of the flows, which is used to find their bi-Hamiltonian formulation. Then we prove the separability of these flows making use of their bi-Hamiltonian structure, and we show that the variables of separation are supplied by the Poisson pair.

In this paper we generalize the Mumford system which describes for any fixed $g$ all linear flows on all hyperelliptic Jacobians of dimension $g$. The phase space of the Mumford system consists of triples of polynomials, subject to certain degree constraints, and is naturally seen as an affine subspace of the loop algebra of $\mathfrak{sl}(2)$. In our generalizations to an arbitrary simple Lie algebra $\mathfrak{g}$ the phase space consists of $\mathrm{dim}\,\mathfrak{g}$ polynomials, again subject to certain degree constraints. This phase space and its multi-Hamiltonian structure is obtained by a Poisson reduction along a subvariety $N$ of the loop algebra $\mathfrak{g}((\lambda-1))$ of $\mathfrak{g}$. Since $N$ is not a Poisson subvariety for the whole multi-Hamiltonian structure we prove an algebraic. Poisson reduction theorem for reduction along arbitrary subvarieties of an affine Poisson variety; this theorem is similar in spirit to the Marsden–Ratiu reduction theorem. We also give a different perspective on the multi-Hamiltonian structure of the Mumford system (and its generalizations) by introducing a master symmetry; this master symmetry can be described on the loop algebra $\mathfrak{g}((\lambda-1))$ as the derivative in the direction of $\lambda$ and is shown to survive the Poisson reduction. When acting (as a Lie derivative) on one of the Poisson structures of the system it produces a next one, similarly when acting on one of the Hamiltonians (in involution) or their (commuting) vector fields it produces a next one. In this way we arrive at several multi-Hamiltonian hierarchies, built up by a master symmetry.