Symmetry gives a conserved quantity — and dm³ is contact, not symplectic, so the quantity is conserved only up to an exponential. That factor is the dissipation, and this chapter is where the corpus admits it.
Emmy Noether lectured at Göttingen for four years under Hilbert's name because the faculty would not habilitate a woman, and Hilbert told the senate that he did not see how a candidate's sex was an argument against her, since the senate was not a bathhouse. She was paid nothing until 1923. In 1933 she was dismissed by letter, taught for a while from her flat, and went to Bryn Mawr, where she died in 1935 at fifty-three. CITED
The reason she is in this gallery is not that biography. It is that two of the things this corpus does every day are hers, and one of them does not survive the move into contact geometry — which is more interesting than if it did.
The 1918 paper, Invariante Variationsprobleme, contains two theorems. The first is the one everyone quotes: to every continuous symmetry of an action there corresponds a conserved quantity. Time-translation gives energy. Space-translation gives momentum. Rotation gives angular momentum. It is the reason a physicist, asked why energy is conserved, can answer with a statement about time rather than a statement about energy.
The second theorem — that a symmetry depending on arbitrary functions produces identities among the equations rather than conservation laws — is the one that turned out to govern gauge theory, and was largely ignored for forty years. She wrote it to settle a problem Hilbert and Klein had run into in general relativity, where energy conservation behaves strangely. It does behave strangely, and her second theorem says exactly why.
Here is the part that matters for this corpus, and it is a limitation, not a triumph.
dm³ is not symplectic. It is contact: an odd-dimensional manifold with $\alpha = dz - p\,dq$, and a contact Hamiltonian flow that does not preserve a symplectic form. Contact systems dissipate; that is what they are for. Noether's theorem in its symplectic form therefore does not transfer. Energy is not conserved along a contact flow, and no amount of symmetry will make it so.
What happens instead is sharper than “it fails”. Take the contact Hamiltonian $H(q,p,z) = \tfrac12(p^2+q^2) + \beta z$ with the standard contact equations
$$\dot q = \frac{\partial H}{\partial p},\qquad\dot p = -\frac{\partial H}{\partial q} - p\,\frac{\partial H}{\partial z},\qquad\dot z = p\,\frac{\partial H}{\partial p} - H$$
which for this $H$ is the damped oscillator $\ddot q + \beta\dot q + q = 0$. Compute $\dot H$ along the flow and the $z$-coupling gives
$$\frac{dH}{dt} = -H\,\frac{\partial H}{\partial z} = -\beta H\qquad\Longrightarrow\qquad H(t) = H(0)\,e^{-\beta t}$$
so $H e^{\beta t}$ is exactly conserved, to $10^{-14}$ relative over two periods under RK4. COMPUTED
And the damping itself has a threshold, which is the corpus's own shape appearing where it was not planted.
The characteristic roots of $\ddot q + \beta \dot q + q = 0$ are $\lambda_\pm = \tfrac12(-\beta \pm \sqrt{\beta^2 - 4})$. For $\beta < 2$ they are a complex pair with real part exactly $-\beta/2$: the system rings while it decays. At $\beta = 2$ the two roots collide at $-1$. Past it they are real and distinct, and the slower one climbs back toward zero — at $\beta = 4$ the roots are $-0.2679$ and $-3.7321$, so the most damped setting of the parameter gives the slowest return. COMPUTED
An earlier draft of this chapter asserted that $\mu_{\max} = -\beta/2$ throughout, and so that $\beta = 4$ would give $\mu_{\max} = -2$ — the corpus's own canonical value. It does not. $-\beta/2$ is the real part only while the roots are complex, that is only for $\beta < 2$. The claim was dropped before the page was written rather than after, and it is recorded here because a number that flatters the framework is exactly the kind that gets kept without checking. OPEN
So $\beta = 2$ is a fold in the parameter, not a scale in it. Nothing about the decay gets gradually less oscillatory; it oscillates, and then at one value it stops, and the eigenvalues that were a conjugate pair become two real numbers going opposite ways.
Her other brick is quieter and this corpus leans on it harder. The ascending chain condition — the definition that makes a ring Noetherian — is what lets finiteness arguments run in commutative algebra at all, and it is a hypothesis in most of what Grothendieck built on top. The K-group that chapter is about needs the categories to be well-behaved, and “well-behaved” is, over and over, “Noetherian”.
She also gave the modern statement of the isomorphism theorems, and the homological reading of Betti numbers as ranks of groups rather than numbers — which is why one says homology group and not homology number. Alexandroff said she taught the topologists what they had been computing. CITED
| Operator | In this chapter | In dm³ |
|---|---|---|
| C | the action functional — a symmetry declared | compression: what is held fixed |
| K | $\beta$ raised toward 2, the roots approaching each other | approach to $\kappa^*$ |
| F | $\beta = 2$: the conjugate pair collides and splits into two reals | the fold — threshold, not scale SHOWN |
| U | $He^{\beta t}$ — the invariant that survives the dissipation | the conserved object of a contact flow COMPUTED |
Every number on this page is produced by book7/ch-noether-verify.py. It records in its own closing block what it establishes and what it does not.
E. Noether, “Invariante Variationsprobleme”, Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, 1918, 235–257.
E. Noether, “Idealtheorie in Ringbereichen”, Math. Ann. 83, 1921 — the ascending chain condition.
Y. Kosmann-Schwarzbach, The Noether Theorems, Springer, 2011 — including the neglect of the second theorem.
A. Bravetti, H. Cruz and D. Tapias, “Contact Hamiltonian mechanics”, Ann. Phys. 376, 2017 — the contact equations used above.
A. Bravetti, “Contact Hamiltonian dynamics: the concept and its use”, Entropy 19, 2017.
P. Alexandroff, address in memory of Emmy Noether, Moscow Mathematical Society, 1935.