Related papers: Helicity invariants in 3D : kinematical aspects
Motivated by the notion of Lagrangian multiforms, which provide a Lagrangian formulation of integrability, and by results of the authors on the role of covariant Hamiltonian formalism for integrable field theories, we propose the notion of…
The one-dimensional modified shallow water equations in Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and Eulerian…
The recently proposed low degree-of-freedom model of Moffat and Kimura [1,2] for describing the approach to finite-time singularity of the incompressible Euler fluid equations is investigated. The model assumes an initial finite-energy…
Kuzmin-Oseledets formulations of compressible Euler equations case are considered. Exact results and physical interpretations are given. One such exact result for the compressible barotropic case is the potential helicity Lagrange…
We prove that the helicity is the only regular Casimir function for the coadjoint action of the volume-preserving diffeomorphism group $\text{SDiff}(M)$ on smooth exact divergence-free vector fields on a closed three-dimensional manifold…
One of the most important conservation laws in atmospheric and oceanic science is conservation of potential vorticity. The original derivation is approximately a century old, in the work of Rossby and Ertel, and it is related to the…
The treatment of exact conservation laws in Lagrangian gauge theories constitutes the main axis of the first part of the thesis. The formalism is developed as a self-consistent theory but is inspired by earlier works, mainly by…
We carry out a parallel study of the covariant phase space and the conservation laws of local symmetries in two-dimensional dilaton gravity. Our analysis is based on the fact that the Lagrangian can be brought to a form that vanishes…
It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equations $$\mathrm{St} \partial_t F + v\cdot…
A relativistic helicity has been formulated in the four-dimensional Minkowski space-time. Whereas the relativistic distortion of space-time violates the conservation of the conventional helicity, the newly defined relativistic helicity…
The equations for a self-similar solution of an inviscid incompressible fluid are mapped into an integral equation which hopefully can be solved by iteration. It is argued that the exponent of the similarity are ruled by Kelvin's theorem of…
We develop an analytic formalism and derive new exact relations that express the short-time dispersion of fluid particles via the single-time velocity correlation functions in homogeneous isotropic and incompressible turbulence. The…
There are significant differences between Helmholtz and Hodge's decomposition theorems, but both share a common flavor. This paper is a first step to bring them together. We here produce Helmholtz theorems for differential 1-forms and…
Helicity, a measure of the linkage of flux lines, has subtle and largely unknown effects upon dynamics. Both magnetic and hydrodynamic helicity are conserved for ideal systems and could suppress nonlinear dynamics. What actually happens is…
The eleven-dimensional gravitational action invariant under local Poincare transformations is given by the dimensional continuation of the Euler class of ten dimensions. Here we show that the supersymmetric extension of this action leads,…
The analysis of the dynamics of a material point perfectly constrained to a submanifold of the three-dimensional euclidean space and subjected to a locally conservative force's field, namely a force's field corresponding to a closed but not…
We address the question whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows being a promising candidate…
This paper addresses the problem of energy conservation for the two- and three-dimensional density-dependent Euler equations. Two types of sufficient conditions on the regularity of solutions are provided to ensure the conservation of total…
Continuum mechanics can be formulated in the Lagrangian frame (addressing motion of individual continuum particles) or in the Eulerian frame (addressing evolution of fields in an inertial frame). There is a canonical Hamiltonian structure…
The barotropic ideal fluid with step and delta-function discontinuities coupled to Einstein's gravity is studied. The discontinuities represent star surfaces and thin shells; only non-intersecting discontinuity hypersurfaces are considered.…