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Noether theorems volume, initial import (M-03)

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{{DISPLAYTITLE:Poisson brackets and the generator identity}}

''Plain-language counterpart: [[Noether's theorems/Explainers/The conserved thing is the thing that does the moving|The conserved thing is the thing that does the moving]].''

In Hamiltonian form the first theorem becomes nearly a tautology. This page
derives it, and it is the shortest complete argument in the volume.

== The bracket ==

For functions <math>F</math> and <math>G</math> of coordinates
<math>q_{k}</math> and momenta <math>p_{k}</math>,

<math>\{F,G\}=\sum_{k}\left(\frac{\partial F}{\partial q_{k}}\frac{\partial G}{\partial p_{k}}-\frac{\partial F}{\partial p_{k}}\frac{\partial G}{\partial q_{k}}\right)\qquad\qquad\mathrm{(1)}</math>

Interchanging <math>F</math> and <math>G</math> interchanges the two products and
reverses the sign, so

<math>\{F,G\}=-\{G,F\}\qquad\qquad\mathrm{(2)}</math>

Antisymmetry is immediate from the definition and is the only property this page
needs.

== Evolution, derived ==

Hamilton's equations are

<math>\frac{dq_{k}}{dt}=\frac{\partial H}{\partial p_{k}};\qquad\frac{dp_{k}}{dt}=-\frac{\partial H}{\partial q_{k}}\qquad\qquad\mathrm{(3)}</math>

Take any <math>F</math> and differentiate along the motion:

<math>\frac{dF}{dt}=\sum_{k}\left(\frac{\partial F}{\partial q_{k}}\frac{dq_{k}}{dt}+\frac{\partial F}{\partial p_{k}}\frac{dp_{k}}{dt}\right)+\frac{\partial F}{\partial t}\qquad\qquad\mathrm{(4)}</math>

Substitute (3):

<math>\frac{dF}{dt}=\sum_{k}\left(\frac{\partial F}{\partial q_{k}}\frac{\partial H}{\partial p_{k}}-\frac{\partial F}{\partial p_{k}}\frac{\partial H}{\partial q_{k}}\right)+\frac{\partial F}{\partial t}=\{F,H\}+\frac{\partial F}{\partial t}\qquad\qquad\mathrm{(5)}</math>

So a quantity with no explicit time dependence is conserved exactly when its
bracket with the Hamiltonian vanishes.

== The generator identity ==

A function <math>G</math> also '''generates''' a flow, by the same formula with
<math>G</math> in the Hamiltonian's place: along that flow,
<math>dF/ds=\{F,G\}</math>. So <math>G</math> is the generator of a
one-parameter family of transformations, in the sense of
[[Noether's theorems/Mathematics/Lie groups and Lie algebras|the Lie page]].

Now put the two readings of the same bracket beside each other.

<math>\{G,H\}=0</math> read one way says <math>G</math> does not change under the
motion. Read the other way — using (2) — it says <math>H</math> does not change
under the flow that <math>G</math> generates.

'''The two statements are the same equation.''' "This quantity is conserved" and
"the Hamiltonian is symmetric under the transformation this quantity generates"
are not two facts connected by a theorem. They are one bracket, read from either
side, and the connection is the antisymmetry (2).

Setting <math>G=H</math> gives <math>\{H,H\}=0</math> identically, so a Hamiltonian
with no explicit time dependence is conserved without any further argument.
Energy conservation is the antisymmetry of the bracket.

== What this does and does not replace ==

'''It is not a shorter proof of the first theorem.''' It presupposes the
Hamiltonian form, which presupposes that the momenta can be solved for the
velocities. Precisely the theories the second theorem is about — those with a local
symmetry — fail that condition, because
[[Noether's theorems/Second theorem/Fewer independent field equations than fields|some
of their equations are not independent]]. For those, the Hamiltonian route is the
hard road, not the easy one.

'''It is not the whole content.''' The bracket argument delivers the on-shell
statement directly and never produces the identity that holds off-shell. It is a
shorter route to less.

It belongs beside Noether's proof, not instead of it — but it is the clearest
statement anywhere of why a symmetry and a conserved quantity are the same object
seen twice.

''Provenance: (1) to (5) are standard and are derived above. The reading of (2) as
the content of the symmetry-conservation correspondence is this volume's own
arrangement. No source was consulted for this page. See
[[Noether's theorems/Sources|Sources]].''

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