The conservation law a symmetry hands over

Three quantities are conserved in elementary mechanics, and they are usually introduced as three separate discoveries. Energy is conserved; momentum is conserved; angular momentum is conserved. Each is demonstrated on its own, each has its own exceptions, and a student learns to check three boxes.

They are one discovery. Each of the three is attached to a way the system can be moved without changing anything: shift the whole problem in time, shift it in space, turn it. The correspondence between the moving and the conserving is exact, it holds for any system whatever, and it was written down by Emmy Noether in 1918.

What a symmetry is, here #

The word does not mean a shape looking the same in a mirror. It means a transformation that leaves the action unchanged, and the transformation has to be continuous — parameterised by a knob that can be turned by any amount, however small.

The symmetry in question is not a property of any particular path or potential. It is the fact that the whole description can be slid along the time axis, or rotated about an origin, or shifted in space, and come out the same. That is a statement about the equations rather than about any solution of them — which is why a conservation law follows from it and why no amount of examining one trajectory reveals where the law comes from.

There are three obvious ones for an isolated system, and their obviousness is what makes them easy to overlook. Nothing in the laws refers to when the experiment is done. Nothing refers to where. Nothing refers to which way it is pointing. Those are the three, and they are statements about the laws rather than about any particular motion.

It is worth being concrete about what “leaves the action unchanged” means, because the phrase does a great deal of work. Take a trajectory, compute its action, then take the same trajectory shifted bodily one second later and compute the action of that. If the two numbers agree for every trajectory, the system has time-translation symmetry. Nothing has been said about any particular motion; the test is applied to the rule that scores motions.

The catch that stops this being trivial is that a symmetry of the action is required — the single number attached to a whole path — rather than a symmetry of the trajectory. A given orbit is not rotationally symmetric; an ellipse has a long axis pointing somewhere. What is rotationally symmetric is the rule that produced it, and the test is that rotating the whole orbit produces another orbit obeying the same rule.

The charges, read off a computation #

The hero figure is the whole theorem in one measurement, and it is worth saying exactly what was and was not put into it. A body was released in an inverse-square attraction and its motion integrated forward by a stepper that was told the force and the initial conditions. Three quantities were then evaluated at every step from the positions and velocities the stepper produced.

None of the three was constrained. Each comes out constant to about a part in ten billion, which is the integrator’s own arithmetic noise.

The first is the energy, and the symmetry behind it is time translation: the force law contains no clock, so a copy of the experiment started an hour later runs identically. The second is the angular momentum, and the symmetry is rotation: the force depends on the distance and not on the direction. Both of those are familiar.

What is conserved when a pendulum swings is the sum of two energies, and the reason is not that energy is a substance being passed between them. It is that nothing in the pendulum’s description mentions the time. Write down a system whose spring constant changes during the day and energy stops being conserved, without anything about energy having changed — only the description’s independence of when it is applied.

The third is not familiar, and it is the one that shows the theorem is doing real work rather than tidying up. The eccentricity vector points from the attracting centre towards the point of closest approach, and its length is the orbit’s eccentricity. It is constant for an inverse-square force and for no other power law, which is exactly why an inverse-square orbit closes and its neighbours do not.

Nobody would guess that quantity from staring at the force law. Its symmetry is a rotation, but not one that can be performed in three-dimensional space: written in the right variables, the Kepler problem has the symmetry of rotations in four dimensions, and the extra rotations are what the eccentricity vector is conserved by. That is a symmetry found by looking for the conserved quantity rather than the other way round, and it is the reason such quantities are called hidden.

The practical consequence is worth stating, because it is the reason this particular charge earned a name. Six numbers specify a body’s state in three dimensions, and each independent conserved quantity cuts the space the motion can wander in by one. Energy and the three components of angular momentum give four; the eccentricity vector’s three components are not all independent of those, but they add one more, and the count is then tight enough that the orbit is confined to a closed curve rather than to a surface. An orbit closes because there is nothing left for it to explore.

What happens when the symmetry is removed #

A theorem stated as a correspondence should be testable by breaking one side and watching the other break. That is a stronger test than any number of examples where both hold.

The perturbation is chosen to destroy exactly one symmetry. Multiplying the potential by a function of the angle means that rotating the whole problem no longer gives back the same problem, so the rotational symmetry is gone. The potential still contains no clock, so the time-translation symmetry is untouched.

The result is the contrast in the figure. The angular momentum stops being conserved and swings through a quarter of its value; the energy stays flat at the noise floor. Both are computed from one trajectory, which is what makes the comparison worth anything: an integration error would spoil both, and only the charge whose symmetry was removed has moved.

The rate is not free either. The theorem’s derivation gives the leak as the explicit derivative of the Lagrangian with respect to the coordinate the symmetry acted on, which here is the torque. Reading the angular momentum’s whole history off the trajectory and comparing it with the accumulated torque — two calculations sharing nothing but the physics — the two agree to within a hundred-thousandth of the swing.

Approximately symmetric, approximately conserved #

Almost no symmetry in the world is exact. A laboratory sits on a planet, which spoils translational symmetry; a galaxy is not quite round; a crystal has a surface. If conservation laws only followed from exact symmetries they would be of no practical use at all.

The slope of one is the quantitative half of the theorem and the more useful half. A symmetry that holds to a part in a thousand gives a quantity conserved to a part in a thousand, and the error is first order in the breaking rather than something worse. That is why angular momentum is a good bookkeeping device for a slightly flattened planet, why momentum is worth using in a laboratory, and why particle physicists spent decades treating isospin as conserved while knowing it was not.

It also says where the useful range stops. The straight line on that chart is a statement about small perturbations; push the asymmetry up and the orbit stops being a perturbed ellipse, the excursion saturates at the whole of the angular momentum, and no power law describes it.

The one nobody counts #

There is a fourth continuous symmetry of ordinary mechanics and it is almost never listed with the others. The laws are the same in a frame moving at constant velocity — a Galilean boost — and by the theorem there must be a conserved quantity attached to it.

The charge is $\mathbf{P}t - M\mathbf{X}$: the total momentum times the time, minus the total mass times the position of the centre of mass. It is the only charge in elementary mechanics that mentions the time explicitly, which is why it is usually stated as a sentence — the centre of mass moves uniformly — rather than listed beside energy and momentum.

Setting it constant and differentiating recovers that sentence. Read the other way, it says two bodies interacting cannot move their common centre, which is why a rocket needs exhaust and why a struck object’s centre of mass carries on as if nothing had happened.

One collision seen from six frames makes the point about the overlooked charge. Every observer disagrees about the momenta and about the energies, and every one measures the same energy loss — because the boost invariance that relates them is itself a symmetry, and its conserved quantity is the motion of the centre of mass. It is the one nobody counts, and it is on the same footing as the three that everybody does.

Why it needs the action rather than the equations #

The theorem is often quoted as symmetries give conservation laws, and the missing word does real work. The symmetry has to be a symmetry of the action, and the two are not the same thing.

The Kepler problem has a scaling symmetry: multiply all lengths by $\lambda$ and all times by $\lambda^{3/2}$ and the equation of motion is unchanged, which is Kepler’s third law. But the action is not unchanged — it picks up a factor — so the theorem supplies no conserved quantity, and there is none.

This is also where the theorem’s premise has to be stated carefully, because a symmetry of the equation of motion is not enough. Orbits of different sizes under the same law are related by a scaling that leaves the equation alone and multiplies the action by a constant — and that scaling has no conserved charge attached to it. What Noether’s theorem requires is a transformation leaving the action invariant, not merely one mapping solutions to solutions. The distinction is easy to lose and it is the difference between a theorem and a slogan.

Going the other way, a discrete symmetry gives nothing either. Reflection is an exact symmetry of gravity and electromagnetism, and there is no conserved quantity attached to it — only a selection rule saying that a mirror-symmetric initial state stays mirror-symmetric. The theorem needs a knob that can be turned by an infinitesimal amount, because its derivation is a first-order variation.

That restriction is not a weakness. It is what makes the theorem sharp enough to be used backwards: finding a quantity that does not change, and going looking for the symmetry that must be there.

The backwards direction has a real history. Baryon number and lepton number were written down as bookkeeping rules — certain reactions were never seen, so a tally was invented that forbade them — and only afterwards were the symmetries they correspond to identified, along with the circumstances in which each is expected to fail. A conservation law with no known symmetry behind it is a standing invitation, and several have been accepted.

What it looks like in a field #

The reason Noether’s theorem is the organising principle of modern physics rather than a nice fact about orbits is that it survives the move from particles to fields without changing shape.

In a field the same machinery gives more. An electromagnetic wave’s energy density, its momentum density and its stresses are all conserved currents following from the field action not mentioning when, where, or which way round it is written — so the whole stress-energy tensor is one theorem applied to one Lagrangian, rather than a collection of separately discovered facts.

For a field, a conserved quantity becomes a conserved current: a density and a flux obeying a continuity equation, so that the amount inside a region changes only by what crosses its boundary. Time-translation symmetry gives the energy density and its flow, which for electromagnetism is the Poynting vector. Space-translation symmetry gives the momentum density. Rotation gives the angular momentum density, including the part stored in fields where nothing at all is moving.

And then there are symmetries with no geometric meaning whatever. Multiplying a complex field by a phase changes nothing observable; the conserved current that follows is electric charge. Every conservation law in the Standard Model was obtained this way, and the machinery is identical to the machinery that gives angular momentum for a planet.

The other way a symmetry can go #

Everything above breaks a symmetry by putting a term into the Lagrangian that spoils it, and the charge then leaks at a rate the term predicts. There is a second and quite different way for a symmetry to stop being visible, in which the Lagrangian keeps it exactly and the world does not.

A block of iron below its Curie temperature is magnetised in some direction. Nothing in the physics of iron prefers a direction — the interaction between two spins depends on the angle between them and not on where either points — so rotational symmetry is exact, and by the theorem angular momentum is conserved exactly. What has happened is that the lowest-energy state is not symmetric even though the law is. There is a continuous family of ground states, one for each direction the magnetisation could have chosen, and the block is sitting in one of them.

The conservation law survives untouched. What appears instead is a new kind of excitation. Because the family of ground states is continuous, a disturbance that slowly rotates the magnetisation from place to place costs an energy that goes to zero as the rotation is stretched out — so there are modes whose frequency vanishes at long wavelength, and there is one for each direction the symmetry could have been broken in. That is Goldstone’s theorem, and it converts a broken symmetry into a spectrum.

The most familiar instance is not magnetic. A crystal breaks continuous translational symmetry: the laws do not care where anything is, and the crystal has decided. The modes that follow are the acoustic phonons — the same standing waves a string carries, whose frequency falls to zero as the wavelength grows. Sound in a solid exists because a symmetry was broken by the solid rather than by the laws.

Why the theorem was written #

The theorem is usually presented as a piece of pure structure, arrived at by someone thinking about actions in general. It was commissioned, and the problem it was commissioned to solve was a specific embarrassment.

Hilbert and Klein brought Noether to Göttingen in 1915 to work on invariance problems in the new general theory of relativity, where the conservation of energy had gone strange. In every previous theory the conservation law was a constraint — a statement that cut down which histories were allowed. In general relativity the corresponding statement turned out to follow from the field equations identically, without excluding anything, and Einstein, Hilbert and Klein spent two years arguing about whether that meant the theory conserved energy improperly or in some new sense.

Noether’s 1918 paper answered it by proving two theorems rather than one. The first is what this essay is about: a symmetry with finitely many parameters gives a conserved current. The second concerns symmetries parameterised by arbitrary functions rather than by numbers, shows that such a theory’s equations of motion are not independent of one another, and shows that its conservation laws are consequences of those dependencies rather than constraints on solutions. General relativity’s peculiar energy law is therefore a theorem about theories of that kind, not a defect of this one.

The second theorem is the harder and the more consequential, and it is a later rung. The circumstances are worth recording here for a different reason: Noether could not hold a position at Göttingen because she was a woman, lectured for four years under Hilbert’s name, and was unpaid throughout the period in which she proved the two results the subject has been organised around ever since.

Where it stops #

A system with friction has no Lagrangian, so it has no action for a symmetry to leave invariant, and the theorem says nothing about it. Energy is not conserved for a sliding block and the theorem does not claim it should be; what is conserved is the energy of the block plus everything the friction is heating, and that larger system does have an action.

Friction is where it stops, and it stops for a stated reason rather than a mysterious one. Friction has no Lagrangian, so the theorem has nothing to act on — and nothing here is a symmetry violation: the missing energy is in the degrees of freedom the model left out. Restore them and both the action and the conservation law come back. A dissipative description is an admission about bookkeeping, not a claim about physics.

Time-dependent laws break the energy law honestly. A pendulum whose length is being shortened by hand has a Lagrangian containing the time, and its energy changes at a rate the Lagrangian’s explicit time derivative predicts in advance. Pumping a swing is exactly this, and the energy arriving comes from whoever is doing the shortening.

The cosmological version is the same statement and is more unsettling. In an expanding universe the metric depends on time, so there is no time-translation symmetry and no globally conserved energy; light stretched by the expansion loses energy with nowhere for it to go. That is not a puzzle awaiting a resolution, it is the theorem’s contrapositive.

A photon climbing out of a static field loses energy, and that is ordinary bookkeeping — the energy goes to the field. In an expanding universe the same loss has no such account, because an expanding spacetime is not time-translation invariant and there is therefore no conserved energy to be accounted to. The reddening of the cosmic background is not energy going somewhere. It is a case where the theorem’s premise fails.

What the pictures cannot show #

The theorem is about the action, and no figure here draws one. Every measurement above is made on trajectories, which are what the action’s stationary condition produces. The demonstrations are consistency checks on a theorem proved by varying a functional, not a proof of it.

Three charges are not all of them. An inverse-square orbit has seven independent conserved quantities in a six-dimensional phase space, which is why every bound orbit closes; a general system has fewer, and a chaotic one may have only the energy. The number of conserved quantities relative to the number of degrees of freedom is what decides whether a system is integrable or not, and that is a question the theorem poses rather than answers.

The broken-symmetry figure breaks one symmetry cleanly, which is a laboratory convenience. Real perturbations usually spoil several at once, and disentangling which leak belongs to which is most of the work in practice.

And the boost charge is drawn in a Newtonian setting. Its relativistic counterpart exists and behaves differently — the centre of energy rather than the centre of mass, with a definition that depends on the frame — and the difference matters for any argument about where a system’s energy is.

The ladder from here #

Later rungs on this anchor: the Euler–Lagrange derivation carried out with its boundary terms; the Hamiltonian formulation, where the conserved charges become the generators of the transformations they are conserved by, and Poisson brackets make that literal; the action as a function of its endpoints; and gauge symmetries, which are a different animal — a redundancy in the description rather than a transformation between distinct states — and whose Noether currents vanish identically.

The neighbouring ladders are the quantity that survives a change of shape, which is angular momentum used rather than derived, least action, which is the principle this theorem is a statement about, and the orbit that does not come back to itself, where the hidden symmetry’s absence is visible as a slow rotation of the whole ellipse.