Noether's theorems/Applications/Continuum mechanics

This page works its case.

The first case in this volume with more than one independent variable, so the first where a divergence is a genuine divergence rather than a total derivative.

The vibrating string

Displacement $u(t,x)$, density $\rho$, tension $T$, both constant:

$f=\frac{1}{2}\rho {\left(\frac{\partial u}{\partial t}\right)}^2-\frac{1}{2}T{\left(\frac{\partial u}{\partial x}\right)}^2\mathrm{(}\mathrm{1}\mathrm{)}$

The Lagrange expression, by (N5), is

$\psi =-\rho \frac{{\partial }^{2}u}{\partial t^2}+T\frac{{\partial }^{2}u}{\partial x^2}\mathrm{(}\mathrm{2}\mathrm{)}$

and $\psi =0$ is the wave equation.

Energy density and flux, worked

$f$ contains neither $t$ nor $x$ explicitly, so translation in either is a symmetry. Take translation in time: $\Delta t=\epsilon$, $\Delta x=0$, $\Delta u=0$. By (N9),

$\overline{\delta }u=-\frac{\partial u}{\partial t}\epsilon \mathrm{(}\mathrm{3}\mathrm{)}$

By (N5) the current has one component per independent variable:

$A^t=-\frac{\partial f}{\partial \frac{\partial u}{\partial t}}\overline{\delta }u=\epsilon \rho {\left(\frac{\partial u}{\partial t}\right)}^2;A^x=-\frac{\partial f}{\partial \frac{\partial u}{\partial x}}\overline{\delta }u=-\epsilon T\frac{\partial u}{\partial x}\frac{\partial u}{\partial t}\mathrm{(}\mathrm{4}\mathrm{)}$

Subtracting $f\cdot \Delta x$, which contributes only to the time component here,

$B^t=\epsilon [\rho {\left(\frac{\partial u}{\partial t}\right)}^2-f]=\epsilon [\frac{1}{2}\rho {\left(\frac{\partial u}{\partial t}\right)}^2+\frac{1}{2}T{\left(\frac{\partial u}{\partial x}\right)}^2];B^x=-\epsilon T\frac{\partial u}{\partial x}\frac{\partial u}{\partial t}\mathrm{(}\mathrm{5}\mathrm{)}$

The time component is the energy density — kinetic plus potential, with the right factors of one half, neither of which was put in. The space component is the energy flux, and on $\psi =0$,

$\frac{\partial B^t}{\partial t}+\frac{\partial B^x}{\partial x}=0\mathrm{(}\mathrm{6}\mathrm{)}$

Energy is not conserved at each point. It flows. Equation (6) says only that what leaves one stretch of string arrives in the next, which is the local statement from the current page appearing for the first time in a case where it is not trivial.

Momentum, and a warning about the name

Translation in $x$ gives a second current by the same route. Its time component is $-\rho (\partial u/\partial t)(\partial u/\partial x)$, often called the momentum density of the string.

It is not the momentum of the material. The string's atoms move up and down and have no net sideways momentum at all. What is conserved is the momentum carried by the wave, which is a different thing wearing the same word, and the two are routinely confused in acoustics.

The theorem is not at fault. It returned a conserved quantity for the symmetry it was given. Naming that quantity is a separate act, performed by people.

Provenance: (1) to (6) are this volume's own derivations in Noether's notation using her (N5), (N9) and (N12). No source was consulted. See Sources.