The action along a family of paths

At its defaults it draws the action along a family of paths. On the left, seven paths between the same two events: the true trajectory of a projectile and six deformations of it, each fixed at both ends and differing by one arch of a sine. On the right, the action of each — the time integral of kinetic minus potential energy — against how much it has been deformed. The true path has the least action, 0.45833 in these units, and every neighbour has more. The curve on the right is a parabola about that minimum with curvature 4.935, so the excess action grows as the square of the deformation and its slope at the true path is zero. Nothing here was minimised: the true path was obtained by solving the equation of motion, and every action on the chart including its own is the same quadrature along a stated curve. What the figure establishes is that the two ways of specifying a trajectory — obey a differential equation at every instant, or make one integral over the whole path stationary — pick out the same curve.

action-paths is one function in lib/figures/mechanics.js — motion, force, energy and rotation. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

On the left, seven paths between the same two events: the true trajectory of a projectile and six deformations of it, each fixed at both ends and differing by one arch of a sine. On the right, the action of each — the time integral of kinetic minus potential energy — against how much it has been deformed. The true path has the least action, 0.45833 in these units, and every neighbour has more. The curve on the right is a parabola about that minimum with curvature 4.935, so the excess action grows as the square of the deformation and its slope at the true path is zero. Nothing here was minimised: the true path was obtained by solving the equation of motion, and every action on the chart including its own is the same quadrature along a stated curve. What the figure establishes is that the two ways of specifying a trajectory — obey a differential equation at every instant, or make one integral over the whole path stationary — pick out the same curve.

On the left, seven paths between the same two events: the true trajectory of a projectile and six deformations of it, each fixed at both ends and differing by one arch of a sine. On the right, the action of each — the time integral of kinetic minus potential energy — against how much it has been deformed. The true path has the least action, 0.45833 in these units, and every neighbour has more. The curve on the right is a parabola about that minimum with curvature 4.935, so the excess action grows as the square of the deformation and its slope at the true path is zero. Nothing here was minimised: the true path was obtained by solving the equation of motion, and every action on the chart including its own is the same quadrature along a stated curve. What the figure establishes is that the two ways of specifying a trajectory — obey a differential equation at every instant, or make one integral over the whole path stationary — pick out the same curve.

The excess action against deformation, for four completely different ways of deforming the same path: one arch, two arches, a narrow local bump and a corner. Each curve has its minimum at the true path and each has zero slope there — measured at -5.5e-7, -5.0e-15, -3.7e-13, 1.4e-14 against an exact zero. That is what stationary means, and it is a much stronger statement than any one of these curves makes on its own: the action does not decrease under *any* small change of the path, including changes with corners in them and changes concentrated in a small part of the flight. The curvatures differ a great deal — a corner costs far more action than a smooth arch of the same height, because the action penalises speed rather than displacement — and that difference is why the second variation is a subject of its own while the first variation is a single equation. Requiring the first variation to vanish for every deformation, including ones localised anywhere, is what turns one integral condition into a differential equation holding at every instant.

The second variation of the action of a harmonic oscillator — how much the action changes when the path is deformed by a small amount — against the duration of the trip measured in radians of the oscillator's own phase, for the first 3 deformation modes. Each curve starts positive, which says the true path is a minimum, and crosses zero at ωT = 3.142, ωT = 6.283, ωT = 9.425 — that is nπ, and past it the deformation *lowers* the action. So for any trip lasting more than half an oscillation the classical path is a saddle point of the action and not a minimum at all: there exist nearby paths with the same endpoints and less action, and one of them is drawn on every textbook page that says 'least'. The point at which this happens is the kinetic focus — for an oscillator, the moment when every path leaving the start returns to the same place — and the correct statement of the principle is that the action is stationary. The computed curves agree with the exact second variation to 4.1e-9.

Each path in the family contributes a unit arrow whose direction is its action divided by Planck's constant — taken as 0.012 of the action's own units here, so that the effect is visible on a page — and the curve is the running sum of those arrows, taken in order of deformation. Where the action changes quickly with the path, successive arrows point in different directions and the sum spirals without going anywhere; near the stationary path the action barely changes, so a whole band of paths contributes arrows pointing the same way and the sum runs straight. That straight run is the whole of the resultant and rather more: the paths within the first phase zone supply 125 per cent of the total, which is over a hundred because the next band subtracts part of what the first contributed — the same overshoot a Fresnel zone plate is built to exploit. Everything outside cancels against its own neighbours. This is why a classical trajectory exists. It is not that the particle chooses the path of stationary action; it is that every path contributes and only the ones near the stationary one fail to cancel — and as ħ is made smaller the surviving band narrows, which is the classical limit arriving.

On the left, seven paths between the same two events: the true trajectory of a projectile and six deformations of it, each fixed at both ends and differing by one arch of a sine. On the right, the action of each — the time integral of kinetic minus potential energy — against how much it has been deformed. The true path has the least action, 0.45833 in these units, and every neighbour has more. The curve on the right is a parabola about that minimum with curvature 4.935, so the excess action grows as the square of the deformation and its slope at the true path is zero. Nothing here was minimised: the true path was obtained by solving the equation of motion, and every action on the chart including its own is the same quadrature along a stated curve. What the figure establishes is that the two ways of specifying a trajectory — obey a differential equation at every instant, or make one integral over the whole path stationary — pick out the same curve.

physicscheck asserts something about action-paths that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.