Related Experiment Video
Updated: May 7, 2025
06:07
Studying Large Amplitude Oscillatory Shear Response of Soft Materials
Published on: April 25, 2019
12.4K
Noether and partial Noether approach for the nonlinear (3+1)-dimensional elastic wave equations.
Akhtar Hussain1, M Usman2, Fiazuddin Zaman1
1Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan.
Plos One
|January 3, 2025
Summary
This study uses the Lie group method to find analytical solutions for nonlinear elastic wave equations. Novel approaches were developed to uncover conservation laws for both standard and damped versions of these equations.
Area of Science:
- Applied Mathematics
- Theoretical Physics
- Nonlinear Dynamics
Background:
- Nonlinear differential equations model complex physical phenomena.
- Analytical solutions are crucial for understanding wave propagation.
- Lie group methods offer powerful tools for solving such equations.
Purpose of the Study:
- To investigate nonlinear elastic wave equations using Lie group invariants.
- To derive and analyze conservation laws for both undamped and damped wave equations.
- To extend the application of variational calculus to these models.
Main Methods:
- Lie group method for symmetry analysis and deriving group-invariant solutions.
- Noether's theorem for conservation laws in the presence of a classical Lagrangian.
- Extended approach using partial Lagrangians for damped equations lacking a classical Lagrangian.
Main Results:
- An eight-dimensional symmetry algebra was derived for the (3+1)-dimensional nonlinear elastic wave equation.
- Optimal systems and group-invariant solutions were successfully obtained.
- Conservation laws for linear momentum and energy were identified for both equation types.
Conclusions:
- The Lie group method provides effective analytical solutions for nonlinear elastic wave equations.
- Novel techniques were established for deriving conservation laws in damped systems.
- This research expands the application of variational calculus in nonlinear wave analysis.
Related Concept Videos
01:21Linear Approximation in Time Domain
53
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
53
01:02Equations of Wave Motion
4.4K
Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a period T. The wave on the string is sinusoidal and is translated in the positive x-direction as time progresses. Sine is a function of the angle θ, oscillating between +A and −A and repeating every 2π radians. To construct a wave model, the ratio of the angle θ and the position x is considered.
4.4K
01:17Differential Form of Maxwell's Equations
356
James Clerk Maxwell (1831–1879) was one of the significant contributors to physics in the nineteenth century. He is probably best known for having combined existing knowledge of the laws of electricity and the laws of magnetism with his insights to form a complete overarching electromagnetic theory, represented by Maxwell's equations. The four basic laws of electricity and magnetism were discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and...
356
01:26Linear Approximation in Frequency Domain
79
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
79
01:24Electromagnetic Wave Equation
921
Maxwell's equations for electromagnetic fields are related to source charges, either static or moving. These fields act on a test charge, whose trajectory can thus be determined using suitable boundary conditions. The objective of electromagnetism is thus theoretically complete.
However, although electric and magnetic fields were first introduced as mathematical constructs to simplify the description of mutual forces between charges, a natural question emerges from Maxwell's equations:...
However, although electric and magnetic fields were first introduced as mathematical constructs to simplify the description of mutual forces between charges, a natural question emerges from Maxwell's equations:...
921
01:18Castigliano's Theorem
321
Castigliano's theorem analyzes displacements and rotations in elastic structures. It relates the derivative of elastic strain energy to the applied forces or moments, allowing for the calculation of deformations. The theorem states that the partial derivative of the total strain energy of a system with respect to a specific load results in the displacement at the point where the load is applied. This principle applies to both forces and moments.
321