Every continuous symmetry of the action hides a conserved quantity. This is the deepest bookkeeping in physics: it is not a coincidence that energy, momentum, and angular momentum are conserved — each is the shadow of a symmetry the world happens to have. Time looks the same tomorrow → energy is conserved. Space looks the same over there → momentum. Space looks the same turned around → angular momentum. Down the center a Lagrangian and a symmetry go in, a symplectic integrator runs, and the conserved current comes out — flat to a millionth. The blue team builds it; the red team breaks the symmetry and watches the current drift.
source Noether, Invariante Variationsprobleme (1918), Nachr. Ges. Wiss. Goettingen, math.-phys. Klasse, S. 235–257 — eudml.org/doc/59024. Rendered, not quoted.
The conservation law is not assumed; it falls out of the symmetry. For a Lagrangian L(q, q̇) invariant under a continuous shift q → q + εδq, the Noether current
J = (∂L/∂q̇) δq is constant along every trajectory.
Three generators, three currents. The highlighted row is what the panel is running:
| symmetry δq | → | conserved current J |
|---|
One theorem sits beneath every conservation law you were ever taught: time → energy, translation → momentum, rotation → angular momentum. It is the hinge between two neighbouring spheres.
the-lagrangian supplies the action S = ∫L dt whose invariances Noether reads; the-hamiltonian is where the same currents reappear as the generators of the very flows that leave H unchanged. Symmetry here is conservation there. Each sphere is the next one's premise.
The blue team's live check: re-run all four Noether laws — energy, momentum, angular momentum, and the broken case — and confirm each current is flat (or, when the symmetry is broken, that it is not). If red tampers, this is where it shows.
Noether's machine eats three things: a Lagrangian L = T − V (natural units, m = 1, k = 1), an initial condition, and a continuous symmetry — a direction you can shift the system a little without changing the action.
| symmetry present in L | what stays fixed |
|---|---|
| L has no explicit t | time-translation |
| L unchanged by x → x + a | space-translation |
| L unchanged by rotation | rotational |
| L depends on absolute x | translation BROKEN |
Feed the symmetry into the panel below; it integrates the real equations of motion and measures the current.
The current is computed from a symplectic (velocity-Verlet) trajectory — never looked up. A symplectic step is used on purpose: plain Euler would bleed energy and fake a violation.
What the machine proves: for each unbroken symmetry the current is flat to < 1e−6 over the whole trajectory — energy for time, total momentum for space, L = r × p for rotation. Break the symmetry with an explicit external field and momentum drifts — exactly the Noether prediction, running backwards.
The blue witness (left) re-derives these live; the red team (right) tries to make a broken symmetry still look conserved.
Sharpest of all: in an expanding universe, time-translation symmetry is genuinely broken by the FRW metric, so the total energy of the cosmos is not conserved. "Energy is always conserved" is a statement about a symmetry that does not always hold.
"Noether's theorem says energy is always conserved." Cut. Only when L has no explicit time dependence. In an expanding universe, or under a time-varying field, energy is not conserved — the symmetry is absent.
"Every symmetry gives a conservation law." Cut. Only continuous ones. Parity and time-reversal are symmetries with no Noether current.
"Noether proved it for quantum mechanics." Kept, corrected. 1918 is classical calculus of variations. The quantum echo — Ward–Takahashi identities — came decades later and is descended from her, not hers.
The red team's move: bolt an explicit external field onto the translation-invariant two-body Lagrangian — so it now depends on absolute position — while the code keeps claiming momentum is conserved. The blue witness (window 7) is watching.
Inject the field and total momentum starts to drift, yet the label still reads "conserved" — the witness recomputes the current, sees it is no longer flat, and turns red. Nothing is faked; the broken symmetry really does break the law, and it is caught.