Separation of variables in multi-Hamiltonian systems: an application to the Lagrange top

Starting from the tri-Hamiltonian formulation of the Lagrange top in a six-dimensional phase space, we discuss the reduction of the vector field and of the Poisson tensors. We show explicitly that, after the reduction on each one of the symplectic leaves, the vector field of the Lagrange top is separable in the sense of Hamilton-Jacobi.Comment: report to XVI NEEDS (Cadiz 2002): 15 pages, no figures, LaTeX. To appear in Theor. Math. Phy

Quasi-BiHamiltonian Systems and Separability

Two quasi--biHamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis coordinates) and its separability is proved.Comment: 10 pages, AMS-LaTeX 1.1, to appear in J. Phys. A: Math. Gen. (May 1997

The quasi-bi-Hamiltonian formulation of the Lagrange top

Starting from the tri-Hamiltonian formulation of the Lagrange top in a six-dimensional phase space, we discuss the possible reductions of the Poisson tensors, the vector field and its Hamiltonian functions on a four-dimensional space. We show that the vector field of the Lagrange top possesses, on the reduced phase space, a quasi-bi-Hamiltonian formulation, which provides a set of separation variables for the corresponding Hamilton-Jacobi equation.Comment: 12 pages, no figures, LaTeX, to appear in J. Phys. A: Math. Gen. (March 2002

Generalized Nijenhuis Torsions and Block-Diagonalization of Operator Fields.

The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhuis and Haantjes, offers new tools for the study of normal forms of operator fields. We prove a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level m vanishes, there exists a local chart where all operators can be simultaneously block-diagonalized. We also introduce the notion of generalized Haantjes algebra, consisting of operators with a vanishing higher-level torsion, as a new algebraic structure naturally generalizing standard Haantjes algebras

Generalized Lenard Chains, Separation of Variables and Superintegrability

We show that the notion of generalized Lenard chains naturally allows formulation of the theory of multi-separable and superintegrable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function defined on a four-dimensional \omega N manifold guarantees the separation of variables. As an application, we construct such chains for the H\'enon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.Comment: 14 pages Revte

On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian

It is shown that a class of dynamical systems (encompassing the one recently considered by F. Calogero [J. Math. Phys. 37 (1996) 1735]) is both quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the separability of these systems; the second one is obtained trough a non canonical map whose form is directly suggested by the associated Nijenhuis tensor.Comment: 11 pages, AMS-LaTex 1.

Reduction of bihamiltonian systems and separation of variables: an example from the Boussinesq hierarchy

We discuss the Boussinesq system with $t_5$ stationary, within a general framework for the analysis of stationary flows of n-Gel'fand-Dickey hierarchies. We show how a careful use of its bihamiltonian structure can be used to provide a set of separation coordinates for the corresponding Hamilton--Jacobi equations.Comment: 20 pages, LaTeX2e, report to NEEDS in Leeds (1998), to be published in Theor. Math. Phy

On the integrability of stationary and restricted flows of the KdV hierarchy.

A bi--Hamiltonian formulation for stationary flows of the KdV hierarchy is derived in an extended phase space. A map between stationary flows and restricted flows is constructed: in a case it connects an integrable Henon--Heiles system and the Garnier system. Moreover a new integrability scheme for Hamiltonian systems is proposed, holding in the standard phase space.Comment: 25 pages, AMS-LATEX 2.09, no figures, to be published in J. Phys. A: Math. Gen.

Detecting Recent Dynamics in Large-Scale Landslides via the Digital Image Correlation of Airborne Optic and LiDAR Datasets: Test Sites in South Tyrol (Italy)

Large-scale slow-moving deep-seated landslides are complex and potentially highly damaging phenomena. The detection of their dynamics in terms of displacement rate distribution is therefore a key point to achieve a better understanding of their behavior and support risk management. Due to their large dimensions, ranging from 1.5 to almost 4 km(2), in situ monitoring is generally integrated using satellite and airborne remote sensing techniques. In the framework of the EFRE-FESR SoLoMon project, three test-sites located in the Autonomous Province of Bolzano (Italy) were selected for testing the possibility of retrieving significant slope displacement data from the analysis of multi-temporal airborne optic and light detection and ranging (LiDAR) surveys with digital image correlation (DIC) algorithms such as normalized cross-correlation (NCC) and phase correlation (PC). The test-sites were selected for a number of reasons: they are relevant in terms of hazard and risk; they are representative of different type of slope movements (earth-slides, deep seated gravitational slope Deformation and rockslides), and different rates of displacement (from few cm/years to some m/years); and they have been mapped and monitored with ground-based systems for many years (DIC results can be validated both qualitatively and quantitatively). Specifically, NCC and PC algorithms were applied to high-resolution (5 to 25 cm/px) airborne optic and LiDAR-derived datasets (such as hillshade and slope maps computed from digital terrain models) acquired during the 2019-2021 period. Qualitative and quantitative validation was performed based on periodic GNSS surveys as well as on manual homologous point tracking. The displacement maps highlight that both DIC algorithms succeed in identifying and quantifying slope movements of multi-pixel magnitude in non-densely vegetated areas, while they struggle to quantify displacement patterns in areas characterized by movements of sub-pixel magnitude, especially if densely vegetated. Nonetheless, in all three landslides, they proved to be able to differentiate stable and active parts at the slope scale, thus representing a useful integration of punctual ground-based monitoring systems

A Left Atrial Appendage Closure Combined Procedure Review: past, present and future perspectives

Atrial fibrillation (AF) represents the most common cardiac arrhythmia worldwide; it poses a great burden in terms of quality of life reduction and yearly stroke risk. Left atrial appendage closure (LAAC) is a stroke prevention strategy that has been proven a viable alternative to anti-thrombotic regimens in non-valvular AF patients. LAAC can be performed as a stand-alone procedure or alongside a concomitant AF trans catheter ablation, in a procedure known as "Combined Procedure". Aim of this study is to summarize the scientific evidence backing this combined strategy