Crash two objects together and almost everything about them changes — speed, direction, shape, temperature. But add up the mass times velocity of everything involved, before and after, and you get the same number to the last decimal place. Always. Add up the electric charge and you get the same number too. Physics is full of quantities like this, and for two centuries nobody knew why they existed. Then in 1918 Emmy Noether proved that every one of them is the same fact told twice: a conservation law is a symmetry wearing a disguise.
Two pucks on a frictionless loop of track — no walls, so nothing outside the pair ever pushes on it. The upper trace is the total momentum of both pucks added together. The lower trace is their total kinetic energy. Every time they hit, the energy trace drops a step and stops somewhere lower. The momentum trace does not move. Not "barely moves" — the drift tile below reports the actual number, and it sits at the last bit of double precision, around 10−16, which is the computer failing to represent the answer rather than the physics failing to hold.
Pull the bounciness slider down to 0 and the pucks stick together on impact. That is the case people find hardest to believe, because so much is obviously lost: the bang, the dent, the warmth. Watch the top trace anyway.
The asymmetry is not that momentum is more fundamental than energy. Energy is conserved too — it just leaves the ledger we are watching. Bang the pucks together and some of the kinetic energy becomes heat, which is nothing but the jiggling of the molecules inside them. Now ask what that jiggling does to the momentum ledger. Nothing: heat is motion in every direction at once, and its vector sum is exactly zero. Heat is a place energy can hide and momentum cannot.
Underneath sits Newton's third law. Whatever force the green puck exerts on the blue one, the blue one exerts an equal and opposite force back, over exactly the same interval. The two impulses cancel term by term, so the total must come out unchanged, no matter how messy the contact is — no matter whether they bounce, crumple, weld, or explode. The energy lost has an exact closed form, and the left-hand chart is that formula, not a fit:
Two things fall out of it immediately. At e = 1 the bracket is zero and no energy is lost at all. And because μ is the reduced mass, a heavy thing hitting a light thing loses almost none of its energy while a light thing hitting a heavy thing loses almost all of it — which is the right-hand chart, and the reason a bullet stops in a wall but the wall does not move.
Momentum is a vector, so in two dimensions there are two ledgers, and both balance independently. The arrow in the middle of the box is the total momentum of every disk added together as vectors; it points the same way and stays the same length no matter how much chaos is going on around it. Drop the bounciness and watch the energy collapse while that arrow does not flinch.
The box has no walls. It wraps: a disk leaving the right edge comes back in on the left. That is not a graphics convenience, it is the whole physics of this page. Switch the walls on and the momentum ledger breaks immediately — and the reason it breaks is the reason section 5 exists. A wall is a place where space stops being the same as everywhere else.
Momentum can at least be handed to the Earth. Charge has nowhere to go at all — no process has ever been observed that changes the total electric charge of an isolated system, and experiments looking for a single decaying electron have pushed its lifetime past 1028 years, which is roughly a quintillion times the age of the universe. Every reaction below is a real one, and the two charge columns are added up live from the particles listed, not typed in.
| process | charge before | charge after | balanced? |
|---|
Notice what charge conservation does not forbid. Beta decay turns a neutron into a proton — a neutral particle becomes a positive one — and that is perfectly legal, because a negative electron comes out with it. Charge is not attached to any particular particle. It is only the total that is protected. Notice too that the law is happy to create matter from nothing: a gamma ray with enough energy becomes an electron and a positron, because 0 = (−1) + (+1).
The same law, worn smooth by everyday use, is Kirchhoff's current rule. Charge cannot pile up at a junction in a wire, so whatever flows in must flow out. Move the sliders: the fourth current is not a free choice, it is whatever the other three leave over.
Charge is also quantised, which is a separate surprise. It comes only in whole multiples of e = 1.602176634 × 10−19 C — a number that since 2019 is exact by definition, because the coulomb is now defined from it rather than the other way round. Quarks carry thirds of e, but they are never found alone, and the combinations that do appear always land on a whole number: a proton is up-up-down, ⅔ + ⅔ − ⅓ = +1, and a neutron is up-down-down, ⅔ − ⅓ − ⅓ = 0, exactly.
Here is the idea that makes this page one page instead of three. In 1915 Hilbert and Einstein had a problem with energy in general relativity and asked Emmy Noether for help. What she gave back in 1918 was far bigger than the problem: a proof that every continuous symmetry of the laws of physics produces a conserved quantity, and every conserved quantity comes from one. Not a coincidence, not an empirical pattern — a theorem.
The symmetries are almost insultingly obvious once stated. Space is the same here as it is a metre to the left: do an experiment, carry the whole apparatus across the room, do it again, get the same answer. That is translation symmetry, and it is where momentum comes from. The laws are the same today as tomorrow — that is time-translation symmetry, and it is where energy comes from. Space has no preferred direction — rotational symmetry — and that is where angular momentum comes from.
The three panels below test exactly this. Each runs a real simulation and plots its quantity live. With the slider at zero every symmetry is intact and all three lines are flat to thirteen or fourteen decimal places. Push the slider and you break one symmetry in each panel — the landscape grows bumps so position starts to matter, the spring's stiffness begins changing with time so when starts to matter, the round bowl is squashed into an ellipse so direction starts to matter — and the matching line stops being flat. Nothing else changes. Nothing was added to make it drift. The symmetry is the conservation law.
The pattern keeps going, and it reaches further than mechanics. The last two rows are the payoff for section 4: conservation of electric charge is not a separate empirical rule at all. It falls out of a symmetry of the quantum field describing the electron — you can rotate the phase of that field by any angle you like, everywhere at once, and no measurement can tell. Noether's theorem turns that freedom into a conserved current, and the current is electricity.
| if this changes nothing… | …then this is conserved | where you meet it |
|---|---|---|
| Moving the whole experiment sideways | momentum | recoil, collisions, rockets |
| Running the experiment tomorrow instead | energy | every energy budget ever drawn |
| Turning the whole experiment to face a new way | angular momentum | spinning skaters, planets, gyroscopes |
| Watching from a frame moving at steady speed | centre-of-mass motion | the centre of mass drifts uniformly, always |
| Rotating the phase of the electron's quantum field | electric charge | section 4 — every circuit and every decay |
| The same, for the fields that bind quarks | colour charge | the strong force holding nuclei together |
And the theorem cuts both ways, which is how it earns its keep. If a quantity is not conserved, some symmetry must be broken — so go and find it. That is not a philosophical remark; it is a working method. When beta decay appeared to lose energy in the 1920s, Bohr was willing to abandon energy conservation. Pauli instead insisted the symmetry was intact and something invisible was carrying the energy off. He was right, it took twenty-six years to catch, and we call it the neutrino.
The collision engine is exact rather than approximate: an impulse along the line of centres with a coefficient of restitution, which is algebra, not integration, so there is no truncation error to accumulate. Momentum is not enforced anywhere in the code — nothing normalises it, nothing corrects it — and yet across three-minute runs with tens of thousands of collisions the total drifts by about one part in 1016, which is double-precision rounding and nothing else. The known cases all come out right on their own: equal masses with one at rest exchange velocities exactly; a perfectly inelastic hit leaves both bodies at the centre-of-mass velocity and loses exactly ½μ(v₁−v₂)² of kinetic energy; and two equal masses colliding elastically in two dimensions always separate at 90°. The three Noether panels use velocity Verlet, a symplectic integrator, which is why energy stays put there too.
What it leaves out is a lot. Every object here is a point mass with a radius and no interior: it cannot spin, so there is no rotational kinetic energy and no friction between surfaces to create any, and the angular momentum on this page is purely orbital. There is no gravity, no drag and no air. When kinetic energy disappears in an inelastic collision the code does not track where it went — real energy becomes heat, sound and permanent deformation, and a real dented car stays dented, whereas these disks are as good as new. A coefficient of restitution is itself a summary of a complicated squashing process, and treating it as one fixed number per material is an approximation that gets worse the faster things hit.
Two deeper caveats. Everything here is non-relativistic: momentum is mv, when the true expression is γmv, and the difference only becomes visible near the speed of light — the conservation law survives, the formula does not. And energy conservation itself is on shakier ground than the other laws in one specific arena. Noether ties energy to time-translation symmetry, and the expanding universe is not time-translation symmetric — which is why the light from distant galaxies genuinely loses energy on its way to us, with no recipient. On cosmological scales, the honest statement is not that energy is conserved but that there is no general law saying it must be.