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NT>NoetherVolumeBuild: Noether theorems volume, initial import (A-02)

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Noether theorems volume, initial import (A-02)

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{{DISPLAYTITLE:The Runge-Lenz vector}}

[[File:NT-F-10.png|thumb|right|360px|alt=An ellipse with the sun at one focus and a planet marked on it. An arrow from the focus along the major axis is labelled A, the Runge-Lenz vector. A faint rotated copy of the ellipse shows what happens for any other force law, where the axis turns.|The closed Kepler orbit and the Runge-Lenz vector]]

'''This page works the conserved quantity and does not derive the symmetry behind
it.''' The second half is flagged below.

== The puzzle ==

A planet moving under an inverse-square attraction traces a closed ellipse that
does not precess. Under any other central force law it precesses: the orbit is
still bounded, but the long axis slowly turns.

Energy and angular momentum are conserved for every central force. So they cannot
be what closes the orbit. Something extra is conserved in the inverse-square case
only, and it was noticed long before anyone could say why.

== The quantity, and the check ==

With <math>V=-k/r</math>, momentum <math>p=m\dot{r}</math>, and angular momentum
<math>L=r\times p</math>, define

<math>A=p\times L-mk\hat{r}\qquad\qquad\mathrm{(1)}</math>

Differentiate along the motion. Since <math>L</math> is conserved,

<math>\frac{dA}{dt}=\dot{p}\times L-mk\frac{d\hat{r}}{dt}\qquad\qquad\mathrm{(2)}</math>

The force is <math>\dot{p}=-k\hat{r}/r^{2}</math>. Using the vector identity for
a triple product,

<math>\dot{p}\times L=-\frac{k}{r^{2}}\,\hat{r}\times(r\times p)=-\frac{k}{r^{2}}\left[r\left(\hat{r}\cdot p\right)-pr\right]=\frac{mk}{r^{2}}\left[r\dot{r}-r\left(\hat{r}\cdot\dot{r}\right)\right]\qquad\qquad\mathrm{(3)}</math>

And differentiating the unit vector,

<math>\frac{d\hat{r}}{dt}=\frac{\dot{r}}{r}-\frac{r\left(\hat{r}\cdot\dot{r}\right)}{r^{2}}=\frac{1}{r^{2}}\left[r\dot{r}-r\left(\hat{r}\cdot\dot{r}\right)\right]\qquad\qquad\mathrm{(4)}</math>

The bracket in (3) and the bracket in (4) are the same. Substituting both into
(2), the two terms cancel exactly and

<math>\frac{dA}{dt}=0\qquad\qquad\mathrm{(5)}</math>

'''Where the inverse square enters.''' At (3), and nowhere else. Any other power
leaves a residue the second term cannot cancel, and the orbit precesses.

The vector points along the long axis of the ellipse, which is why its constancy
is the statement that the axis does not turn.

== The symmetry, not derived ==

'''UNVERIFIED.''' The symmetry responsible is stated in the literature to be a
rotational symmetry of a four-dimensional space, larger than the ordinary
rotations of the three-dimensional problem, and acting on a combination of
position and momentum rather than on position alone. Exhibiting it and running it
through (N12) is not done here.

Two things are worth saying without it.

'''It is not visible in the Lagrangian.''' The transformation mixes the
coordinates with their derivatives. That is exactly the case Noether takes trouble
to allow — see
[[Noether's theorems/First theorem/Prolongation to derivatives|prolongation]] — and
the case most textbook statements of the theorem exclude by assumption.

'''This is the reverse direction earning its keep.''' A conserved quantity turned
up with no known symmetry. The converse says one exists. It did.

''Provenance: (1) to (5) are this volume's own derivation. The symmetry is flagged
UNVERIFIED and no source was consulted. See
[[Noether's theorems/Sources|Sources]].''

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