Least action, except that it is not least

A trajectory can be specified in two quite different ways. One is a rule about each instant: at every moment, the acceleration is the force divided by the mass, and the path is what follows from integrating that rule forward. The other is a rule about the path as a whole, applied to all of it at once.

The quantity being computed #

The action of a path is

$$S = \int_{t_1}^{t_2} \left(T - V\right)\,dt$$

the time integral of kinetic energy minus potential energy — the two quantities that trade against one another as a body moves through a field. That combination is called the Lagrangian, and it is worth saying at once that it has no independent physical meaning: it is not an energy, not a rate of anything, and nothing measures it directly. The total energy $T+V$ is the meaningful combination; $T-V$ is the one that makes this construction work, which is a different kind of justification.

The quantity being computed is worth pausing on, because it is not the obvious one. Kinetic and potential energy trade places continuously in any oscillation; their sum is what is conserved, and their difference is what gets integrated to give the action. Nothing in the picture of the trade suggests the difference should be the useful combination, and no amount of staring at it makes the choice look natural. It is justified afterwards, by what it produces.

To evaluate the action, a path is needed. Any path will do, provided it starts and ends at the specified events; it does not have to obey any equation, and most of the ones on the figure above do not. What is being computed is a number attached to a curve.

The true path is not found by minimising anything here. It is found by solving the equation of motion, and then handed to the same integrator as the false ones — which is what makes the comparison a test rather than a definition.

Stationary, not least #

The parabola in the first figure has its minimum at the true path, and a first reading of that says the true path is the shortest, or the cheapest, or the most efficient. It says none of those things.

What is true, and what the whole construction rests on, is that the slope is zero at the true path in every direction. The action is stationary: a small change to the path, of any shape whatever, changes the action only at second order.

That is what turns one integral condition into a differential equation. Requiring the first variation to vanish for arbitrary deformations — including ones concentrated in a small part of the flight, like the bump in the figure — forces the integrand’s variation to vanish at every instant separately. Carrying that through gives

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q}$$

the Euler–Lagrange equation, which for a projectile is $\ddot{y} = -g$. The rule about the whole path and the rule about every instant are the same rule.

Near the true path the action is quadratic in the deformation, which is what any smooth function is near a stationary point — and the parabola in the first figure is that statement drawn.

The curvatures in that figure are worth noticing too. A corner costs far more action than a smooth arch of the same height, because the action penalises speed rather than displacement, and a corner is a sudden speed. That is why paths with kinks are never near-winners and why the winning path is smooth.

Where the minimum stops being a minimum #

The name “least action” is older than the theorem and it is wrong. There is a duration past which the true path is a saddle point, with nearby paths of lower action, and it is not an exotic case.

For a harmonic oscillator the second variation under a deformation $\lambda\sin(n\pi t/T)$ is exactly

$$\delta^2 S = \frac{\lambda^2 T}{4}\left[\left(\frac{n\pi}{T}\right)^2 - \omega^2\right]$$

which turns negative for $\omega T > n\pi$. Any trip lasting more than half an oscillation therefore has a deformation that lowers its action.

The geometric statement behind that is the kinetic focus. Launch a family of paths from one point with slightly different initial conditions and they generally spread apart; for an oscillator they come back together after half a period, at a point conjugate to the start. Once the endpoint is past that conjugate point, the path is no longer a local minimum.

The moment at which a family of neighbouring paths reconverges has a name — the conjugate point — and it is where the minimum stops being one. Trajectories launched at slightly different energies come back to the same place at different times, and once a trip is long enough to reach past the first such reconvergence, there are nearby paths with smaller action. The stationary point is still stationary; it has simply stopped being a bottom.

The same thing happens in optics and has been visible there for longer: a ray between two points can take the longest time rather than the shortest, and the boundary between the two cases is a caustic, which is the optical name for the same conjugate point.

Optics got to the correct statement a century before mechanics did. Fermat’s principle at three different mirror curvatures makes the same reflection point a minimum of the path length for one, a maximum for another, and neither for the third — and opticians said stationary while mechanicians were still saying least. The physics is identical and only the vocabulary lagged.

What the deformations cost, and why smooth wins #

The four curves in that figure have very different curvatures, and reading them is worth a paragraph because it explains the shape of every real trajectory.

The action’s second variation splits into two pieces: the extra kinetic energy the deformation requires, which is always positive and grows as the square of the deformation’s slope, and the change in potential energy, which can be of either sign and depends on the deformation’s height. A corner has a large slope for a small height, so it is expensive; a broad arch has a small slope for the same height, so it is cheap. The cheapest deformation of a given height is the smoothest one, and the winner is the path for which no deformation of any smoothness is cheap enough to be free.

That is why the true path is smooth even though nothing in the principle asked it to be, and why a numerical search over paths — which is one practical way of solving mechanics problems — converges to a smooth curve without any smoothing being imposed. It is also why the harmonic oscillator’s crossover comes at half a period: the potential term wins over the kinetic one exactly when the deformation is slow enough to have less kinetic cost than potential gain, and for a spring that comparison is between $n\pi/T$ and $\omega$.

Why bother, if the equation is equivalent #

For a single particle in a uniform field, the action principle is a longer route to a shorter answer. Its value shows up in four places where the equation of motion is awkward.

Coordinates stop mattering. The action is one number attached to a path, and a number does not care what coordinates the path is described in. Changing to polar, or to a rotating frame, or to the angle of a pendulum bob, means rewriting one scalar rather than resolving vector equations along moving directions — and constraints are imposed by choosing coordinates that respect them rather than by introducing forces that enforce them.

Symmetries become conservation laws mechanically. If the Lagrangian does not change when time is shifted, energy is conserved; if it does not change when space is shifted, momentum is; if it does not change under rotation, angular momentum is. That is Noether’s theorem, and it is a statement about the action that has no equally direct counterpart in the force law. What stays the same becomes a question about the symmetries of a single function.

Fields and relativity inherit it unchanged. Every field theory in physics is specified by writing an action — electromagnetism, general relativity, the Standard Model — because relativistic invariance is easy to impose on a scalar and hard to impose on a set of component equations. The action is how a theory is stated in modern physics, and the equations of motion are derived output.

And it is the classical limit of something. This is the deepest of the four.

The first of those is worth one concrete illustration, because “coordinates stop mattering” sounds like a convenience and is closer to a change of method. A bead sliding on a wire feels a normal force from the wire at every point, of a size that depends on the bead’s speed and the wire’s curvature and that has to be solved for alongside the motion. In the Lagrangian treatment the wire is imposed by using distance along it as the coordinate, the constraint force never appears, and the equation of motion falls out in one line. The force is still there and can be recovered afterwards; what has changed is that it is an output rather than an unknown. The slope, and the two directions that make it easy is the same trick done by hand for one problem.

The sum over paths #

Quantum mechanics assigns every path an amplitude, not just the classical one. The amplitude is a unit arrow whose direction is the action of that path divided by $\hbar$, and the total amplitude is their sum.

Away from the stationary path, neighbouring paths have actions differing by many multiples of $\hbar$, so their arrows point in every direction and cancel in pairs. Near the stationary path the action barely changes — that is what stationary means — so a whole band of paths contributes arrows that add rather than cancel.

So the classical path is not chosen. It is the only one that survives the cancellation. As $\hbar$ is made smaller the surviving band narrows, and in the limit the particle appears to follow one curve. The principle of stationary action is therefore not an axiom of mechanics; it is what quantum mechanics looks like from far away.

The cancellation between paths is visible directly when only two of them are allowed. Arrivals accumulate one at a time into an interference pattern, and that pattern is the same cancellation the sum over paths performs — made visible because the alternatives have been cut down to a number a reader can hold. The classical limit is what happens when the paths available are so numerous that all but a narrow band cancel each other out.

The bridge between the two descriptions is that action divided by $\hbar$ is a phase. The number of radians a path accumulates is its action in units of $\hbar$ — so for anything with a macroscopic mass, whose associated wavelength is short beyond any ordinary comparison, two neighbouring paths differ by an enormous number of radians and cancel. Only near a stationary point do neighbouring paths agree in phase, and that is why the stationary path is the one that survives.

Feynman’s formulation makes that precise, and it inverts the usual pedagogy: the sum over paths is the physics, and the stationary path is an approximation to it that happens to be exact enough for anything larger than a molecule.

Two events, not a state #

There is a structural difference between the two formulations that is easy to miss and changes what kind of question is being asked.

The equation of motion is an initial-value problem: give a position and a velocity now, and the future follows. The action principle is a boundary-value problem: give a position now and a position later, and it selects the path between them. Those are not the same question, and they do not always have the same number of answers — a projectile can reach a given target at a given time by one path, and reach it at a later time by two, one lobbed and one flat.

That difference is why the action formulation is natural in quantum mechanics, where the amplitude to go from one event to another is exactly the object of interest, and awkward in a simulation, where the future is what is wanted and the endpoint is not known. Each formulation is convenient for the question shaped like it.

What the action is not #

It is not energy, and not a cost. Its units are energy times time — the same as angular momentum, and the same as $\hbar$, which is not a coincidence. Nothing is spent and nothing is conserved.

It is not unique. Adding a total time derivative to the Lagrangian changes the action by a constant depending only on the endpoints, so it changes no equation of motion. Two quite different-looking Lagrangians can describe the same system, and there are systems with no Lagrangian at all — anything with friction, for a start, unless the dissipation is modelled explicitly.

It is worth saying what the action principle is not. It does not replace a potential-energy picture: the potential is an input to the Lagrangian, and what the principle supplies is the rule that turns it into a path. Nothing is explained about why the potential is what it is, and nothing is removed from the ordinary business of writing down forces — the principle is a different way of getting the same trajectory, chosen for what it makes easy rather than for what it explains.

And “nature is economical” is not what it says. Maupertuis proposed the principle in 1744 with exactly that theological reading, and was wrong about the quantity, about the minimisation, and about the argument. Euler and Lagrange supplied the correct version within a decade. The teleological flavour has survived in popular accounts for two hundred and fifty years, and the figure of the saddle above is the shortest refutation of it.

Orbits make the conjugate-point business concrete. Each orbit is a stationary path of its own action, and whether it is a minimum depends on whether neighbouring orbits have reconverged before the trip ends — which for a planetary orbit lasting more than half a revolution, they have. So the most familiar trajectory in physics is a stationary path that is not a least one, which is the essay’s title stated as a fact about the solar system.

Two principles, one name #

Maupertuis was dismissed in a sentence above, which is fair on the physics and hides a story and a genuine technical distinction.

His 1744 proposal was that in any change in nature the quantity of action — which he took to be the integral of mass times speed along the path — is a minimum, and he offered it as a demonstration of the economy of a wise Creator. Euler published a careful variational treatment the same year, independently and correctly, and generously gave Maupertuis the credit for the idea.

What followed was one of the more spectacular academic quarrels of the century. In 1751 Samuel König claimed Leibniz had stated the principle decades earlier and produced a fragment of a letter to prove it. Maupertuis, then president of the Berlin Academy, had the Academy formally declare the letter a forgery — a use of institutional authority to settle a priority dispute that Voltaire found irresistible. His satire of Maupertuis circulated widely enough that Frederick the Great had copies burned by the public executioner, and Maupertuis’s reputation never recovered.

The technical point is more useful than the story. Maupertuis’s principle and Hamilton’s are not the same principle, and running them together is a common confusion.

Maupertuis’s quantity is the abbreviated action, the integral of momentum along the path, and it is made stationary at fixed energy, with the time taken left free to vary. Hamilton’s, the one this essay computes, is the integral of $T-V$ over time, made stationary at fixed endpoints in time, with the energy left free. The two are related by a Legendre transform and they answer different questions: one asks which geometrical curve a particle of a given energy follows, the other which motion joins two events at two given times.

The abbreviated version survives in exactly the places where energy is the natural constant — orbital mechanics, ray optics, and the semiclassical quantisation of the next section — and Hamilton’s is the one that generalises to fields.

There is a small irony in this essay’s existence. The definitive statement of the subject is Lagrange’s Mécanique analytique of 1788, whose preface boasts that the reader will find no diagrams in it at all, the whole of mechanics having been reduced to algebra. Every figure on this page is a drawing of a principle whose author regarded the absence of drawings as its chief merit.

The quantum of action #

The remark that the action has the same units as $\hbar$ was left as an aside, and it is the most consequential fact on the page.

The sum over paths gives each path a phase equal to its action divided by $\hbar$. So $\hbar$ is a conversion factor between an action and an angle, and an action of one $\hbar$ is a phase of one radian. That is why Planck named it the Wirkungsquantum — the quantum of action — before anybody knew about paths at all.

The relation was exploited before it was understood. Bohr and Sommerfeld’s quantisation rule, which held the field from 1913 to 1925, is

$$\oint p\,\mathrm{d}q = nh,$$

which is Maupertuis’s abbreviated action, taken round a closed orbit and required to be a whole number of Planck constants. It gives hydrogen’s spectrum exactly, gives the fine structure when relativity is included, and fails completely on helium — which is what a rule with the right kinematics and the wrong dynamics does.

The semiclassical treatment that replaced it makes the connection explicit. Write the wavefunction as $e^{iS/\hbar}$ with $S$ the classical action and expand in $\hbar$: the leading term reproduces the classical equation of motion, and the requirement that the wavefunction be single-valued round a closed orbit reproduces the quantisation rule — with a correction. The condition comes out as $\oint p\,\mathrm{d}q = (n + \tfrac12)h$, and the extra half is a quarter-cycle of phase picked up at each of the two turning points, where the classical description fails and the quantum one has to be matched through.

There is one further place the same constant appears, and it has no trajectories in it at all. Counting quantum states in statistical mechanics requires dividing phase space into cells, and the cell has a volume of $h$ per degree of freedom — which is what makes an entropy a pure number rather than a quantity with arbitrary units, and what fixes the constant in the Sackur–Tetrode expression for the entropy of an ideal gas.

Three uses, one constant: the phase per unit action, the quantum of a closed orbit’s action, and the volume of a phase-space cell. They are the same statement seen three ways, and each of them is a statement that action is the quantity nature counts.

What the pictures cannot show #

Every family drawn here is one-parameter or four-parameter. The space of paths between two events is infinite-dimensional, and a figure can show a line through it. The claim that the slope vanishes in every direction is established by the Euler–Lagrange equation rather than by any finite set of pictures; the four directions drawn are a check on that, not a proof of it.

The action integrals are quadratures. Each is a midpoint sum over a thousand steps, and the numbers quoted are converged to the digits shown; the curvature of the projectile’s action under one arch of a sine is asserted against its exact value of π²/2 before anything is drawn.

ħ is not ħ. The phase figure uses a Planck constant of 0.012 in the action’s own units, chosen so that the spiral is visible on a page. The real value in those units would be about 10⁻³⁴, and the surviving band would be narrower than a line.

And the kinetic focus is drawn for a harmonic oscillator. Its conjugate point is at exactly half a period because every path has the same period, which is special. For a general system the conjugate point has to be found by integrating the deviation equation, and there may be several.

The ladder from here #

Later rungs on this anchor: the Euler–Lagrange derivation carried out, with the boundary terms that the fixed endpoints kill; Noether’s theorem, which turns each symmetry of the Lagrangian into a conserved quantity; the Hamiltonian formulation and the canonical transformations it makes possible; the action as a function of its endpoints, which satisfies the Hamilton–Jacobi equation and is the object whose gradient is the momentum; and constrained systems, where the choice of coordinates does the work that forces would otherwise have to.

The neighbouring ladders are what a system actually minimises, which is the thermodynamic version of the same question and has a different answer, Fermat’s principle, which is the optical case and reached the correct wording first, and one arrival at a time, where the cancellation between paths is directly visible.