scholarly journals Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 907 ◽  
Author(s):  
Oğul Esen ◽  
Miroslav Grmela ◽  
Hasan Gümral ◽  
Michal Pavelka
Keyword(s):  
Kinetic Theory ◽  
Lie Algebra ◽  
Particle Motion ◽  
Evolution Equations ◽  
Poisson Brackets ◽  
Hamiltonian Approach ◽  
Symmetric Tensors ◽  
Field Equations ◽  
Algebra Homomorphisms

Geometrical and algebraic aspects of the Hamiltonian realizations of the Euler’s fluid and the Vlasov’s plasma are investigated. A purely geometric pathway (involving complete lifts and vertical representatives) is proposed, which establishes a link from particle motion to evolution of the field variables. This pathway is free from Poisson brackets and Hamiltonian functionals. Momentum realizations (sections on T * T * Q ) of (both compressible and incompressible) Euler’s fluid and Vlasov’s plasma are derived. Poisson mappings relating the momentum realizations with the usual field equations are constructed as duals of injective Lie algebra homomorphisms. The geometric pathway is then used to construct the evolution equations for 10-moments kinetic theory. This way the entire Grad hierarchy (including entropic fields) can be constructed in a purely geometric way. This geometric way is an alternative to the usual Hamiltonian approach to mechanics based on Poisson brackets.

A cosmological study of Einstein–Skyrme model in anisotropic Kantowski–Sachs spacetime using Lie and Noether symmetries

2018 ◽  
Vol 15 (06) ◽  
pp. 1850089 ◽  
Author(s):  
Santu Mondal ◽  
Sourav Dutta ◽  
Manjusha Tarafdar ◽  
Subenoy Chakraborty
Keyword(s):  
Lie Algebra ◽  
Evolution Equations ◽  
Einstein Gravity ◽  
Lie Symmetry ◽  
Noether Symmetry ◽  
Noether Symmetries ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Unknown Parameters ◽  
Augmented Space

The present work deals with anisotropic (but zero heat flux) Skyrme fluid in a locally rotational Kantowski–Sachs (KS) spacetime in the background of Einstein gravity. The Lie point symmetry is imposed to the system of Einstein field equations and unknown parameters are either determined or interrelated by the imposition of the symmetry. Subsequently, Noether symmetry, a point-like symmetry of the Lagrangian is used and it is found that the Lie algebra of the Noether symmetry is a subalgebra of the corresponding Lie algebra of the Lie symmetry. Then a point transformation in the 2D augmented space is taken in such a manner that one of the variables becomes cyclic and hence the Lagrangian as well as the evolution equations are simplified to a great extent. Finally, solutions to the physical system are presented and are analyzed physically.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

scholarly journals PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

2021 ◽  
Author(s):  
DMITRI I. PANYUSHEV ◽  
OKSANA S. YAKIMOVA
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Invariant Theory ◽  
Cyclic Permutation ◽  
Poisson Brackets ◽  
Enveloping Algebra ◽  
Order Automorphism ◽  
Finite Dimensional ◽  
Compatible Poisson Brackets ◽  
Commutative Subalgebras

AbstractLet 𝔮 be a finite-dimensional Lie algebra. The symmetric algebra (𝔮) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (𝔮) to any finite order automorphism ϑ of 𝔮. We study related Poisson-commutative subalgebras (𝔮; ϑ) of 𝒮(𝔮) and associated Lie algebra contractions of 𝔮. To obtain substantial results, we have to assume that 𝔮 = 𝔤 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If 𝔤 = 𝔥⊕⋯⊕𝔥 (sum of k copies), where 𝔥 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (𝔮; ϑ) is polynomial and maximal. Furthermore, we quantise this (𝔤; ϑ) using a Gaudin subalgebra in the enveloping algebra 𝒰(𝔤).

Comment on "Kinetic theory of single-particle motion in a fluid"

1979 ◽  
Vol 19 (1) ◽  
pp. 416-418 ◽  
Author(s):  
H. van Beijeren ◽  
J. R. Dorfman
Keyword(s):  
Kinetic Theory ◽  
Single Particle ◽  
Particle Motion ◽  
Single Particle Motion

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

scholarly journals SPIN-1 GRAVITATIONAL WAVES: THEORETICAL AND EXPERIMENTAL ASPECTS

2006 ◽  
Vol 03 (03) ◽  
pp. 451-469 ◽  
Author(s):  
F. CANFORA ◽  
L. PARISI ◽  
G. VILASI
Keyword(s):  
Physical Properties ◽  
Exact Solutions ◽  
Gravitational Waves ◽  
Lie Algebra ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Gravitational Fields ◽  
Killing Fields

Exact solutions of Einstein field equations invariant for a non-Abelian bidimensional Lie algebra of Killing fields are described. Physical properties of these gravitational fields are studied, their wave character is checked by making use of covariant criteria and the observable effects of such waves are outlined. The possibility of detection of these waves with modern detectors, spherical resonant antennas in particular, is sketched.

Constructing Poisson and Dissipative Brackets of Mixtures by using Lagrangian-to-Eulerian Transformation

2010 ◽  
Vol 26 (2) ◽  
pp. 219-228
Author(s):  
K.-C. Chen
Keyword(s):  
Evolution Equations ◽  
Poisson Brackets ◽  
State Variables ◽  
Component Mixture ◽  
Dissipative Part ◽  
Canonical Variables ◽  
Lagrangian Variable ◽  
Two Component ◽  
Lagrangian Form

AbstractThis paper aims to construct the bracket formalism of mixture continua by using the method of Lagrangian- to-Eulerian (LE) transformation. The LE approach first builds up the transformation relations between the Eulerian state variables and the Lagrangian canonical variables, and then transforms the bracket in Lagrangian form to the bracket in Eulerian form. For the conservative part of the bracket formalism, this study systematically generates the noncanonical Poisson brackets of a two-component mixture. For the dissipative part, we deduce the Eulerian-variable-based dissipative brackets for viscous and diffusive mechanisms from their Lagrangian-variable-based counterparts. Finally, the evolution equations of a micromorphic fluid, which can be treated as a multi-component mixture, are derived by constructing its Poisson and dissipative brackets.

INTEGRABLE HAMILTONIAN HIERARCHIES ASSOCIATED WITH THE EQUATION OF HEAT CONDUCTION

2010 ◽  
Vol 24 (14) ◽  
pp. 1573-1594 ◽  
Author(s):  
YUFENG ZHANG ◽  
HONWAH TAM ◽  
JIANQIN MEI
Keyword(s):  
Heat Conduction ◽  
Quadratic Form ◽  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Arbitrary Parameter ◽  
Higher Dimensional ◽  
Soliton Hierarchy ◽  
Equation Of Heat Conduction ◽  
Hamiltonian Functions

Using a 4-dimensional Lie algebra g, an isospectral Lax pair is introduced, whose compatibility condition is equivalent to a soliton hierarchy of evolution equations with three components of potential functions. Its Hamiltonian structure is obtained by employing the quadratic-form identity proposed by Guo and Zhang. In order to obtain explicit Hamiltonian functions, a detailed computing formula for the constant appearing in the quadratic-form identity is obtained. One type of reduction equations of the hierarchy is also produced, which is further reduced to the standard equation of heat conduction. By introducing a loop algebra of the Lie algebra g, we obtain a soliton hierarchy with an arbitrary parameter which can be reduced to the previous equation hierarchy obtained, whose quasi-Hamiltonian structure is also worked out by the quadratic-form identity. Finally, we extend the Lie algebra g into a higher-dimensional Lie algebra so that a new integrable Hamiltonian hierarchy, which comprise integrable couplings, is produced; its reduced equations in particular contain two arbitrary parameters.

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