We introduce a rotation invariant short distance cut-off in the theory of an ideal fluid in three space dimensions, by requiring momenta to take values in a sphere. This leads to an algebra of functions in position space is non-commutative. Nevertheless it is possible to find appropriate analogues of the Euler equations of an ideal fluid. The system still has a hamiltonian structure. It is hoped that this will be useful in the study of possible singularities in the evolution of Euler (or Navier-Stokes) equations in three dimensions.
Guided by ordinary quantum mechanics we introduce new fuzzy spheres of dimensions d=1,2: we consider an ordinary quantum particle in D=d+1 dimensions subject to a rotation invariant potential well V(r) with a very sharp minimum on a sphere of unit radius. Imposing a sufficiently low energy cutoff to `freeze' the radial excitations makes only a finite-dimensional Hilbert subspace accessible and on it the coordinates noncommutative à la Snyder; in fact, on it they generate the whole algebra of observables. The construction is equivariant not only under rotations - as Madore's fuzzy sphere -, but…
Within the Hamiltonian framework, the propositions about a classical physical system are described in the Borel σ-algebra of a symplectic manifold (the phase space) where logical connectives are the standard set operations. Considering the geometric formulation of quantum mechanics we give a description of quantum propositions in terms of fuzzy events in a complex projective space equipped with Kähler structure (the quantum phase space) obtaining a quantized version of a fuzzy logic by deformation of the product t-norm.
Arnold showed that the Euler equations of an ideal fluid describe geodesics in the Lie algebra of incompressible vector fields. We will show that helicity induces a splitting of the Lie algebra into two isotropic subspaces, forming a Manin triple. Viewed another way, this shows that there is an infinitesimal quantum group (a.k.a. Lie bi-algebra) underlying classical fluid mechanics.
We explore the relationship between mechanical systems describing the motion of a particle with the mechanical systems describing a continuous medium. More specifically, we will study how the so-called intermediate integrals or fields of solutions of a finite dimensional mechanical system (a second order differential equation) are simultaneously Euler's equations of fluids and conversely. This will be done both in the classical and relativistic context. A direct relationship will be established by means of the so-called time constraint (classical unsteady case, static or not) and the relativis…
Projecting a quantum theory onto the Hilbert subspace of states with energies below a cutoff $\overline{E}$ may lead to an effective theory with modified observables, including a noncommutative space(time). Adding a confining potential well $V$ with a very sharp minimum on a submanifold $N$ of the original space(time) $M$ may induce a dimensional reduction to a noncommutative quantum theory on $N$. Here in particular we briefly report on our application of this procedure to spheres $S^d\subset\mathbb{R}^D$ of radius $r=1$ ($D=d\!+\!1>1$): making $\overline{E}$ and the depth of the well depend …
This is an idiosyncratic survey of statistical fluid mechanics centering on the Hopf functional differential equation. Using the Burgers equation for illustration we review several functional integration approaches to theory of turbulence. We notice in particular that some important contributions have been brought about by researchers working on wave propagation in random media, among which Uriel Frisch is not an exception. We also discuss a particular finite-dimensional approximation for the Burgers equation.
We briefly report our recent construction of new fuzzy spheres of dimensions d=1,2 covariant under the full orthogonal group O(D), D=d+1. They are built by imposing a suitable energy cutoff on a quantum particle in D dimensions subject to a confining potential well V(r) with a very sharp minimum on the sphere of radius r=1; furthermore, the cutoff and the depth of the well depend on (and diverge with) a natural number L. The commutator of the coordinates depends only on the angular momentum, as in Snyder noncommutative spaces. When L diverges, the Hilbert space dimension diverges, too; S^d_L c…
This article attempts to use the ideas from the field of complexity sciences to revisit the classical field of fluid mechanics. For almost a century, the mathematical self-consistency of Navier-Stokes equations has remained elusive to the community of functional analysts, who posed the Navier-Stokes problem as one of the seven millennium problems in the dawn of 21st century. This article attempts to trace back the historical developments of fluid mechanics as a discipline and explain the consequences of not rationalising one of the commonly agreed upon tenets - continuum hypothesis - in the co…
Painleve analysis, Backlund transformation, Lax pair and periodic wave solutions for a generalized (2+1)-dimensional Hirota-Satsuma-Ito equation in fluid mechanics
