The action that knows where every path ends

Hamilton’s principle hands out one number per path. Draw a curve between two events, integrate kinetic minus potential energy along it, and the true motion is the curve for which that number is stationary. The principle compares paths, and each path is a separate candidate with its own integral.

Hamilton had a second idea, published in 1834 in the same paper, and it is the one that turned out to carry the weight. Fix where a motion starts. For every place and every later time, find the true trajectory that gets there, and record its action. The result is no longer a number per path; it is a function of the endpoint, $S(\mathbf{x}, t)$, defined over the whole of space and time. It contains every trajectory the starting condition allows, and it contains them as a landscape rather than as a list.

Three things follow from treating the action that way, and each of them is checked below against trajectories computed independently rather than against the formula. The slope of the landscape in space is the momentum of the particle arriving there. Its slope in time is minus that particle’s energy. And its level curves are wavefronts — surfaces the trajectories cross at right angles — which is how mechanics acquired an equation that looks exactly like the one optics uses for light.

A map of one instant #

The simplest case where the map can be drawn exactly is a swarm of particles launched at the same moment from one point, in every direction and at several speeds, into a uniform gravitational field. In units where the mass and the field strength are both one, the action of the true path from the origin to the point $(x, y)$ in a time $t$ is

$$S(x, y, t) = \frac{x^2 + y^2}{2t} - \frac{y\,t}{2} - \frac{t^3}{24}.$$

That is not assumed. It is what the integral of $\tfrac12 v^2 - y$ comes to along the unique parabola joining the origin to that point in that time, and the figure recomputes it by quadrature along each drawn trajectory.

The first thing to read off the figure is the geometry. The level curves are circles, and every arriving velocity points straight out across them, so every trajectory meets every level curve at a right angle. That is the defining property of a wavefront and its rays, and nothing about light was used to get it.

The second thing is where the circles are centred, and it is the detail most worth stopping on. Every particle in the swarm has fallen by the same amount, half the field strength times the time squared, so the swarm as a whole is a ring of particles centred on a point below the launch. The level curves of the action are centred the same distance above it. No particle is there and none ever was.

The reason is a short argument about velocities. In a frame falling with the field, every particle moves in a straight line at constant speed from the falling launch point, so its position relative to that point is its velocity multiplied by the elapsed time. Transforming back to the ground adds the frame’s own downward velocity, $gt$, to every particle at once. A velocity equal to “displacement from the falling point, divided by $t$, minus $gt$” is the same as “displacement from a point raised by $\tfrac12 gt^2$, divided by $t$” — and the second form says that extrapolating any arriving velocity backwards in a straight line, for the elapsed time, lands exactly on the raised point. The two points are mirror images about the launch, and the map knows about the upper one because the map is built from velocities.

The third thing is the claim the figure actually tests. At the tip of every trajectory the arriving momentum was compared with the gradient of $S$ at that point, differenced numerically, and they agree to a part in ten thousand. The momentum of a particle is the slope of the action at the place the particle has reached. A map of one number over space encodes a vector at every point, and the vector is the one Newton’s law would have delivered after a step-by-step integration.

Why the gradient should be the momentum is a one-line argument once the action is thought of as a function of the endpoint. Move the endpoint by a small displacement and the true path to the new endpoint differs from the old one by a small deformation. The action is stationary along the old path, so to first order the deformation in the middle contributes nothing, and the only change that survives is the boundary term from moving the end — which is the momentum dotted into the displacement. Stationarity of the action is precisely what makes its derivative with respect to the endpoint so simple.

How fast the map changes is the energy #

The same argument applied to the endpoint’s time rather than its position gives the other half. Hold the destination fixed and allow the journey a little longer. The true path to the same place in the new time is a small deformation of the old one, the middle contributes nothing, and the change in the action is minus the energy times the extra time.

That turns an abstract identity into something that can be drawn for a single target. Fix a point at unit distance on level ground. Every journey time corresponds to exactly one throw that lands there — quick arrivals need a flat, fast launch, slow ones a lob — and each throw has its own energy.

The tangents in the upper panel are the claim. Each is drawn with a slope of minus the energy read from the lower panel, where that energy came from the launch velocity of the throw and not from the action; they touch the curve to six decimal places. The action itself has no special value anywhere along the curve — it falls steadily as the journey is lengthened — and it is the rate of fall that carries the physics.

The lower panel holds a result that deserves its own sentence. The energy has a minimum, at an arrival time of $\sqrt{2}$ in these units, and the throw that achieves it is launched at 45 degrees. That is the angle that throws furthest, arrived at here from the other direction: the throw that reaches a given range most cheaply is the throw that, at a given cost, reaches furthest. On the action’s own curve, that throw is simply the point where the curve’s slope is least steep.

One equation that holds every trajectory #

The two slopes together make an equation. The momentum is the spatial gradient of $S$; minus the energy is its time derivative; and the energy of a particle is its kinetic energy, $p^2/2m$, plus its potential energy. Substituting the first two into the third,

$$\frac{\partial S}{\partial t} + \frac{|\nabla S|^2}{2m} + V(\mathbf{x}) = 0.$$

That is the Hamilton–Jacobi equation, and the striking thing about it is what it lacks. There is no trajectory in it, no particle, and no second derivative. It is a single first-order partial differential equation for one scalar function, and Newton’s law — a second-order ordinary differential equation for each particle separately — has disappeared into it. The trajectories reappear only when the equation is solved: they are the curves along which information travels through the solution, what the mathematics of such equations calls characteristics, and following one of them is the same computation as integrating one particle.

For a time-independent potential the time can be separated off. Writing $S = W(\mathbf{x}) - Et$ leaves

$$|\nabla W|^2 = 2m\,\bigl(E - V(\mathbf{x})\bigr),$$

and that is, symbol for symbol, the eikonal equation of geometrical optics, $|\nabla \phi|^2 = n^2$, with an index proportional to $\sqrt{E - V}$. The optical–mechanical correspondence that Maupertuis’ principle produced for rays is here produced for fronts. A ray in a medium of varying index bends without a surface because the front it is perpendicular to turns; a particle curves in a potential for the same reason, and the function whose level sets are its fronts is $W$.

The fronts were not found by solving the equation. Each is built by walking every throw of a fan of 181 to the point where its own integral of momentum reaches a chosen value and joining the points in order of launch angle, and the right angles were then checked at more than seven hundred places. So the drawing tests the eikonal statement rather than illustrating it: a function whose gradient is the momentum must have level curves perpendicular to the motion, and these are. The small front near the launch is a plain arc. The larger ones do something an optical front does at a caustic, which is the subject of the next section, and they do it at exactly the curve the throws themselves pile up against.

The equation also does the thing for which it is mostly used, which is to solve problems that the trajectory picture makes hard. Jacobi’s move was to look for a solution in which each coordinate appears in a separate term. When that works — when the equation separates — every separation constant is a conserved quantity — the symmetry that hands over a conservation law appearing as a term that splits off — the family of solutions is complete, and the trajectories follow by differentiation rather than by integration of the equations of motion.

The Kepler problem separates. In spherical coordinates it splits into a radial part, an angular part and the energy, and the separation constants are the energy, the total angular momentum and its component along an axis. It also separates in parabolic coordinates, with a different set of constants, and a problem that separates in two different coordinate systems has more conserved quantities than a generic one. The extra one is the vector that points at the perihelion, and the double separability is the same fact seen from the partial differential equation rather than from the orbit. When Sommerfeld and Epstein quantised the hydrogen atom in 1916 they did it by imposing a condition on each separated term, and the parabolic separation is how Epstein got the splitting of hydrogen’s lines in an electric field right a decade before quantum mechanics.

Where two trajectories arrive at the same place #

The map in the first figure is single-valued because every point of the plane is reached, at a given instant, by exactly one trajectory from the origin. That is special to fixing the time. Fix the energy instead — throw everything at one speed, in every direction — and the region the throws can reach has an edge, the envelope no throw at that speed crosses. Every point inside it is reached twice, by a flat throw and by a lob, and every point outside not at all.

The right-hand panel is what the Hamilton–Jacobi function looks like when it is computed honestly from the trajectories. It has two branches. Along the dashed line each point has one value of $\int \mathbf{p}\cdot\mathrm{d}\mathbf{s}$ for the flat throw and a larger one for the lob, and the two approach each other and meet at the envelope, where the flat throw and the lob become the same throw. Beyond the envelope there is nothing to draw.

The slope of each branch was checked against the horizontal momentum of the throw on that branch, and they agree. So at a point reached twice, the gradient of the function — the momentum — has two values as well, which is exactly right: two particles really do pass through that point with two different velocities. What cannot be right is any single smooth function claiming to be the solution there. The equation’s solution, started as a single-valued front from the launch point, has folded over on itself, and a fold in the front is a caustic: the same structure that makes a bright curve at the bottom of a coffee cup, arrived at in mechanics.

The fold matters well beyond projectiles, because the Hamilton–Jacobi equation has a second life under another name. The equation that dynamic programming produces for the cost of reaching a state optimally — the Hamilton–Jacobi–Bellman equation of control theory — has the same structure, with the least cost playing the part of the action. There, two branches arriving at one point cannot both be kept, because only the cheaper route is the optimal one. The accepted solution takes the lower branch everywhere and has a corner where the two cost the same, and an entire mathematical theory of non-smooth solutions exists to say which corners are allowed. In mechanics both branches are real motions; in control only one is a decision. The figure is the mechanical case and draws them both.

The focus, where the action forgets the start #

A fold is one way for the map to fail. A focus is a sharper one, and an oscillator produces it with no approximation at all.

Launch particles from the centre of a harmonic well, all at the same instant and at different speeds. Each oscillates with the same period whatever its amplitude — the property a real pendulum only approximates — so at half a period every one of them is back at the centre simultaneously. In units where the mass and the angular frequency are one, the action of the true path from the centre to position $x$ at time $t$ is $S = \tfrac12 x^2 \cot t$, and its gradient is $x\cot t$, which the figure checks against the arriving momentum of each trajectory.

The parabola steepens as half a period approaches because a small displacement of the endpoint requires an enormous change in launch speed — every speed lands near the centre at that moment, so reaching a point a little way off needs a very large one. At exactly half a period the function has no value to take. Every launch reaches the centre, so the action to the centre is not one number but a whole family of them, and no launch reaches anywhere else, so the action elsewhere is undefined. Past half a period the parabola reappears opening the other way, which is the change of sign that a quantity passing through infinity leaves behind.

This is the same point in time at which the action stops being least. A path that runs past a focus is still a stationary path, but it is no longer a minimum; there the second variation acquires a negative direction. The Hamilton–Jacobi picture sees the same event as a blow-up of the function’s curvature, and the two descriptions are a single fact. When the classical action is used as the phase of a wave, which is what quantum mechanics does once every particle is given a wavelength, the passage through a focus leaves a trace in the wave as well — a quarter-cycle shift in phase for each focus passed, the mechanical counterpart of the shift light acquires passing through the focus of a lens — and the same bookkeeping of quarter-cycle shifts, taken at the turning points of a bound orbit, is what puts the half into the oscillator’s quantised energies, $(n + \tfrac12)\hbar\omega$.

What the function depends on that the motion does not #

The function belongs to a family of trajectories, not to the motion. Every figure here fixes a launch point, and the action is built from the trajectories leaving it. Launch from a line instead, or from a plane front with all velocities parallel, and the same physics produces a different function with different level sets. The trajectories of any one particle are identical in both descriptions; what differs is which neighbours it is grouped with. A Hamilton–Jacobi function is therefore a choice of initial front as much as a statement about dynamics, and that choice is where both the folds and the foci come from.

The gradient is the canonical momentum, which is not always mass times velocity. In a magnetic field the momentum that appears as the gradient of $S$ is $m\mathbf{v} + q\mathbf{A}$, and the fronts are then not perpendicular to the velocity. The right angles in the first figure are a property of forces that come from a scalar potential, and they fail in exactly the way that a potential nobody can measure makes visible in quantum interference.

Nothing dissipative has an action. Friction, drag and every other force that turns ordered motion into heat has no Lagrangian of the ordinary kind, so there is no action to make a function of, and the whole construction is a statement about conservative mechanics.

The closed forms are an accident of the two problems chosen. A uniform field and a harmonic well are the two cases where the action is a quadratic in position, which is why the fronts are circles and the curves at fixed time are parabolas. For almost any other potential the function has to be computed numerically along the trajectories, which is what the second and third figures do anyway, and in a chaotic system it cannot be written as a single-valued function over any appreciable region at all.

A surface in space and time, drawn as slices #

Each map is drawn at one instant or along one line. The function lives in the plane and in time, and what the figures show are slices: circles at one moment, a curve at one target, two branches along one height. The fold in particular is a surface folded over itself in three dimensions, and the pair of curves in the third figure is a single cut across it.

The drawings also show the function only where it is defined by trajectories that have actually been launched. Outside the envelope the third figure has nothing, and the right answer there is that the particles do not arrive — but the wave that quantum mechanics builds on this function does arrive, faintly, decaying past the envelope in the way light leaks past the bright edge of a caustic. The classical map has no way to represent that tail, and it is the first thing the wave description adds.

Still open: what the action does when every path is summed #

Hamilton–Jacobi theory is usually introduced not as a map but as a method: find a change of variables in which the new coordinates and momenta are all constant, so that the solved motion is trivial, and the function that generates that change of variables is $S$. That is the reading in which the separation of the Kepler problem, the action variables of the old quantum theory and the perturbation theory of planetary orbits all live, and it is where the freedom to redescribe a problem — canonical transformation — becomes the whole of the technique rather than a convenience.

The habit worth carrying away is to ask what a quantity is a function of. A number defined along one path becomes a field once it is recorded at every endpoint, and a field has derivatives that the path never had. The action as a number made mechanics a variational principle; the action as a function of where a path ends made it a theory of fronts, and every appearance of a wavefront in mechanics — in optics, in control, in the phase of a quantum wave — is that one change of viewpoint.

The same function has a second life that is not settled. In quantum mechanics every path contributes, each with a phase equal to its action divided by Planck’s constant, and the classical trajectories are the paths whose phases are stationary. Where two of them merge at a fold, the simple approximation that adds one wave per trajectory diverges, and the repair is known — an Airy function across the fold, and a quarter-cycle phase lost for every focus a path passes through, which is the conjugate point of this essay reappearing as a measurable shift in energy levels. What is not known is how far that repair can be carried. For motion that is chaotic, the number of classical paths between two points grows exponentially with the time allowed, and the sums over periodic orbits that should give a quantum system’s energy levels from its classical trajectories do not converge in general. Which quantum properties of a chaotic system its classical action fixes, and which it cannot, is still being worked out.