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Noether theorems volume, initial import (A-01)

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{{DISPLAYTITLE:Conserved quantities in point mechanics}}

'''This page works its case.'''

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

== The recipe ==

[[Noether's theorems/First theorem/Energy, momentum and angular momentum|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
<math>\bar{\delta}u</math> from <math>\Delta u</math> and <math>\Delta x</math> by
(N9). Assemble <math>A</math> by (N4) and subtract <math>f\cdot\Delta x</math> to
get <math>B</math>. Then (N12) is the identity, and setting
<math>\psi_{i}=0</math> gives the conservation law.

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

== A many-particle system, worked ==

Take <math>N</math> particles with positions <math>u_{i}</math>, indexed so that
each particle contributes several, and

<math>f=\sum_{i}\tfrac{1}{2}m_{i}u_{i}'^{2}-V\qquad\qquad\mathrm{(1)}</math>

with <math>V</math> 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 <math>V</math> unchanged and
therefore leaves <math>f</math> unchanged.

Take <math>\Delta x=0</math> and <math>\Delta u_{i}=\varepsilon</math> for every
<math>i</math> belonging to one spatial direction. By (N9),
<math>\bar{\delta}u_{i}=\varepsilon</math>, and by (N4),

<math>A=B=-\varepsilon\sum_{i}\frac{\partial f}{\partial u'_{i}}=-\varepsilon\sum_{i}m_{i}u'_{i}\qquad\qquad\mathrm{(2)}</math>

so (N12) gives the identity

<math>\sum_{i}\psi_{i}=-\frac{d}{dx}\left(\sum_{i}m_{i}u'_{i}\right)\qquad\qquad\mathrm{(3)}</math>

On <math>\psi_{i}=0</math>,

<math>\sum_{i}m_{i}u'_{i}=\mathrm{const}\qquad\qquad\mathrm{(4)}</math>

'''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 <math>N</math> of them. Translating a single particle is not a
symmetry, because <math>V</math> 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 <math>f</math> not containing
<math>x</math> explicitly, the energy result of the worked-cases page reads

<math>f-u'\frac{\partial f}{\partial u'}=\mathrm{const}\qquad\qquad\mathrm{(5)}</math>

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
[[Noether's theorems/First theorem/Centre-of-mass motion and electric charge|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
[[Noether's theorems/Sources|Sources]].''

Noether's theorems/Applications/Conserved quantities in point mechanics - Revision history

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