Noether's theorems/Applications/Conserved quantities in point mechanics

This page works its case.

Notation. Across this track, $\psi$, $\mu$ and $\epsilon$ keep Noether's meanings, and spacetime indices are written $a,b,c$ rather than with her reserved letters.

The recipe

The worked-cases page took three symmetries one at a time. The procedure behind all three is the same and is worth stating once as a recipe.

Given an integrand, find a transformation leaving it invariant. Compute $\overline{\delta }u$ from $\Delta u$ and $\Delta x$ by (N9). Assemble $A$ by (N4) and subtract $f\cdot \Delta x$ to get $B$. Then (N12) is the identity, and setting ${\psi }_i=0$ gives the conservation law.

Every case in this track is that recipe with different inputs.

A many-particle system, worked

Take $N$ particles with positions $u_i$, indexed so that each particle contributes several, and

$f=\sum _i\frac{1}{2}{m}_i{u}_i{\text{'}}^2-V\mathrm{(}\mathrm{1}\mathrm{)}$

with $V$ depending on the positions only through the separations between particles. Then a rigid translation of the whole system, the same displacement applied to every particle, leaves $V$ unchanged and therefore leaves $f$ unchanged.

Take $\Delta x=0$ and $\Delta u_i=\epsilon$ for every $i$ belonging to one spatial direction. By (N9), $\overline{\delta }{u}_i=\epsilon$, and by (N4),

$A=B=-\epsilon \sum _i\frac{\partial f}{\partial u{\text{'}}_i}=-\epsilon \sum _i{m}_{i}u{\text{'}}_i\mathrm{(}\mathrm{2}\mathrm{)}$

so (N12) gives the identity

$\sum _i{\psi }_i=-\frac{d}{dx}\left(\sum _i{m}_{i}u{\text{'}}_i\right)\mathrm{(}\mathrm{3}\mathrm{)}$

On ${\psi }_i=0$,

$\sum _i{m}_{i}u{\text{'}}_i=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{(}\mathrm{4}\mathrm{)}$

The total momentum, not each particle's. The symmetry was a rigid translation of everything at once, so the theorem returns one conserved quantity for the system, not $N$ of them. Translating a single particle is not a symmetry, because $V$ depends on separations.

Read (3) before imposing anything and it says something else, worth noticing: the sum of the Lagrange expressions is a total derivative whatever the particles are doing. Newton's third law is that identity, arriving before the equations of motion rather than as a separate postulate.

The one-variable special case has a name

For a single dependent variable with $f$ not containing $x$ explicitly, the energy result of the worked-cases page reads

$f-u^{\prime }\frac{\partial f}{\partial u^{\prime }}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{(}\mathrm{5}\mathrm{)}$

which is the Beltrami identity, standard in the calculus of variations and usually derived there without reference to symmetry at all. It is the first theorem applied to translation in the independent variable, and nothing else.

The count for Newtonian mechanics

An isolated system in empty space admits ten symmetries: one time translation, three spatial translations, three rotations, three boosts. Ten conserved quantities, and the theorem says there is no eleventh of this kind.

The boosts need the weaker invariance condition and are on the centre-of-mass page, attributed there.

Provenance: (1) to (5) are this volume's own derivations, carried out in Noether's notation with her (N4), (N9) and (N12). Beltrami's name for (5) is standard usage and no source was consulted. See Sources.