Module 08

Least action

7 minute read

Watch light cross from air into water: at the surface it bends, sharply, as if it changed its mind. Measure the bent route and a pattern falls out. Of all the ways to get from the lamp above the surface to the pebble below it, the route light takes is the fastest one, given that light moves slower in water.

A lifeguard does the same thing. Sprinting beats swimming, so the smart line to a drowning swimmer hugs the sand longer than the straight line would. But the lifeguard stands still for a moment and plans. That is what fastest-path behavior seems to require: a survey of the options.

Light has no options and no moment. It leaves the lamp already committed. How does it know, at departure, anything about routes it will never take?

The takeaway

Mechanics carries the same secret. Of every path a body could take between two fixed events, the real one is the path whose action is stationary: nudge it any way you like and, to first order, the score refuses to move. Newton's laws are the local shadow of that global fact.

One number per path

Everything so far has been local. Newton's rule is a next-tick machine: state now, forces now, acceleration now, repeat. This module hands you the same physics in an alien shape, and the shape is the point.

Fix two events, not one: leave here at this time, arrive there at that time. Between them the body could, in imagination, do anything: the direct route, a soaring detour, a frantic wiggle. Call each complete history a path. Now grade them. For any moment along a path, take the kinetic energy and subtract the potential energy. That odd difference, motion's account minus position's account, is the Lagrangian. Add it up along the whole path, instant by instant, and the total is one number for that path: its action. Do it again for another path; that path gets a different number.

Figure 8.1 Every candidate route gets one number. The route the ball actually takes sits at the flat bottom of that list, where nudging the path no longer moves the score.

Here is the claim, and it is the deepest sentence in this volume. The path the body actually follows is the one where the grading goes flat: shift that path slightly, any shift at all, and its action does not change to first order. It is stationary, the way the bottom of a valley is flat. Usually the real path has the least action of its neighbors, which is why the principle carries the name least action, but flatness, not smallness, is the real criterion.

Nature does not follow rules step by step. It picks a whole path. Every next-tick law you have learned, force, response, update, is the view through a keyhole of something that is not local at all: a comparison across histories that were never lived, settled in favor of the one that was. The demon has been integrating its way through time like a clerk; the world, meanwhile, grades journeys whole.

Be careful with the word "picks," and here is where the choosing analogy breaks: nothing surveys the losing paths, the way the lifeguard surveys the beach. The comparison happens in the mathematics, not in the world; classical mechanics is silent about any mechanism, and the flat-spot rule and Newton's rule are provably the same rule worn two ways. Why the universe is arranged so that a global grading and a local push agree, classical physics cannot say. The next physics after this volume, quantum mechanics, finally answers it: every path is in some sense taken, and the flat spot is where the alternatives stop cancelling each other out. That sentence is a promissory note; this course will not cash it.

The math

Define the Lagrangian of a body with position and velocity :

L = \frac{1}{2} m \overset{x}{˙}^{2} - U (x)

Define the action of a path between fixed events and :

S = \int_{0}^{T} L (x (t), \overset{x}{˙} (t)) d t

Take the real path and disturb it: , where is any wiggle with . The endpoints are appointments; the wiggle must keep them.

Feed the disturbed path into . Keep only terms linear in . The velocity of the disturbed path is , so:

δ S = ϵ \int_{0}^{T} (\frac{\partial L}{\partial x} η + \frac{\partial L}{\partial x ˙} \overset{η}{˙}) d t

The second term contains , and the wiggle's rate is not free to vary independently of the wiggle. Trade it away. One rule is needed: the derivative of a product, which module 03 promised to earn on arrival. Multiply two changing quantities; the product's change collects one term from each: , because exceeds by and a second-order crumb. Apply it with and , integrate both sides over the flight, and rearrange:

\int_{0}^{T} \frac{\partial L}{\partial x ˙} \overset{η}{˙}; d t = [\frac{\partial L}{\partial x ˙} η]_{0}^{T} - \int_{0}^{T} \frac{d}{d t} (\frac{\partial L}{\partial x ˙}) η; d t

The boundary term dies: is zero at both appointments. What remains assembles into one integral:

δ S = ϵ \int_{0}^{T} (\frac{\partial L}{\partial x} - \frac{d}{d t} \frac{\partial L}{\partial x ˙}) η; d t

Stationary means for every allowed . An integral that vanishes against every wiggle can do so only if the bracket is zero at every instant:

\frac{d}{d t} \frac{\partial L}{\partial x ˙} - \frac{\partial L}{\partial x} = 0

This is the Euler-Lagrange equation. Now cash it. With : the first slot gives , whose time derivative is ; the second gives . The equation reads:

m \overset{x}{¨} = - U^{'} (x)

Mass times acceleration equals the downhill pull of the potential. Newton's second law walked out of a grading scheme for paths, and nothing about forces was assumed on the way in.

Play with it

The two events are fixed: leave the ground now, be back on it two seconds later. Bend the path between them with the three posts and watch its action recompute. Try to beat -32.0. When you give up, reveal what nature drew.

Check yourself

Problem 1. For the two-second throw with g = 9.8, compute the action of the laziest path: stay on the ground the whole time. The real parabola scores about -32.0. Which is smaller, and what does that mean?

Reveal the solution

On the ground, kinetic and potential are both zero the whole time, so S = 0. The parabola's -32.0 is smaller. Between these two appointments the real path is the lower-action one; sitting still fails to be stationary, which you can feel in the sim: from the flat path, small bends lower the score.

Problem 2. Run the Euler-Lagrange equation on L = ½mẋ² - ½kx².

Reveal the solution

∂L/∂ẋ = mẋ, whose time derivative is mẍ. ∂L/∂x = -kx. The equation gives mẍ = -kx: the harmonic oscillator, with the spring force arriving unasked.

Problem 3. A free particle (U = 0) must cover 10 m in 2 s. Compare the action of steady 5 m/s against 8 m/s for the first second and 2 m/s for the second.

Reveal the solution

Steady: S = ½ × 25 × 2 = 25 (per unit mass). Two-speed: ½(64)(1) + ½(4)(1) = 34. Steady wins, and it wins against every split, because squares punish unevenness. Least action is why free bodies coast at constant velocity: module 05's first law, rederived by a grading scheme.

What this did to the demon

The demon's rulebook was local, one tick at a time; the world has now shown it a single global principle that contains the whole rulebook as a shadow. Module 09 asks that principle one question, what happens to the action when the world is shifted sideways or the clock is started late, and the two conserved currencies the demon has been carrying since modules 06 and 07 start falling out as answers.

Next · module 09
Symmetry and conservation