The third law that falls out of a change of scale

Least action, except that it is not least restated mechanics as a rule about whole paths: of all the ways a system could get from one configuration to another in a given time, it takes one that makes the action — the time integral of kinetic minus potential energy — stationary. The conservation law a symmetry hands over drew the first great dividend from that restatement. A change that leaves the action unchanged, such as a shift in time or a rotation in space, hands over a conserved quantity, and energy, momentum and angular momentum are three instances of one theorem.

There is a second kind of change worth trying, one that does not leave the action unchanged but multiplies it by a constant. Multiplying the action by a constant does not change which path makes it stationary — the stationary point of a function and of twice that function are the same point — so such a change also maps real motions to real motions. It does not hand over a conserved quantity. What it hands over is a law relating motions of different sizes, and for a force that is a single power of distance the law is exact. Lev Landau and Evgeny Lifshitz put it near the front of their Mechanics under the name mechanical similarity, and it is one of the most economical arguments in physics.

Stretching space and time together #

Take a potential energy that is a single power of distance, $V(\alpha\mathbf r) = \alpha^k V(\mathbf r)$: gravity and electrostatics have $k = -1$, a spring $k = 2$, a uniform field $k = 1$. Now stretch every length in a motion by a factor $\alpha$ and every time by a factor $\beta$. The potential energy is multiplied by $\alpha^k$. The kinetic energy, which goes as velocity squared, is multiplied by $(\alpha/\beta)^2$. If the two factors are chosen equal — $\beta = \alpha^{1 - k/2}$ — the whole Lagrangian is multiplied by $\alpha^k$, and the stretched path is a real motion.

So for every motion under such a force there is a family of similar motions, enlarged copies traversed at a related pace. Times along them scale as lengths to the power $1 - k/2$. That single exponent is the content of the argument, and it gives, without solving any equation of motion:

• for gravity, $k = -1$: periods go as size to the $3/2$ — Kepler’s third law;
• for a spring, $k = 2$: periods go as size to the zero — the period does not depend on the amplitude;
• for a uniform field, $k = 1$: times go as size to the $1/2$ — the time to fall from rest goes as the square root of the height, as Galileo found with his inclined planes.

The figure checks the first of these by doing what the argument avoids: integrating three orbits from the force law, step by step, and timing them. The orbits are exact enlargements of each other, and dots placed at twelfths of each period sit at the same places on each ellipse. The periods come out in the ratio $1 : 2.828 : 5.196$, which is $1 : 2^{3/2} : 3^{3/2}$ to the four figures printed.

Kepler found the law in 1618 from Tycho Brahe’s observations, after years of trying other relations between period and distance. Newton derived it in 1687 by solving for the orbits. The scaling argument gets it in the time it takes to count powers, and it gets more than Kepler did: the law holds not only for circles and not only for the ratio of a planet’s year to its mean distance, but for any pair of similar motions, eccentric or not, and for any time along them — the time to fall from aphelion to perihelion, the time to cross a given angle.

The uniform-field case repays a moment, because it is the one everybody already uses without noticing. With $k = 1$, lengths scale by $\alpha$, times by $\alpha^{1/2}$ and velocities by $\alpha^{1/2}$. So two throws at the same angle, one twice as fast as the other, are similar motions with every length four times larger: the faster throw goes four times as far, rises four times as high and stays up twice as long. Range proportional to the square of the launch speed, which the angle that throws furthest found by solving for the parabola, is scaling, and so is the fact that the best angle is the same at every speed — similar motions have the same angles.

Similarity is not dimensional analysis #

It is tempting to think all of this is dimensional analysis, and for the simplest cases the two give the same answer. A planet’s period can depend only on its orbit’s size $a$, the Sun’s mass $M$ and the gravitational constant $G$, and the only combination of those with the dimensions of time is $\sqrt{a^3/GM}$. Kepler’s law again, in one line.

The difference shows in what each needs and what each yields. Dimensional analysis needs to be told which quantities matter, and it cannot tell a period from the time to cross a particular angle, or one eccentricity from another, since angles and eccentricities have no dimensions. It has nothing to say about averages and gives no virial theorem. Mechanical similarity starts from the action and so knows which motions are possible; it says that every feature of a motion — every time, every angle, every eccentricity — maps to its counterpart in a similar motion, and it applies to systems of any number of bodies, interacting in any arrangement, as long as every force is the same power of distance. Dimensional analysis is the shadow similarity casts on the units.

A table of exponents #

The figure extends the check to six potentials, measuring each period at two sizes by integrating the motion and reading off the exponent. Every measurement lands on the line $1 - k/2$ to three decimal places.

The line crosses zero at $k = 2$, and that crossing is why a spring keeps time. A harmonic oscillator’s period is the same at every amplitude because the scaling exponent for a quadratic potential is zero, and for no other reason; it would be just as true of a spring that was large or small, stiff or soft. That is why the harmonic approximation is so useful and why it eventually fails. Every smooth minimum looks quadratic close enough to its bottom, so every small oscillation keeps time, and the clocks of the world are built on that. But a pendulum’s restoring force is a sine of its angle, not a single power, so the scaling argument does not apply to it at all, and the pendulum’s isochronism is a small lie that holds only while the swing is small enough to be a spring.

Beyond $k = 2$ the exponent goes negative. A particle in a quartic well, with potential energy proportional to the fourth power of distance, swings back and forth faster the further it is pulled: period inversely proportional to amplitude. Steeper walls mean a particle released far out hits them harder, and the fall more than makes up for the distance.

Where scaling has nothing to say #

The argument needs the potential to be a single power. A well with two terms, such as $x^2/2 + x^4/4$, has no stretching of space and time that multiplies its Lagrangian by a constant, because the two terms scale differently, and it has no exact scaling law. What it has instead is a crossover. At small amplitudes the quadratic term dominates and the period is nearly constant; at large amplitudes the quartic term dominates and the period falls as one over the amplitude. In between, near the amplitude where the two terms are equal, the local slope passes from one exponent to the other, and at that amplitude the motion has a scale of its own.

That is the general situation, and it is worth stating because it says when to expect a law and when not to. A system has a scaling law when nothing in it sets a size — when its potential is a single power and so has no characteristic length. Put in a second power and the ratio of their coefficients defines a length, below which one law holds and above which another does. The same reasoning tells a biologist that an animal’s jump does not get higher as it gets bigger and tells a physicist where a pendulum stops keeping time.

Half the potential, and a star that heats as it cools #

The same stretching gives a second result, about averages rather than periods. Apply it to an infinitesimal stretch, and follow what it does to the quantity $\mathbf p \cdot \mathbf r$. Its rate of change is twice the kinetic energy minus $\mathbf r \cdot \nabla V$, and for a single-power potential $\mathbf r \cdot \nabla V = kV$. Averaged over a long time for a bounded motion, $\mathbf p \cdot \mathbf r$ cannot grow without limit, so its average rate of change is zero:

$$2\langle K\rangle = k\,\langle V\rangle.$$

That is the virial theorem, Rudolf Clausius’s of 1870, and it is mechanical similarity applied to averages.

For a spring the two averages are equal: a swinging mass spends as much of its energy, on average, moving as stretched. For gravity the kinetic energy is minus half the potential, and the consequence is stranger than it looks. The total energy is then $\langle K\rangle + \langle V\rangle = -\langle K\rangle$: a bound orbit’s energy is minus its kinetic energy. Take energy away and the kinetic energy goes up. A satellite dragged by the thin upper atmosphere loses energy and speeds up as it sinks. A ball of gas held together by its own gravity, radiating heat into space, gets hotter as it loses energy — which is how a contracting cloud reaches the temperature at which hydrogen burns, and why a star’s heat capacity is negative.

The same theorem with $k = -1$ for the electrostatic attraction in an atom says the electrons’ average kinetic energy is minus half their potential energy, which is the balance that sets the size of an atom: squeeze the atom and the kinetic energy rises by more than the potential energy falls.

Levels, from the same stretch #

Mechanical similarity has a quantum counterpart, and it arrives through the action. The quarter cycle a turning point costs found the old quantum rule in its corrected form: a periodic motion is allowed if its action round one cycle, $\oint p\,dx$, is $n + \tfrac12$ Planck constants. Under the scaling, the action of a motion in a $|x|^k$ well at energy $E$ is proportional to $E^{1/2 + 1/k}$ — a fact about similar motions, needing no solution — so the allowed energies go as $(n + \tfrac12)^{2k/(k+2)}$.

For a spring the exponent is one, so the levels are evenly spaced — the even spectrum of the quantum harmonic oscillator, which is exact. For steeper wells the levels spread apart and approach the $n^2$ of a particle in a box, which is the limit $k \to \infty$. For gentler ones they crowd together: a linear well, $k = 1$, the potential of a particle bouncing on a floor in uniform gravity, has levels rising as $n^{2/3}$. And for gravity’s own $k = -1$ the formula gives $n^{-2}$ — the Rydberg series of hydrogen, $-13.6\ \text{eV}/n^2$, falling into place as one more instance of the same exponent.

Scaling the masses and the charges too #

The stretch need not stop at space and time. Multiply the masses or the strength of the force by a factor as well, and the same bookkeeping says how motions with different ingredients compare. For an inverse-square attraction between a light body of mass $m$ and a heavy one, the Lagrangian’s two terms scale with $m$ and with the coupling, and a hydrogen-like atom with its electron replaced by something heavier, or its nucleus’s charge raised, is a similar system with predictable sizes and energies.

Replace the electron by a muon, 207 times heavier, and the atom is 207 times smaller and 207 times more tightly bound. Give the nucleus a charge $Z$ and a single orbiting electron’s orbit shrinks as $1/Z$ while its binding grows as $Z^2$, which is why the innermost electrons of a heavy atom are bound by tens of thousands of electronvolts and give off X-rays where hydrogen gives off visible light. Go the other way, to a level with a huge quantum number, and the orbit swells as $n^2$, which is how an atom can be made the size of a bacterium while remaining, in every proportion, the same atom.

None of these needs a new calculation. Each is the hydrogen atom stretched, and the stretch is legitimate because Coulomb’s law, like Newton’s, contains no length of its own.

Why the argument is legitimate #

The step that does all the work — multiplying the action by a constant changes nothing — deserves a second look, because it seems too cheap. Hamilton’s principle says the real path is the one for which small changes in the path do not change the action to first order. If $S$ has that property at a path, so does $cS$, for any constant $c$; the condition “the first-order change vanishes” is blind to an overall factor. So any transformation that maps paths to paths while multiplying every path’s action by the same constant maps real paths to real paths. That is all mechanical similarity uses.

Two things are needed for it to apply, and both are often hidden. The transformation must act on the whole problem, so the boundary conditions — where the motion starts and ends, any walls or other bodies — must scale with it. And the Lagrangian must be homogeneous. Planetary orbits satisfy both, because nothing in the solar system sets a length other than the orbits themselves, at least as far as Newtonian gravity is concerned. Relativity adds a length — the Schwarzschild radius $2GM/c^2$ — and with it, close to a black hole, Kepler’s third law stops being exact.

What the pictures cannot show #

The figures check the scaling argument by integrating motions, which is the opposite of the argument’s point: the argument needs no integration. What they show is that the predictions are exact, to the precision of the integrator, for every power tried, and that they fail cleanly when the potential is not a single power. They cannot show the argument’s real strength, which is that it applies to motions nobody can solve — three bodies in a chaotic dance, or a gas of a thousand particles — as long as the force is a single power. Two similar configurations of a chaotic three-body system, one twice the size of the other, evolve identically with time stretched by $2^{3/2}$, whatever the motion is.

The domain is classical mechanics with a potential that is a single power of position, in a system whose boundaries scale with it. Inside it, every motion has a family of similar motions with times scaling as $(\text{size})^{1 - k/2}$, and every bounded motion obeys $2\langle K\rangle = k\langle V\rangle$. With two powers, or a fixed boundary, or relativity, the argument gives a crossover or nothing.

Still open: how far similarity reaches into chaos #

For a single power-law force, similarity maps each motion to a family of similar ones, and it maps chaotic motions too. What it does not say is how the statistics of chaotic motion change with the potential’s power — how fast nearby orbits separate, what fraction of the phase space is chaotic — because those are properties of each family rather than relations between families. For gravitating systems with many bodies, similarity says that a cluster of stars twice as large and with the same mass evolves exactly as the original but slower by $2^{3/2}$; whether a cluster’s long-term fate, such as the collapse of its core, can be read off from scaling arguments of this kind alone, or needs the full statistics of close encounters, which bring in a second length, is a question that has occupied stellar dynamics for decades.

The argument itself is three lines long. If the potential energy is a single power of distance, r^k, then stretching every length by α and every time by α^(1 − k/2) multiplies the action by a constant and leaves its stationary paths stationary, so motions come in similar families: periods go as size to the 3/2 under gravity — Kepler’s third law, 1.000 : 2.828 : 5.196 for sizes 1 : 2 : 3 — and not at all for a spring; averages obey 2⟨K⟩ = k⟨V⟩, which makes a radiating star heat up; and quantum levels rise as (n + ½)^(2k/(k+2)). No orbit needs to be solved for any of it.