Illustrated Physics
physics, drawn
Series

Least action — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · mechanics
  2. 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.

    part 2 · mechanics
  3. 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.

    part 3 · mechanics
  4. 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.

    part 4 · mechanics
  5. The action at one instant, drawn as a map. Trajectories leaving one point at the same moment, at launch speeds 0.6, 1 and sixteen directions each, in a uniform field pulling downward, drawn up to time 1. Behind them, dashed, are the level curves of the action at that instant, regarded as a function of where a trajectory ends. They are circles, and their common centre is neither the launch point nor anywhere the particles have reached: it is 0.500 above the launch point, while the whole swarm has fallen by the same 0.500. Every arriving velocity points straight out from that centre, so every trajectory crosses the level curves at right angles, and the arriving momentum equals the gradient of the action to one part in ten thousand. The action integrated along each path agrees with the map's value at its end.

    The action that knows where every path ends

    The action is usually a number attached to one path. Treat it instead as a function of where the true path ends, and a single function of position and time holds every trajectory at once — its slope is the momentum, its rate of change is the energy, and its level curves are wavefronts, drawn about a point that sits above the source while everything falls.

    part 5 · mechanics
  6. The area each quantum state is allowed. Phase-space orbits of the four lowest quantum levels, computed as eigenvalues of the discretised Schrödinger equation, for a harmonic well (left) and a box with hard walls (right), in units ħ = m = 1. Each orbit encloses an area that is a whole number of Planck constants plus a fixed offset: for the harmonic well 0.500, 1.500, 2.500, 3.500 h — n + ½ — and for the box 1.000, 2.000, 3.000, 4.000 h — n + 1. The harmonic orbit turns round smoothly at both ends and loses a quarter of a cycle at each; the box orbit is reflected sharply by each wall and loses half a cycle there.

    The quarter cycle a turning point costs

    The old quantum rule said an orbit's action is a whole number of Planck constants. It gets the harmonic oscillator wrong by exactly half a unit, the box by exactly one, and a neutron bouncing on a mirror by three quarters. Each is the same correction: a wave loses a quarter of a cycle every time it turns round smoothly, and half a cycle every time it hits a wall.

    part 6 · mechanics
  7. Three orbits that are one orbit, rescaled. Three orbits round the same mass, each an ellipse of eccentricity 0.6, with sizes in the ratio 1 : 2 : 3, integrated from the inverse-square force. The dots mark twelfths of each orbit's period, at the same fractions of the way round. The shapes are exact enlargements of one another and so are the motions, with time stretched as size to the power 3/2: the measured periods are 1.000, 2.828, 5.196 times the smallest, against 1.000, 2.828, 5.196 from the rule. That is Kepler's third law, obtained without solving for any orbit.

    The third law that falls out of a change of scale

    Kepler spent a decade of tables to find that the squares of the planets' periods go as the cubes of their distances. The law can be had in three lines, without solving for a single orbit. Gravity's potential energy halves when distances double; stretch every length of an orbit by some factor and every time by that factor to the power three-halves, and the action changes only by a constant, so the stretched orbit is an orbit too. The same move gives the swing of a spring that keeps time at any amplitude, the virial theorem that makes a star heat up as it loses energy, and the spacing of quantum levels.

    part 7 · mechanics
  8. The same damped motion in two phase planes. A cloud of damped oscillators that start on a small circle of positions and velocities, drawn at times 0, 2, 4, 6 (outlines) with the centre's path. Left: position against the physical momentum mẋ. The cloud spirals inward and shrinks: its area falls as e^(−γt), to 0.30 of its start by t = 6, as friction squeezes every motion towards rest. Right: position against the canonical momentum of the Caldirola–Kanai Lagrangian, p = e^(γt)mẋ. The same cloud keeps exactly its starting area while it is stretched out of shape, as Liouville's theorem requires of any Hamiltonian flow. The friction has been hidden in the definition of momentum: the canonical momentum grows exactly as fast as the real one decays.

    The friction an action can hold only by changing its clock

    The principle of least action generates every conservative law of motion from a single integral, and it has no place for friction: there is no ordinary Lagrangian whose equation of motion is a damped oscillator's. Multiply the Lagrangian by an exponential of time and friction appears — but only by redefining momentum so that it grows as fast as the real momentum decays. The phase-space area the principle conserves is then no longer the one that counts motions, and quantised, the damped oscillator becomes more sharply known than the uncertainty principle allows. Friction belongs to a system and the surroundings it loses energy to, and an action for the system alone can only fake it.

    part 8 · mechanics

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