scholarly journals Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 24
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Conservation Laws ◽  
Graphical Representation ◽  
Power Series Expansion ◽  
Multiplier Method ◽  
Local Conservation ◽  
Noether’S Theorem ◽  
Conservation Of Energy ◽  
Closed Form Solutions ◽  
Noether's Theorem ◽  
Weierstrass Elliptic Function

In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum.

Introduction

2017 ◽  
Author(s):  
Laurent Baulieu ◽  
John Iliopoulos ◽  
Roland Sénéor
Keyword(s):  
Conservation Laws ◽  
Lagrangian Mechanics ◽  
General Introduction ◽  
Noether’S Theorem ◽  
Noether's Theorem

General introduction with a review of the principles of Hamiltonian and Lagrangian mechanics. The connection between symmetries and conservation laws, with a presentation of Noether’s theorem, is included.

scholarly journals Conservation Laws in Physics – Emmy Noether’s Theorem

2014 ◽  
Vol 29 ◽  
pp. 55-61
Author(s):  
Daniela Manolea
Keyword(s):  
Conservation Laws ◽  
Celestial Body ◽  
Practical Importance ◽  
Microscopic Level ◽  
Human Knowledge ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
The World ◽  
Physical Laws ◽  
Practical Impact

The study is explanatory-interpretative and argues the practical character of Physics. It starts from premise that formation of a correct conception of the world begins with the understanding of physics. It is one of the earliest chapters of human knowledge, studying the material world from the microscopic level of the particles to the macroscopic level of the celestial body. As an example for the practical importance of applying the laws of physics take the set of physical laws of conservation, in particular, it explains the practical impact of Emmy Noether's Theorem.

Conservation laws of linear elasticity in stress formulations

2005 ◽  
Vol 461 (2053) ◽  
pp. 99-116 ◽  
Author(s):  
Shaofan Li ◽  
Anurag Gupta ◽  
Xanthippi Markenscoff
Keyword(s):  
Conservation Laws ◽  
Linear Elasticity ◽  
Three Dimensional ◽  
Cauchy Stress ◽  
General Boundary Conditions ◽  
Cauchy Stress Tensor ◽  
Equilibrium Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Three Dimensional Elasticity

In this paper, we present new conservation laws of linear elasticity which have been discovered. These newly discovered conservation laws are expressed solely in terms of the Cauchy stress tensor, and they are genuine, non–trivial conservation laws that are intrinsically different from the displacement conservation laws previously known. They represent the variational symmetry conditions of combined Beltrami–Michell compatibility equations and the equilibrium equations. To derive these conservation laws, Noether's theorem is extended to partial differential equations of a tensorial field with general boundary conditions. By applying the tensorial version of Noether's theorem to Pobedrja's stress formulation of three–dimensional elasticity, a class of new conservation laws in terms of stresses has been obtained.

Conservation Laws in Systems With Variable Mass

1993 ◽  
Vol 60 (4) ◽  
pp. 954-958 ◽  
Author(s):  
L. Cveticanin
Keyword(s):  
Variational Principle ◽  
Dynamic System ◽  
Conservation Laws ◽  
Dynamic Systems ◽  
Conservation Law ◽  
Variable Mass ◽  
Mass Variation ◽  
Noether’S Theorem ◽  
Noether's Theorem

In this paper, a method for obtaining conservation laws of dynamic systems with variable mass is developed. It is based on Noether’s theorem to the existence of conservation laws and D’Alembert’s variational principle. In the general case, a dynamic system with variable mass is purely nonconservative. Noether’s identity for such a case is expanded by the terms that describe the mass variation. If Noether’s identity if satisfied, a conservation law exists. Two groups of systems with variable mass are considered: a nonlinear vibrating machine and a rotor with variable mass. For these systems, conservation laws are obtained using the procedure developed in this paper.

scholarly journals SYMMETRIES, NEWTONOID VECTOR FIELDS AND CONSERVATION LAWS IN THE LAGRANGIAN k-SYMPLECTIC FORMALISM

2012 ◽  
Vol 24 (10) ◽  
pp. 1250030 ◽  
Author(s):  
LUCÍA BUA ◽  
IOAN BUCATARU ◽  
MODESTO SALGADO
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Conservation Law ◽  
Vector Fields ◽  
Partial Differential ◽  
Noether’S Theorem ◽  
Dynamical Symmetries ◽  
Noether's Theorem ◽  
Configuration Manifold

In this paper, we study symmetries, Newtonoid vector fields, conservation laws, Noether's theorem and its converse, in the framework of the k-symplectic formalism, using the Frölicher–Nijenhuis formalism on the space of k1-velocities of the configuration manifold.For the case k = 1, it is well known that Cartan symmetries induce and are induced by constants of motions, and these results are known as Noether's theorem and its converse. For the case k > 1, we provide a new proof for Noether's theorem, which shows that, in the k-symplectic formalism, each Cartan symmetry induces a conservation law. We prove that, under some assumptions, the converse of Noether's theorem is also true and we provide examples when this is not the case. We also study the relations between dynamical symmetries, Newtonoid vector fields, Cartan symmetries and conservation laws, showing when one of them will imply the others. We use several examples of partial differential equations to illustrate when these concepts are related and when they are not.

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