Noether's theorems/Mathematics/Poisson brackets and the generator identity

Plain-language counterpart: 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 $F$ and $G$ of coordinates $q_k$ and momenta $p_k$,

$\{F,G\}=\sum _k(\frac{\partial F}{\partial q_k}\frac{\partial G}{\partial p_k}-\frac{\partial F}{\partial p_k}\frac{\partial G}{\partial q_k})\mathrm{(}\mathrm{1}\mathrm{)}$

Interchanging $F$ and $G$ interchanges the two products and reverses the sign, so

$\{F,G\}=-\{G,F\}\mathrm{(}\mathrm{2}\mathrm{)}$

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

Evolution, derived

Hamilton's equations are

$\frac{d{q}_k}{dt}=\frac{\partial H}{\partial p_k};\frac{d{p}_k}{dt}=-\frac{\partial H}{\partial q_k}\mathrm{(}\mathrm{3}\mathrm{)}$

Take any $F$ and differentiate along the motion:

$\frac{dF}{dt}=\sum _k(\frac{\partial F}{\partial q_k}\frac{d{q}_k}{dt}+\frac{\partial F}{\partial p_k}\frac{d{p}_k}{dt})+\frac{\partial F}{\partial t}\mathrm{(}\mathrm{4}\mathrm{)}$

Substitute (3):

$\frac{dF}{dt}=\sum _k(\frac{\partial F}{\partial q_k}\frac{\partial H}{\partial p_k}-\frac{\partial F}{\partial p_k}\frac{\partial H}{\partial q_k})+\frac{\partial F}{\partial t}=\{F,H\}+\frac{\partial F}{\partial t}\mathrm{(}\mathrm{5}\mathrm{)}$

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

The generator identity

A function $G$ also generates a flow, by the same formula with $G$ in the Hamiltonian's place: along that flow, $dF/ds=\{F,G\}$. So $G$ is the generator of a one-parameter family of transformations, in the sense of the Lie page.

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

$\{G,H\}=0$ read one way says $G$ does not change under the motion. Read the other way — using (2) — it says $H$ does not change under the flow that $G$ 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 $G=H$ gives $\{H,H\}=0$ 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 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 Sources.