Gaetano Vilasi

The local structure of n-Poisson and n-Jacobi manifolds☆

Abstract n -Lie algebra structures on smooth function algebras given by means of multi-differential operators, are studied and their canonical forms are obtained. Necessary and sufficient conditions for the sum and the wedge product of two n -Poisson structures to be again a multi-Poisson are found. It is proven that the canonical n -vector on the dual of an n -Lie algebra g is n -Poisson iff dim g ⩽ n +1. The problem of compatibility of two n -Lie algebra structures is analyzed and the compatibility relations connecting hereditary structures of a given n -Lie algebra are obtained. ( n +1)-dimensional n -Lie algebras are classified and their “elementary particle-like” structure is discovered. Some simple applications to dynamics are discussed.

A new characterization of completely integrable systems

SummaryA characterization of separability, projectability and integrability of dynamical systems in terms of the spectral properties of invariant mixed tensor fields with vanishing Nijenhuis tensor is given. In addition, some preliminary results on the inverse problem (from Liouville integrability to Lax representation) are illustrated.RiasuntoSi presenta una caratterizzazione della separabilità, proiettabilità ed integrabilità di sistemi dinamici in termini delle proprietà spettrali di campi di tensori misti invarianti con tensore di Nijenhuis nullo. Sono inoltre illustrati alcuni risultati preliminari sul problema inverso (dall’integrabilità à la Liouville alla rappresentazione di Lax).РезюмеПредлагается характеристика разделимости, проектируемости и интегрируемости динамических систем в терминах спектральных свойств инвариантных смешанных полей, с нулевым тензором Нидженуиса. Кроме того, обсуждаются некоторые предварительные результаты, касающиеся обратной проблемы (от интегрируемости Лиувилля к представлению Лакса).

On the stability of the Einstein Static Universe in Massive Gravity

We consider static cosmological solutions along with their stability properties in the framework of a recently proposed theory of massive gravity. We show that the modifcation introduced in the cosmological equations leads to several new solutions, only sourced by a perfect fluid, generalizing the Einstein Static Universe found in General Relativity. Using dynamical system techniques and numerical analysis, we show that the found solutions can be either neutrally stable or unstable against spatially homogeneous and isotropic perturbations.

A geometrical setting for the Lax representation

Abstract A geometrical interpretation of the Lax representation is suggested; it is read as the condition on a section of a fiber bundle based on the phase manifold to be covariantly constant, with respect to a suitable connection, along the dynamics.

Vacuum Einstein metrics with bidimensional Killing leaves ∗ I-Local aspects.

Abstract The solutions of vacuum Einsteins field equations, for the class of Riemannian metrics admitting a non-Abelian bidimensional Lie algebra of Killing fields, are explicitly described. They are parametrized either by solutions of a transcendental equation (the tortoise equation ), or by solutions of a linear second order differential equation in two independent variables. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a 3-dimensional Lie algebra of Killing fields with bidimensional leaves.

When do recursion operators generate new conservation laws

Abstract Two hamiltonian structures for the Kepler problem are constructed. However, the recursion operator T does not lead to a new functionally independent hamiltonian. The result holds for any vector field satisfying the energy-period theorem hypotheses, and admitting a tensor field T which factorizes via two symplectic structures.

Noncommutative integrability and recursion operators

Abstract Geometric structures underlying commutative and noncommutative integrable dynamics are analyzed. They lead to a new characterization of noncommutative integrability in terms of spectral properties and of Nijenhuis torsion of an invariant (1,1) tensor field. The construction of compatible symplectic structures is also discussed.

SYMPLECTIC STRUCTURES AND QUANTUM MECHANICS

Canonical coordinates for the Schrodinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schrodinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a recursion operator which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket. The approach may provide a good starting point to get a clear interpretation of Quantum Mechanics in the general setting, provided by Stone–von Neumann theorem, of Symplectic Mechanics. It may give new tools to solve in the general case the inverse problem of quantum mechanics whose solution is given up to now only for one-dimensional systems by the Gel’fand-Levitan-Marchenko formula.

On the hamiltonian structures of the Korteweg-de Vries and sine-Gordon theories

Abstract We construct a Miura-like transformation which transforms symplectic forms, Poisson brackets and conservation laws of KdV into those of SG, and then, transforms the hamiltonian flow of KdV into a one-parameter symmetry group of SG.

A geometrical approach to the integrability of soliton equations

A separability criterion in one-degree-of-freedom dynamics, suitable for soliton equations, is given in terms of a geometrical structure on the phase manifold. For solitonic degrees of freedom, i.e., those corresponding to the discrete spectrum of the associated Lax operator, integrability is a priori proved.