The principle that fixes the energy instead of the clock

The phrase “the principle of least action” names two different principles. They were published forty years apart, they compare different collections of paths, they are stationary at the same trajectory, and one of them turns mechanics into a problem in optics. Which of the two is meant is almost never said, and the difference is not a technicality: it is the difference between holding a clock fixed and holding an energy fixed.

Hamilton’s principle, the one that appears in modern courses, compares paths that start and end at the same places and take the same time. Along each, the action is the time integral of kinetic minus potential energy. The true trajectory makes that integral stationary.

Maupertuis’ principle, published in 1744 and made precise by Euler in the same year, compares paths that start and end at the same places and carry the same energy. Along each, the abbreviated action is the integral of momentum along the path — a purely geometric quantity, with no clock in it at all. The true trajectory makes that stationary too.

The right-hand panel is the whole distinction. Two of those curves are flat at the origin and two are sloped. Each principle is stationary in its own quantity exactly because it allows the other one to move — and a statement that a trajectory “minimises the action” without saying which quantity was held fixed has not said which of the two flat curves it means.

Why there have to be two #

The reason a single principle cannot serve both purposes is a counting argument. A path between two fixed points in space is specified by a curve and by a schedule along it. Fixing the duration and letting the shape vary is one problem; fixing the energy and letting the shape vary is another, because at a given energy the schedule along a given curve is already determined — the speed at each point is $\sqrt{2(E-V)}$ and nothing is free.

So Maupertuis’ principle is a statement about shapes. Give it a curve from A to B, and the energy fixes how fast the particle moves at every point of it, hence how long the journey takes and what the momentum integral comes to. The variational problem is over curves in space, with time nowhere in it. Hamilton’s principle is a statement about motions: a curve and a schedule together, with the duration pinned and the energy free to be whatever the schedule implies.

A concrete case makes the difference tangible. Take a projectile thrown from one point to another over level ground. Hamilton’s competitors are all the ways of getting there in exactly 0.92 seconds — arched higher, arched lower, wobbling on the way — and among them the parabola is the one whose action is stationary. Maupertuis’ competitors are all the ways of getting there while moving at the speed the energy allows at each height, which means a high arch is traversed slowly near the top and takes longer than a low one. The parabola is stationary among those. The two families overlap in exactly one curve, and it is not a coincidence that the curve is the same: it is the theorem below.

That the two pick out the same trajectory is not obvious and is the content of a Legendre transform, which is the same operation that turns a Lagrangian into a Hamiltonian and the same one that turns an energy at fixed volume into one at fixed pressure. Along the true path,

$$S = W - E T,$$

with $S$ Hamilton’s action, $W$ the abbreviated action, $E$ the energy and $T$ the duration. Each is a separate quadrature over a separate integrand, so the identity is a real constraint rather than a definition, and the figures below are computed in a way that lets it be checked.

The transform explains why the older principle is the one with the geometry in it. Subtracting $ET$ is precisely the operation that removes the time from the description, and what is left is a statement about the shape of the path alone. Maupertuis was reaching for a metaphysical principle — nature is economical — and Euler gave him a quantity that turned out to be the one worth having, for a reason neither of them could have stated: it is the generator of the transformation that makes mechanics geometrical.

The order of events is worth having straight, because the usual telling reverses it. The abbreviated action came first and was argued about on theological grounds for a generation; Lagrange put it on a proper footing in the 1780s; Hamilton’s principle, the one now taught as the principle of least action, arrived in 1834 and 1835, ninety years later. The modern course teaches the newer statement first because it is the general one, and then attributes the older name to it. What is lost in that compression is precisely the geometry: Hamilton’s action is a functional on motions and does not turn into a length, and everything in the second half of this essay follows from the fact that Maupertuis’ does.

Mechanics as optics #

Write the abbreviated action out. The momentum at a point of the path is $\sqrt{2m(E-V)}$, so

$$W = \int \sqrt{2m\left(E - V\right)}\;\mathrm{d}s,$$

an integral along the curve of a scalar that depends only on position. That is exactly the form of an optical path length, $\int n\,\mathrm{d}s$, and the principle that it is stationary is exactly Fermat’s principle with

$$n(\mathbf{r}) \;\propto\; \sqrt{E - V(\mathbf{r})}.$$

Jacobi put it this way in the 1830s and Hamilton had seen the analogy from the optical side before that. It is not a loose resemblance. It means that every result about rays carries over to particles with a change of what the index means: a particle in a region of smoothly varying potential curves like a ray in a medium of varying index, a particle crossing a boundary refracts, and a particle can be totally reflected.

The conserved quantity that produces the refraction is worth naming, because it makes the analogy exact rather than formal. The potential in the figure depends only on the coordinate across the boundary, so the momentum component along the boundary is conserved — this is the conservation law a symmetry hands over, applied to a translation the potential does not notice. Conservation of $p\sin\theta$ is Snell’s law, whatever $p$ happens to be a function of. For light, $p$ is proportional to $n$; for a particle, to $\sqrt{E-V}$; the law is the same law.

The index that is upside down #

There is a sign in the correspondence, and the sign is the most consequential thing on this page.

For light, the refractive index is inversely proportional to the speed: a medium of high index is one in which light travels slowly. For a particle, the effective index is proportional to the momentum, which is to say proportional to the speed. A particle entering a region of lower potential speeds up and bends toward the normal; light entering a medium where it slows down bends toward the normal. Both bend the same way for opposite reasons.

That is not a curiosity. It is the fork in the road that the whole seventeenth- and eighteenth-century argument about the nature of light ran into. Newton’s corpuscular theory explained refraction by supposing that glass attracts the corpuscles, pulling them across the boundary and speeding them up — which gives Snell’s law correctly, with the index as the ratio of speeds in glass over air. Huygens’ wave theory gave the same law with the ratio the other way up. The two theories agreed on every refraction anybody could measure and disagreed absolutely on one number: whether light goes faster or slower in water.

Foucault measured it in 1850 with a rotating mirror and found light slower in water. The corpuscular index is the reciprocal of the measured one, and the reason it is a reciprocal is the reason drawn in the figure above: the mechanical index goes as momentum and the optical one goes as inverse speed. Interference had made a wave theory necessary decades earlier; this is the measurement that made the corpuscular alternative arithmetically impossible.

The step-up case produces the mechanical version of the angle past which light cannot leave, and it is a real and useful effect rather than an analogy. A slow neutron approaching a surface sees a step in potential — the average of the nuclear interaction over the material, of order a hundred nanoelectronvolts — and a neutron whose energy is above the step still fails to enter it at grazing incidence. Sufficiently slow neutrons are reflected at every angle and can be stored in a bottle for minutes at a time, which is the basis of every measurement of the neutron’s lifetime and of its electric dipole moment. The instrument is total internal reflection with a matter wave, and its design equation is the figure above.

What the analogy was missing #

Hamilton’s optical–mechanical analogy sat in the literature for ninety years, admired and unused. It is exact and it is incomplete, and the missing piece is a number.

Optics has two levels of description. Fermat’s principle is the ray level, and it is an approximation to a wave theory that becomes exact when the wavelength is small compared with everything in the problem. The analogy says mechanics has a ray level too — the trajectory — and that its stationary principle has the same form. What it does not say is what the mechanical wavelength is, or whether there is one. Nothing in classical mechanics answers that, because the ray level is all classical mechanics has.

De Broglie supplied the number in 1924 and Schrödinger completed the analogy in 1926 by writing the wave equation whose short-wavelength limit is the Hamilton–Jacobi equation. The relation $\lambda = h/p$ is exactly what makes the correspondence work: the mechanical index goes as the momentum, and the wavelength goes as the reciprocal of the momentum, so a region of high mechanical index is a region of short wavelength — which is what a region of high optical index is. Every particle has a wavelength is the sentence that finishes a piece of nineteenth-century mechanics.

The historical shape of that is worth stating. The analogy was not a heuristic that happened to be suggestive; it was a statement that classical mechanics is the ray limit of something, sitting in print with the something unnamed. The way a quantum description hands the old one back is the same statement read forwards.

Why either of them is stationary #

Neither principle explains itself. Both say that a particular integral is flat at the true path, and neither says why nature should care about an integral over a path it does not take. The answer is a wave answer, and it arrives from the same direction as the wavelength did.

The argument is the one that makes Fermat’s principle intelligible, transplanted. A ray is not chosen; it is where the wave’s contributions fail to cancel, and they fail to cancel exactly where the optical path length is stationary, because that is where neighbouring paths have equal phase. If a particle has a phase that accumulates as its action divided by a constant with the units of action, the same argument gives the same conclusion, and the constant is Planck’s.

Two things are worth noticing about that. It explains the stationary, not the least: a maximum and a saddle both have neighbours in phase, which is why the principle has to be stated as stationarity and why a path past a focus is still a path the argument selects. And the size of the constant decides how sharply. Rays are sharp because optical wavelengths are small compared with lenses; trajectories are sharp because Planck’s constant is small compared with the actions of everyday objects, and the same statement in both cases is the same limit taken twice.

Which of the two actions belongs in the exponent depends on which is being held fixed, and the answer is Hamilton’s for a propagator between two events and Maupertuis’ for one at fixed energy. That is the same pairing as before, arriving at the quantum level as the difference between a time-dependent amplitude and an energy eigenfunction.

The geodesic underneath #

There is a third way to write the same principle and it is worth a paragraph because it is the one that makes the geometry unavoidable.

An integral of the form $\int n\,\mathrm{d}s$ is a length — a length measured in a geometry where a metre near a large $n$ counts for more than a metre where $n$ is small. Stationary length is a geodesic. So the trajectory of a particle of energy $E$ in a potential $V$ is a straight line, in the geometry whose metric is the ordinary one multiplied by $2m(E-V)$. Nothing is left of the force; the potential has been absorbed into the shape of the space, and the particle goes straight.

That sentence is the mechanical rehearsal for the one general relativity makes about gravity, and the differences between the two are as instructive as the similarity. The Jacobi metric depends on the energy of the particle, so each energy gets its own geometry and there is no single space in which all trajectories are straight; a gravitational metric does not, which is the geometrical statement of the equivalence principle. And the Jacobi metric describes only the shape of the path, with the schedule recovered afterwards from the energy, where the relativistic one describes the whole worldline at once. The mechanical version is a geometry of space; the relativistic one is a geometry of spacetime, and the extra dimension is what removes the dependence on the particle.

Where the model stops #

Maupertuis’ principle needs the energy to be conserved, which needs the potential to be independent of time. A time-varying potential has no fixed-energy family of paths to compare, and the abbreviated action is not defined; Hamilton’s principle is untroubled by that and remains the general statement. So the two principles are not interchangeable even where both are usually applicable, and the older one is the more restricted.

Two trajectories reach the target, and the figures use one. At a given energy above the minimum, a projectile has a flat launch and a lobbed one that both land at the same range, and both are stationary points of the abbreviated action. The figures take the flat one. The other is a stationary point too, and past the first focus it is a saddle rather than a minimum — which is the sense in which least action is not least, arriving here from the other principle.

The refraction figures are a discontinuous step, and no potential is discontinuous. The step stands for a transition over a distance short compared with everything else, and the criterion for “short” is not obvious in the mechanical case. It is short compared with the de Broglie wavelength for a quantum particle, which for a neutron at a metal surface is a few tenths of a nanometre against a surface layer of a few atoms — comfortable but not enormous, and the discrepancy is what surface-roughness corrections in neutron optics are about.

And the identity S = W − ET is evaluated on the true path only. It holds nowhere else, and the figures do not claim that it does. Off the true path, S and W are integrals over different comparison classes and there is no relation between them to check.

What the pictures cannot show #

The right-hand panel of the first figure plots four quantities against a deformation and has to scale each by its own range to fit them on one axis. What that hides is how different their sizes are: the change in duration across the family is a few per cent, the change in energy a fraction of a per cent, and the changes in the two actions smaller again. The shape near zero is the claim; the heights are not comparable and are not drawn to be.

The refraction figures draw straight trajectories on each side of a step, which is right for a uniform potential and wrong for every real one. In a potential that varies smoothly the path curves continuously and there is no boundary to apply a law at — the trajectory is a geodesic of a metric that changes from point to point, and the refraction picture is what that becomes when the change is concentrated. The smooth case is the more general one and is harder to draw, which is the usual relationship between the two.

Where the ladder goes next #

The least-action ladder began with the action being stationary rather than least, went through the conservation law a symmetry hands over and the force a coordinate cannot see. This rung is the other principle. The rungs after it: the Hamilton–Jacobi equation, where the action is treated as a function of its endpoint and mechanics becomes a first-order partial differential equation; canonical transformations, where the freedom to redescribe a problem is the whole method; and the sum over paths, where the stationary path stops being selected and starts being the place where the phases of all the others fail to cancel.

The habit worth carrying away is to ask, of any variational statement, what is held fixed and what is left free. Two principles over the same system with different constraints are different statements even when they select the same answer, and the one worth using is the one whose fixed quantity is the one actually known.