Paolo Maria Santini

Integrable symplectic maps

Abstract In this paper, we first give a terse survey of symplectic maps, their canonical formulation and integrability. Then, we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices. A number of illustrative examples are provided.

Multidimensional quadrilateral lattices are integrable

Abstract The notion of a multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The geometric construction of the lattice is also discussed and, in particular, the number of initial-boundary data is clarified, which define the lattice uniquely.

An elementary geometric characterization of the integrable motions of a curve

Abstract We show that the following elementary geometric properties of the motion of a curve select hierarchies of integrable dynamics: (i) the curve moves in an N -dimensional sphere of radius R ; (ii) the motion is nonstretching; (iii) the dynamics does not depend explicitly on the radius of the sphere. For N = 2 we obtain the modified Korteweg-de Vries hierarchy, for N = 3 the nonlinear Schrodinger hierarchy and for N > 3 we obtain integrable multicomponent generalizations of the above hierarchies.

Transformations of quadrilateral lattices

Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogs: the multidimensional quadrilateral lattices, i.e., lattices x:ZN→RM, N⩽M, whose elementary quadrilaterals are planar. Our investigation is based on the discrete analog of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogs of the Laplace, Combescure, Levy, radial, and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between “transformations” and “discretizations” is also investigated for quadrilateral lattices. We finally interpret these results within the ∂ formalism.

The integrable discrete analogues of orthogonal coordinate systems are multi-dimensional circular lattices

Abstract We show that the integrable discrete analogues of the Lame orthogonal systems of coordinates are given by multidimensional circular lattices, i.e. by multidimensional lattices whose elementary quadrilaterals are inscribed in circles.

Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation

We construct the formal solution to the Cauchy problem for the dispersionless Kadomtsev-Petviashvili equation as an application of the inverse scattering transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy problem for the Kadomtsev-Petviashvili equation, associated with the inverse scattering transform of the time-dependent Schrödinger operator for a quantum particle in a time-dependent potential.

Dynamics of multibody systems in space environment; Lagrangian vs. Eulerian approach

Abstract The paper describes the motion of a multibody in space environment: by space environment we mean space-varying gravity, gradient forces, control forces, if any. (1) In the Eulerian approach, the motion of each individual member is described through kinematic parameters: (a) position of its CM with respect to the inertial frame; (b) rotation of the members with respect to the inertial frame; amplitude of the elastic modes (free–free). The said parameters are of different order of magnitudes, and therefore an adequate separation of them is highly desirable. Therefore, individual positions are replaced by overall position of the system (of the order of Earths radius), and by the motion of each bar relative to it (of the order of members dimension), and for modes amplitudes modal equations are used. It should be noted, however, that the above-described motion parameters are redundant, and we must introduce: (a) reactions between members, (b) equations of compatibility of the same number of reactions. In summary, (i) the set of unknowns is: motion parameters, reactions, control forces; (ii) the equations are equilibrium, compatibility, control. Control is introduced by prescribing the motion of some members, produced by control moments of forces. By simple matrix algebra, it is reduced to a system with motion parameters (overall + local) only. (2) In the Lagrangian approach, motion parameters are selected which are already consistent with compatibility conditions. In this case, as customarily, the expression of kinetic, potential, elastic energy is written, and the application of Lagrangian techniques provides directly the solving system. No reactions and compatibility equations appear here, however; for control purpose, prescribed motion law must again be introduced. Comparison of the two approaches shows perfect agreement (as one should have expected), since they are both exact models referring to the same physical system. In general, however, the Eulerian approach lends itself to a better understanding of physical facts, in particular, of the entity of the reactions and of the corresponding structural stresses.

Darboux transformations for multidimensional quadrilateral lattices. I

Abstract The vectorial Darboux transformation for the multidimensional quadrilateral lattice equations is constructed. Its particular reduction to the case of trivial background gives Gramm, Wronski and Casorati type representations for the solutions. Some examples of the quadrilateral lattices are constructed explicitly.

Stability of flexible spacecrafts

Abstract The paper deals with the general equations of a flexible spacecraft in a gravitational field. The motion of a generic point of the body is described as the superposition of a rigid motion plus a combination of structural modes. The general equations for the motion of a particle of the body are written; as a consequence, the equations of momentum, and of moment of momentum are derived. All the terms appearing in such equations are very general; in particular, gravitational torques include the effect of flexural deformations and of higher harmonics of the gravitational field. Similarly, the equations for the generic vibration mode amplitude include all the above effects. Stability of the motion around a given configuration is studied, for both nonspinning, and for spinning satellites. Numerical examples complete the work.

The inflammatory circuitry of miR-149 as a pathological mechanism in osteoarthritis

Osteoarthritis (OA) is a multifactorial degenerative pathology, whose progression is exacerbated by pro-inflammatory cytokines signaling. Among the changes triggered in chondrocytes during inflammation, modified expression of tiny epigenetic regulators as microRNAs was shown having deleterious implications for articular cartilage. Aim of the present study was to identify differentially expressed microRNAs in human OA cartilage and to determine their relevance to pathological progression. An OA model based on inflammatory stimulation of a chondrocytic human cell line was used to analyze microRNAs deregulation, and results revealed miR-149 severely down-regulated by IL1β and TNFα. Real-time PCR analysis of miR-149 was exerted also in human primary chondrocytes isolated from cartilage of OA donors and postmortem from subjects with no known history of OA, confirming down-regulation in osteoarthritis. Moving on a functional study, miR-149 regulatory effect on tumor necrosis factor alpha (TNFα), interleukin 1 beta (IL1β) and interleukin 6 (IL6) 3′UTRs was evaluated by luciferase assays, and chondrocytes production of TNFα upon miR-149 transfection was measured by enzyme-linked immuno sorbent assay. We found that miR-149 is down-regulated in OA chondrocytes, and this decrease seems to be correlated to increased expression of pro-inflammatory cytokines such as TNFα, IL1β and IL6. OA is a multifactorial disease and we think that our results give new insights for understanding the complex mechanisms of osteoarthritic pathogenesis.