Force That ClicksBonus: Force and Mathematics 2 (Part 2 of the trilogy)

Math 1: action takes the lead → Math 2: the symmetry of that action gives birth to conservation laws

Noether's Theorem Neither energy conservation nor momentum conservation is a rule handed down from on high.
If the action has a symmetry, there is always a corresponding conserved quantity ── one of the most beautiful theorems in twentieth-century physics.

Tools you'll need: the action from Math 1, symmetry from Episode 9 symmetry → conservation law

In Math 1 we saw that "nature makes the action \(S\) stationary." This time, precisely because that action takes the lead, we can state a surprisingly deep theorem ── Emmy Noether's theorem. Energy is conserved; momentum is conserved ── these things you learn in school as "rules" are, in fact, not handed down from on high. Whenever the action is unchanged under some transformation (has a symmetry), there is always a corresponding "conserved quantity." And which symmetry corresponds to which conserved quantity is pinned down exactly. In Episode 9 we saw that "symmetry gives birth to force"; here is its sibling ── "symmetry gives birth to conservation laws."

01Three symmetries, three conservation laws

A "symmetry" means that performing some operation leaves the physics (the action) unchanged. Noether's theorem assigns one conserved quantity to each continuous symmetry. The headline three are these.

Operation that leaves the action unchanged (symmetry)Conserved quantity
The same after shifting in time (time-translation symmetry)Energy
The same after shifting in space (space-translation symmetry)Momentum
The same after turning the orientation (rotational symmetry)Angular momentum

To put it another way ── because "the result is the same whenever you run the experiment," energy is conserved; because "it's the same wherever you run it," momentum is conserved; and because "it's the same in any orientation," angular momentum is conserved. Conservation laws turned out to be the shadow of the uniformity (symmetry) the universe possesses ── "the same everywhere, at every time, in every direction."

02Why? ── along a direction with no distinctions, there is no force

Let's grasp it by intuition. Space-translation symmetry (the same after shifting in place) means that along that direction there are no "distinctions" anywhere. If there are no distinctions, there is no slope of potential energy along that direction ── that is, no force (Episode 6, \(F=-\nabla V\); no slope means zero force). No force means the momentum in that direction does not change = conserved. Conversely, if you break the symmetry and add a slope (differences from place to place), a force is born, and momentum is no longer conserved. Broken symmetry = force = broken conservation. Check it in the figure below.

Figure: break the terrain's symmetry and conservation breaks. When flat (translation-symmetric, green), momentum stays constant. Tilt it (break the symmetry, red) and a force arises, and momentum changes.
momentum (constant if symmetric) momentum (changes once symmetry is broken)

03The sibling relationship with Episode 9 ── force and conservation law from the same symmetry

Here the series ties together into one. In Episode 9 we saw that "requiring a local symmetry (freely re-choosing at each point) gives birth to a force (a gauge field)." Noether, this time, says "when there is a (global) symmetry, a conservation law is born." From the single idea of symmetry, both the "existence" of force (Episode 9) and conservation laws (Noether) emerge. Symmetry gives birth to force, and gives birth to conservation ── this means that the backbone of this series (force = relation = symmetry) turned out, at its deepest level, to be the very framework of mechanics itself.

A connecting voice ── in an expanding universe, is energy not conserved? Noether also has a powerful "honesty" ── if a symmetry is broken, the conservation law is broken too. An expanding universe is not "the same after shifting in time" (the size of the universe differs between then and now = time-translation symmetry is broken). Then, by Noether's theorem, energy conservation does not hold straightforwardly for the universe as a whole. In fact, light from distant galaxies is redshifted by cosmic expansion and appears to "lose" energy ── the story from the sister series "Cosmology That Clicks" can be explained here in Noether's language. Conservation laws are not absolute commandments; they depend on symmetry.

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04What the peeling revealed ── conservation laws are the shadow of symmetry

The conclusion of Force and Mathematics 2. The conservation of energy, momentum, and angular momentum is not a rule handed down from on high, but the shadow of the universe's symmetry (the same at every time, everywhere, in every direction). Along a direction that has a symmetry there are no distinctions, and no force, and so that quantity is conserved. Break the symmetry and a force is born, and conservation breaks too. Combined with Episode 9's "symmetry → force," we've seen that symmetry itself is the common parent of both force and conservation laws.

The honest line

Noether's theorem (1918) yields conservation laws for the continuous symmetries the action possesses (discrete symmetries such as reflection are handled separately). What we treated here is the "first theorem" for global symmetries; the local symmetries of Episode 9 correspond to the "second theorem," and yield not so much conservation laws as constraints (identities) among fields ── the two are related but not the same. "Along a direction with no distinctions there is no force" is an intuitive rewording; strictly, it is derived from the invariance of the action by way of the Lagrange equations.

The figure is a schematic of the key point ── "symmetric (flat) means constant momentum / broken (tilted) means changing momentum" ── and shows the linear change of momentum under a constant tilt (a uniform force).

Practice problems
  1. Of which symmetry is "energy is conserved" the shadow?
    See the answer
    Time-translation symmetry (the physics is the same whenever you run the experiment). By Noether's theorem, energy conservation corresponds to the action being unchanged under a shift in time.
  2. Explain, in the language of force, why space-translation symmetry (the same after shifting in place) gives rise to momentum conservation.
    See the answer
    No distinctions in place = no slope of potential energy = zero force (F=−∇V). With no force, the momentum in that direction does not change = conserved. Break the symmetry to add a slope and a force arises, and conservation breaks.
  3. Why does (total) energy conservation not hold straightforwardly in an expanding universe?
    See the answer
    Because expansion breaks "the same after shifting in time" = time-translation symmetry. By Noether's theorem, if a symmetry is broken, the corresponding conservation law (energy conservation) does not hold straightforwardly either. The redshift of light is one manifestation.

SummaryConservation laws were the shadow of symmetry

Noether's theorem: if the action is invariant (symmetric) under some continuous transformation, there is always a corresponding conserved quantity. Time-translation symmetry → energy, space-translation symmetry → momentum, rotational symmetry → angular momentum. Conservation laws are not rules handed down from on high, but the shadow of the universe's uniformity ── "the same at every time, everywhere, in every direction." Along a direction with a symmetry there are no distinctions and no force, and so that quantity is conserved ── break the symmetry and a force is born, and conservation breaks.

Episode 9's "local symmetry → the existence of force" and Noether's "symmetry → conservation law" are siblings. Symmetry itself was the common parent of force and conservation laws. With the action of Math 1 as the soil, symmetry gives birth to the framework of mechanics ── next time (Math 3) is the finale that sees that force as geometry. Gauge = connection, gravity = curvature, groups classify the forces. It's the closing of "Force and Mathematics."

This document is installment 2 of "Force and Mathematics," a bonus of the "Force That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. That Noether's theorem (E. Noether, 1918) assigns a conserved quantity to each of the continuous global symmetries of the action (time translation ↔ energy, space translation ↔ momentum, rotation ↔ angular momentum) is established content. Local (gauge) symmetries correspond to Noether's second theorem and yield constraints and identities rather than conservation laws. In an expanding universe, global energy conservation does not hold in general because of the absence of time-translation symmetry, and this manifests as cosmological redshift (in general relativity, local conservation holds covariantly). "Along a direction with no distinctions there is no force" is an intuitive rewording; strictly it is derived from the invariance of the action. The figure is a schematic of the change of momentum under a uniform force. ── To print, use your browser's "Print" → "Save as PDF" (in the print version, the slider and the answers are frozen and hidden).

Print / save as PDF: ⌘+P (on Windows, Ctrl+P). On screen, the slider lets you see how breaking the symmetry breaks momentum conservation. Click "See the answer" to open a solution.