Illustrated Physics
physics, drawn
Concept

Lagrangian — where it appears

The kinetic energy minus the potential energy, written in whatever coordinates are convenient, whose path integral is stationary along the true motion. Choosing coordinates that satisfy the constraints removes the constraint forces from the equations, which is its great convenience and its one blind spot.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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.

Least action, except that it is not least

Mechanics can be stated twice over. Once as a rule about every instant — force equals mass times acceleration — and once as a rule about the whole path at once, which says that one number computed along it is stationary. The two pick out the same trajectory, and the second name for it is wrong — past a certain duration the real path has more action than its neighbours, not less.

mechanics · Least action Three quantities that do not move while everything else does. On the left, an orbit in an inverse-square attraction, integrated from its equation of motion over 2.4 revolutions at an eccentricity of 0.55. On the right, three quantities computed from that same trajectory at every step and plotted against time: the energy, the angular momentum, and the length of the eccentricity vector that points at periapsis. Every one is flat to better than 1.2e-11 in units where the circular speed at r = 1 is 1, and none of them was constrained to be — the integrator was given the force and nothing else. Each is a symmetry seen sideways. The energy is constant because the force law does not mention the time; the angular momentum is constant because it does not mention the direction; and the eccentricity vector is constant because of a symmetry that is not a motion of space at all, which is why the inverse square closes its orbits and its neighbours do not. What the picture cannot show is the direction of the argument: it demonstrates that these three are constant here, and the theorem says something much stronger, that a constant exists for every continuous symmetry whatever the system.

The conservation law a symmetry hands over

Energy, momentum and angular momentum are usually presented as three separate empirical facts that happen to hold. They are one fact three times: every continuous symmetry of a system's action supplies a quantity that does not change, the correspondence is exact, and it runs both ways.

mechanics · Least action The reaction that reaches zero, and where the bead lets go. The force the sphere pushes back with, in units of the bead's weight, against the angle from the top. It starts at 1.000 and falls, because the speed the bead has gained needs more centripetal force than gravity's component along the radius can supply. At 48.19° it reaches zero, and past that the surface would have to pull inward to keep the bead on it — which a surface cannot do. So the bead leaves there, and the departure angle is a statement about the sign of a constraint force rather than about a speed or a height. The multiplier is what carries that sign: solve the motion in the angle alone and the reaction is absent from every equation, so nothing in the solution knows that the constraint has stopped holding, and the bead is drawn happily circling a sphere it has already left.

The force a coordinate cannot see

Writing a pendulum in terms of its angle is the first good move anybody learns, and it deletes the tension from every equation that follows. The string still breaks. Recovering the force that the clever choice of coordinate threw away turns out to be a computation with a sign in it, and the sign is where the bead leaves the sphere.

mechanics · Least action Two principles, two classes of path, two things left free. On the left, five curves from the same launch point to the same target: the true trajectory of a particle of energy 0.7 in a uniform field, and four deformations of it that share both ends. On the right, four quantities computed along that family and plotted as departures from their values on the true path. Maupertuis' abbreviated action ∫p·ds, computed at fixed energy, is stationary — flat at the centre. Hamilton's action ∫L dt, computed at fixed duration, is stationary too. The other two are not: the time a fixed-energy path takes changes at first order in the deformation, and so does the energy a fixed-duration path carries. That is the whole difference between the two principles. Each holds one of those quantities fixed and lets the other vary, and neither can hold both.

The principle that fixes the energy instead of the clock

There are two principles of least action, they compare different sets of paths, and they are not the same statement. One holds the duration fixed and lets the energy vary; the other holds the energy fixed and lets the duration vary — and written that way, mechanics turns into optics with a refractive index.

mechanics · Least action

Named alongside it

The objects these essays reach for when they reach for this one.

ActionConserved quantityEnergyAngular momentumBifurcationCalculus of variationsConstraintConstraint forceDegrees of freedomEffective potentialEquation of motionFermat's principle

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