Related papers: An Infinitesimal Quantum Group Underlies Classical…
We present a consistent scheme of quantization of chiral flows (flows with extensive vorticity) in ideal hydrodynamics in two dimensions. Chiral flows occur in rotating superfluid, rotating turbulence and also in electronic systems in the…
Using the Lie derivative of the metric we define a class of Lie algebras of vector fields by generalising the concept of Killing vectors. As a Lie algebra they define locally a group action on the pseudo-Riemannian manifold through…
We consider the vorticity formulation of the Euler equations describing the flow of a two-dimensional incompressible ideal fluid on the sphere. Zeitlin's model provides a finite-dimensional approximation of the vorticity formulation that…
A hierarchy of integrable hamiltonian nonlinear ODEs is associated with any decomposition of the Lie algebra of Laurent series with coefficients being elements of a semi-simple Lie algebra into a sum of the subalgebra consisting of the…
We consider $L^2$ minimizing geodesics along the group of volume preserving maps $SDiff(D)$ of a given 3-dimensional domain $D$. The corresponding curves describe the motion of an ideal incompressible fluid inside $D$ and are (formally)…
Euler hydrodynamics of perfect fluids can be viewed as an effective bosonic field theory. In cases when the underlying microscopic system involves Dirac fermions, the quantum anomalies should be properly described. In 1+1 dimensions the…
We briefly review a theoretical picture of the quantum Hall effect in which gauge symmetry of non-relativistic quantum mechanics, the representation theory of U(1)-current algebra, and the study of odd, positive integral quadratic forms on…
Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on…
In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra $\textbf{${\mathfrak g}_{\mathsf u}$}$ that extends $\mathbf{e_9}$. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds…
This paper surveys results found by the authors in the previous papers (see for example, A. Duyunova, V. Lychagin, S. Tychkov, Differential invariants for spherical layer flows of a viscid fluid, Journal of Geometry and Physics, 130,…
The effective field theory of the Calogero-Sutherland model represents a universality class of quantum hydrodynamic fluids in one spatial dimension. It describes quantum compressible fluids involving both chiralities in which the chiral…
This paper is a study of the Lie groups of point symmetries admitted by a system describing a non-stationary planar flow of an ideal plastic material. For several types of forces involved in the system, the infinitesimal generators which…
The Euler equation for an inviscid, incompressible fluid in a three-dimensional domain M implies that the vorticity is a frozen-in field. This can be used to construct a symplectic structure on RxM. The normalized vorticity and the…
A stochastic Euler equation is proposed, describing the motion of a particle density, forced by the random action of virtual photons in vacuum. After time averaging, the Euler equation is reduced to the Reynolds equation, well studied in…
Streamlines of a relativistic perfect isentropic fluid are geodesics of a Riemannian space whose metric is defined by enthalpy of the fluid. This fact simplifies the solution of some problems, as is also of interest from the point of view…
The conservation of helicity in ideal barotropic fluids is discussed from a group theoretical point of view. A new symmetry group is introduced i.e. the alpha group of translations. It is proven via the Noether theorem that this group…
The state space of a homogeneous body is derived under two different assumptions: infinitesimal reducibility and irreducibility. The first assumption leads to a real vector space, used in classical mechanics, while the second one leads to a…
It is shown that the quantum mechanics of a charged particle moving in a uniform magnetic field in the plane (Landau) or on a planar lattice (Peierls) is described in all detail by the projective representation theory of the "euclidean"…
The irreducible decomposition technique is applied to the study of classical models of metric-affine gravity (MAG). The dynamics of the gravitational field is described by a 12-parameter Lagrangian encompassing a Hilbert-Einstein term,…
A generalized Euler equation of fluid dynamics is derived for describing many-body states of quantum mechanics. The Eulerian Eq. can be viewed as representing the interaction of two substates, where each substate has its own velocity and…