SYMMETRY REDUCTION IN THE VARIATIONAL APPROACH TO LIOUVILLE DYNAMICS

2005 ◽  
Vol 02 (04) ◽  
pp. 657-674 ◽  
Author(s):  
GIUSEPPE GAETA ◽  
PAOLA MORANDO ◽  
OSMAN TEOMAN TURGUT
Keyword(s):  
Variational Principle ◽  
Conservation Laws ◽  
Variational Approach ◽  
Vector Fields ◽  
Symmetry Reduction ◽  
Conserved Quantities ◽  
Volume Form ◽  
Liouville Dynamics

In a recent paper, a variational principle for Liouville dynamics (vector fields preserving a volume form) was introduced. In the present paper, we discuss the relation between symmetries and conservation laws in Liouville dynamics. We recall a result by Crampin relating symmetries and conserved quantities in Liouville dynamics, and discuss how symmetry reduction is implemented in the frame of the variational principle for Liouville dynamics.

CONSERVATION LAWS OF SOME NON-VARIATIONAL PERTURBED PDE'S VIA A PARTIAL VARIATIONAL APPROACH

2010 ◽  
Vol 24 (22) ◽  
pp. 4253-4267 ◽  
Author(s):  
A. G. JOHNPILLAI ◽  
A. H. KARA ◽  
F. M. MAHOMED
Keyword(s):  
Partial Differential Equations ◽  
Variational Principle ◽  
Differential Equations ◽  
Conservation Laws ◽  
Variational Approach ◽  
Partial Differential ◽  
Noether Operators

We show how one can construct approximate conservation laws of systems of perturbed or approximate partial differential equations that are not derivable by the variational principle but are approximate Euler–Lagrange in part by using approximate partial Noether operators associated with partial Lagrangians. We investigate perturbed partial differential equations that arise in several important physical applications. Finally, we obtain new approximate conservation laws for these equations.

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

Conservation laws

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Dynamical System ◽  
Angular Momentum ◽  
Conservation Laws ◽  
Total Energy ◽  
Superposition Principle ◽  
Momentum Conservation ◽  
Conserved Quantities ◽  
Angular Momentum Conservation ◽  
Third Law ◽  
The Law

This chapter defines the conserved quantities associated with an isolated dynamical system, that is, the quantities which remain constant during the motion of the system. The law of momentum conservation follows directly from Newton’s third law. The superposition principle for forces allows Newton’s law of motion for a body Pa acted on by other bodies Pa′ in an inertial Cartesian frame S. The law of angular momentum conservation holds if the forces acting on the elements of the system depend only on the separation of the elements. Finally, the conservation of total energy requires in addition that the forces be derivable from a potential.

Spacetime symmetries

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Riemannian Manifold ◽  
Conservation Laws ◽  
Vector Fields ◽  
Killing Vector ◽  
Geodesic Equation ◽  
Spacetime Symmetries ◽  
Tensor Fields ◽  
Killing Vector Fields ◽  
Covariant Derivatives ◽  
Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

scholarly journals Pairings between bounded divergence-measure vector fields and BV functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Graziano Crasta ◽  
Virginia De Cicco ◽  
Annalisa Malusa
Keyword(s):  
Vector Field ◽  
Conservation Laws ◽  
Bounded Variation ◽  
Vector Fields ◽  
Hyperbolic Conservation Laws ◽  
Divergence Measure ◽  
Compensated Compactness ◽  
Hyperbolic Conservation ◽  
Bv Functions ◽  
Bounded Functions

AbstractWe introduce a family of pairings between a bounded divergence-measure vector field and a function u of bounded variation, depending on the choice of the pointwise representative of u. We prove that these pairings inherit from the standard one, introduced in [G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], all the main properties and features (e.g. coarea, Leibniz, and Gauss–Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in \mathrm{BV}. We remark that the standard pairing in general does not share this property.

Automated generation of turbulence shell models by the methods of computer algebra

2021 ◽  
pp. 65-80
Author(s):  
Глеб Михайлович Водинчар ◽  
Любовь Константиновна Фещенко
Keyword(s):  
Conservation Laws ◽  
Computer Algebra ◽  
Quadratic Forms ◽  
Main Idea ◽  
Physical Space ◽  
Turbulence Models ◽  
Three Dimensions ◽  
Conserved Quantities ◽  
Automated Generation ◽  
Shell Models

Описана разработанная методика генерации уравнений каскадных моделей турбулентности с помощью систем компьютерной алгебры. Методика позволяет варьировать размер масштабной нелокальности модели, вид квадратичных законов сохранения и спектральных законов, знаменатель геометрической прогрессии масштабов. Ее использование позволяет быстро и безошибочно генерировать целые классы моделей. Может использоваться для разработки каскадных моделей гидродинамических, магнитогидродинамических и конвективных турбулентных систем. There is a great variety of shell turbulence models. Such models reproduce certain characteristics of turbulence. A model that could reproduce all turbulence regimes does not exist at the moment. Information about a particular model is contained in a set of persistent quantities, which are some quadratic forms of turbulent fields. These quadratic forms should be formal analogs of the exact conserved quantities. It is important to note that the main idea of Shell models presupposes a refusal to describe the geometric structure of movements. At the same time, it is well known that turbulent processes in spaces of two and three dimensions behave differently. Therefore, the provision of certain combinations of conserved quantities allows indirect introducing into the shell model the information about the dimension of the physical space in which the turbulent process develops. Purpose. The aim of this work was to create software tools that would quickly generate classes of models that satisfy one or another set of conservation laws. The choice of a specific model within these classes can then be specified using additional physical considerations, for example, the existence of a given probability distribution for the interaction of certain shells. Methods. The developed technique for generating equations of shell turbulence models is carried out using symbolic computation systems (computer algebra systems - CAS). Note that symbolic packages are used not for studying ready-made shell models, but for the automated generation of the equations of these models themselves. The technique allows varying the value of the scale nonlocality of the model, the form of the quadratic conservation laws and spectral laws, the denominator of the geometric progression of scales. It allows quickly and accurately generating the entire set of classes of the models. It can be used to develop shell models of hydrodynamic, magnetohydrodynamic and convective turbulent systems. Findings. It seems that the proposed technique will be useful for studying the properties of turbulence in the framework of cascade models

Conservation laws corresponding to the Noether symmetries of the geodetic Lagrangian in spherically symmetric spacetimes

2017 ◽  
Vol 26 (05) ◽  
pp. 1741006 ◽  
Author(s):  
Bismah Jamil ◽  
Tooba Feroze
Keyword(s):  
Conservation Laws ◽  
Noether Symmetry ◽  
Complete List ◽  
Spherically Symmetric ◽  
Conserved Quantities ◽  
Noether Symmetries ◽  
Symmetric Spacetimes ◽  
Spherically Symmetric Spacetimes

In this paper, we present a complete list of spherically symmetric nonstatic spacetimes along with the generators of all Noether symmetries of the geodetic Lagrangian for such metrics. Moreover, physical and geometrical interpretations of the conserved quantities (conservation laws) corresponding to each Noether symmetry are also given.

Shock Waves, Variational Principle and Conservation Laws of a Schamel–Zakharov–Kuznetsov–Burgers Equation in a Magnetised Dust Plasma

2018 ◽  
Vol 73 (8) ◽  
pp. 693-704 ◽  
Author(s):  
O.H. EL-Kalaawy ◽  
Engy A. Ahmed
Keyword(s):  
Variational Principle ◽  
Conservation Laws ◽  
Acoustic Waves ◽  
Burgers Equation ◽  
Special Functions ◽  
Nonlinear Propagation ◽  
Ion Acoustic Waves ◽  
New Exact Solutions ◽  
Kudryashov Method ◽  
Plasma Dust

AbstractIn this article, we investigate a (3+1)-dimensional Schamel–Zakharov–Kuznetsov–Burgers (SZKB) equation, which describes the nonlinear plasma-dust ion acoustic waves (DIAWs) in a magnetised dusty plasma. With the aid of the Kudryashov method and symbolic computation, a set of new exact solutions for the SZKB equation are derived. By introducing two special functions, a variational principle of the SZKB equation is obtained. Conservation laws of the SZKB equation are obtained by two different approaches: Lie point symmetry and the multiplier method. Thus, the conservation laws here can be useful in enhancing the understanding of nonlinear propagation of small amplitude electrostatic structures in the dense, dissipative DIAWs’ magnetoplasmas. The properties of the shock wave solutions structures are analysed numerically with the system parameters. In addition, the electric field of this solution is investigated. Finally, we will study the physical meanings of solutions.

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