scholarly journals Symmetry and conservation laws in non-holonomic mechanics

2021 ◽  
Vol 62 (5) ◽  
pp. 052901
Author(s):  
Enrico Massa ◽  
Enrico Pagani
Keyword(s):  
Conservation Laws ◽  
Symmetry And Conservation Laws

scholarly journals On Galilean Invariant and Energy Preserving BBM-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 878
Author(s):  
Alexei Cheviakov ◽  
Denys Dutykh ◽  
Aidar Assylbekuly
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Local Symmetry ◽  
Symmetry And Conservation Laws ◽  
Higher Order ◽  
Special Equation ◽  
Long Wave ◽  
Local Symmetries

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

Hamiltonian Fluid Dynamics

1998 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Variational Principle ◽  
Conservation Laws ◽  
Variational Principles ◽  
Symmetry And Conservation Laws ◽  
The Body ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Symmetry Properties

In this final chapter, we return to the subject of the first: the fundamental principles of fluid mechanics. In chapter 1, we derived the equations of fluid motion from Hamilton’s principle of stationary action, emphasizing its logical simplicity and the resulting close correspondence between mechanics and thermodynamics. Now we explore the Hamiltonian approach more fully, discovering its other advantages. The most important of these advantages arise from the correspondence between the symmetry properties of the Lagrangian and the conservation laws of the resulting dynamical equations. Therefore, we begin with a very brief introduction to symmetry and conservation laws. Noether’s theorem applies to the equations that arise from variational principles like Hamilton’s principle. According to Noether’s theorem : If a variational principle is invariant to a continuous transformation of its dependent and independent variables, then the equations arising from the variational principle possess a divergence-form conservation law. The invariance property is also called a symmetry property. Thus Noether’s theorem connects symmetry properties and conservation laws. We shall neither state nor prove the general form of Noether’s theorem; to do so would require a lengthy digression on continuous groups. Instead we illustrate the connection between symmetry and conservation laws with a series of increasingly complex and important examples. These examples convey the flavor of the general theory. Our first example is very simple. Consider a body of mass m moving in one dimension. The body is attached to the end of a spring with spring-constant K. Let x(t) be the displacement of the body from its location when the spring is unstretched.

Symmetry and conservation laws for tuberculosis model

2017 ◽  
Vol 10 (03) ◽  
pp. 1750042 ◽  
Author(s):  
Maba Boniface Matadi
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Three Dimensional ◽  
Symmetry And Conservation Laws ◽  
Second Order ◽  
Dimensional System ◽  
First Order ◽  
Tuberculosis Model ◽  
Nonlinear Lagrangian ◽  
Three Dimensional System

In this paper, three-dimensional system of the tuberculosis (TB) model is reduced into a two-dimensional first-order and one-dimensional second-order differential equations. We use the method of Jacobi last multiplier to construct linear Lagrangians of systems of two first-order ordinary differential equations and nonlinear Lagrangian of the corresponding single second-order equation. The Noether's theorem is used for determining conservation laws. We apply the techniques of symmetry analysis to a model to identify the combinations of parameters which lead to the possibility of the linearization of the system and provide the corresponding solutions.

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