Module 09
Symmetry and conservation
Slide a coffee cup across a table in Miami. Fly to Boston, borrow an identical table, slide the cup again: identical physics, down to the scratch it leaves. Nobody has ever been surprised by this. Repeat the experiment tomorrow instead of today; nothing changes then either. Turn the table forty degrees; still nothing.
These are the least interesting facts in all of science. Where you stand does not matter, when you start does not matter, which way you face does not matter. Pure yawns.
Except that module 06 found a number no collision could touch, and module 07 found a total no journey could move, and both looked like accounting miracles. They are not miracles. They are invoices, and the yawns above are what paid them. Why should boring sameness pay out in untouchable numbers?
The takeaway
Every symmetry of the Lagrangian buys one conserved quantity. Indifference to place pays momentum; indifference to starting time pays energy. The two currencies of modules 06 and 07 stop being miracles and become consequences.
The yawns, itemized
Give the yawns their technical name. A symmetry is a change you could make to a whole setup that the Lagrangian does not register: shift every position a meter to the left, and if comes out identical, translation is a symmetry. Start the experiment an hour later; if has no clock of its own, delay is a symmetry. A symmetry is not a feature the system has. It is a question the Lagrangian refuses to distinguish from the original. Module 02 built the machinery for asking it, the change of frame; what was bookkeeping there becomes physics here.
In 1918, Emmy Noether proved the exchange rate: each such indifference forces one quantity to be conserved. Not suggests. Forces. The logic, in one breath: the action grades whole paths, module 08 showed the real path is where the grade goes flat, and if a shift of the world cannot change any grade, then the flatness travels with the shift, and something measurable must be carrying it. That something is the conserved quantity. The law's blind spots are exactly its treasures.
Every symmetry gives a conserved quantity. Conservation laws become consequences. Sit with what that trade does to the two big results you already own: momentum conservation is no longer a fact about collisions, it is a fact about space, that one place is as good as another; energy conservation is no longer a fact about hills, it is a fact about time, that one Tuesday is as good as the next. The bookkeeping miracles of modules 06 and 07 were the universe failing to care about location and date, converted into arithmetic. Module 02 called a frame-proof number an invariant; a conserved quantity is the same honor, awarded along time instead of across grids.
The theorem earns its keep equally fast when a symmetry dies. Near Earth's surface, up is not interchangeable with down; the Lagrangian registers every meter of height through . Vertical translation is broken, and, exactly on cue, vertical momentum is not conserved: dropped things speed up. Meanwhile the horizontal directions stay interchangeable and horizontal momentum stays honest, and time stays interchangeable and energy stays honest. Conservation laws fail precisely where symmetries fail. When a physicist meets a quantity that leaks, the first question is no longer what pushed it, but which indifference broke.
A bargain analogy, and its crack in the same breath: Noether's theorem is a vending machine, symmetry in, conserved quantity out. The crack is that a vending machine could in principle be stocked with anything; this one cannot. The pairing is fixed by the mathematics, space to momentum, time to energy, rotation to angular momentum, and no other pairings exist to buy. The machine has exactly one shelf per coin.
The math
Two cases, both run to the end.
Case 1: space does not matter. Take two bodies joined by any interaction that depends only on their separation:
Shift both bodies by the same : the difference is untouched, so is untouched. Translation is a symmetry. Now write the Euler-Lagrange equation for each coordinate:
The two right sides are the same number with opposite signs; that is module 05's third law, reappearing uninvited. Add the equations:
Total momentum, conserved. Module 06 assumed the pair rule; here it fell out of the shape of . Break the symmetry instead, with for a single body: then and the same machinery gives . The leak rate of the conserved quantity equals the degree to which the symmetry fails.
Case 2: time does not matter. Build the quantity
For this evaluates to : module 07's total energy, reassembled from parts. Differentiate it and use the Euler-Lagrange equation to replace with :
The bracket is the equation of motion itself. Energy is conserved because carries no clock: nothing in it says what time it is. Had depended on directly, a spring stiffening overnight, the cancellation would fail and would leak, again at exactly the rate the symmetry breaks.
Play with it
Two masses, one spring, drifting through space. The parts trade momentum furiously; the green total ignores them. Now drag the tilt slider and watch which line bends. The wiggles never change. Only the world's indifference does.
Check yourself
Problem 1. A free particle, m = 2 kg at 3 m/s, with U = 0 everywhere. What is conserved, and why is the answer everything on the menu?
Reveal the solution
With U = 0 the Lagrangian registers neither place nor time. Both symmetries hold, so both currencies are frozen: p = 6 kg·m/s and E = ½ × 2 × 9 = 9 J, forever.
Problem 2. Two bodies joined by a spring: m₁ = 1 kg at 2 m/s, m₂ = 3 kg at -1 m/s. No outside forces. What is the total momentum at every future time?
Reveal the solution
1 × 2 + 3 × (-1) = -1 kg·m/s, forever. The spring depends only on the separation, so translation is a symmetry and the total cannot move, however wild the individual traces get.
Problem 3. Near Earth's surface, U = mgx with m = 2 kg and g = 9.8. Which symmetry died, which survived, and at what rate does the dead one's currency leak?
Reveal the solution
Vertical translation died: the Lagrangian registers height. Vertical momentum leaks at dp/dt = -mg = -19.6 newtons, which is nothing but the weight. Time symmetry survived, so total energy holds while momentum drains. One broken symmetry, one leaking account, and the leak rate is the breakage.
What this did to the demon
The demon's list of never-recompute quantities turned out to be a dictionary of the world's indifferences, one entry per yawn, and it now knows how to read a leak as a broken symmetry rather than a mystery. Module 10 redraws its entire map: every possible world becomes a single point, and the shortcuts it has been collecting become geometry.
Hamiltonian mechanics