The friction an action can hold only by changing its clock

Least action, except that it is not least found the principle that underlies all of mechanics: of every path a system might take between two configurations at two times, the one it actually takes makes the integral of kinetic minus potential energy stationary. From that single statement come Newton’s laws, the conservation of energy and momentum, the equations of electromagnetism and general relativity, and the starting point of quantum mechanics. The conservation law a symmetry hands over found the deepest part of it: every symmetry of the Lagrangian brings a conserved quantity, and since the laws do not change with time, energy is conserved.

That last sentence is the problem with friction. A swinging pendulum in air, a car coasting to rest, a plucked string falling silent — none of them conserves energy, and none of them can come from a Lagrangian that does not depend on time, because such a Lagrangian always conserves an energy. Friction is the most familiar force there is, and the principle of least action, in its standard form, cannot express it. The damped oscillator is the test case because it is the simplest dissipative system with a clean answer: its swing decays as $e^{-\gamma t/2}$, and the width that is a lifetime found that the same $\gamma$ sets the width of its resonance when it is driven, so a single number captures everything friction does to it. What happens when it is made to is a lesson in what the principle is really about.

The path friction takes is not stationary #

The first figure tests it. Take a damped oscillator — a mass on a spring with friction proportional to its speed, its swing decaying exponentially — and compute the ordinary action, kinetic minus potential energy integrated over time, along its true path and along nearby paths with the same start and end that bulge slightly to one side or the other. If the true path were the one the principle picks, the action would be stationary there, flat to first order in the size of the bulge. It is not. The curve crosses the true path with a definite slope, and the path the principle would pick lies off to one side: a path of an undamped oscillator, roughly, which is what the ordinary Lagrangian describes. There is nothing in kinetic and potential energy to know about friction.

Now multiply the Lagrangian by $e^{\gamma t}$, where $\gamma$ is the friction rate. The Euler–Lagrange equation of the new Lagrangian is the damped oscillator’s: the derivative of $e^{\gamma t}$ in the momentum term produces exactly the friction force. Piero Caldirola in 1941 and Eiji Kanai in 1948 proposed it, and the figure confirms that the true damped path is stationary for it, to the precision of the arithmetic. The weight makes later times count more in the action, and that tilt is enough to make the stationary path one that decays. So friction can be put into a principle of least action after all — at a price the rest of this essay counts.

What the weight does to momentum #

The trouble lies in what a Lagrangian means by momentum. The canonical momentum of a Lagrangian is its derivative with respect to velocity, and for the Caldirola–Kanai Lagrangian that is $e^{\gamma t}m\dot x$ — the ordinary momentum multiplied by a factor that grows exponentially with time. As the real momentum decays, the canonical momentum stays the same size. The friction has not been put into the dynamics so much as hidden in the definition of the variables.

The second figure shows the consequence in phase space. Take a small cloud of oscillators, a range of starting positions and velocities, and follow it. In the plane of position and ordinary momentum, the cloud spirals inward towards rest and shrinks as it goes: friction draws every motion towards the same state of rest, so motions that started apart become harder and harder to tell apart, and the area the cloud covers falls exponentially. In the plane of position and canonical momentum, the same cloud is distorted but keeps its area exactly. That is Liouville’s theorem, which holds for every flow generated by a Hamiltonian: the Caldirola–Kanai Lagrangian generates one, so its phase-space area must be conserved, and it is — in its own variables.

The third figure puts the two areas side by side. Both are correct descriptions of the same motion; they differ in what counts as a unit of phase space. The physical area is the one that counts distinguishable motions — the one that enters statistical mechanics, where entropy is a count of states and a shrinking phase-space volume is the signature of a system losing information to its surroundings. The second law, with a probability attached found that the entropy produced by a dissipative process is measured by exactly that kind of contraction. The Caldirola–Kanai flow keeps its canonical area fixed by letting the canonical momentum blow up while the real one dies, and so it describes a damped oscillator with no entropy production at all, which is a contradiction in terms.

A mass that grows, or a clock that runs #

The weighted Lagrangian has an exact reading that makes its strangeness concrete. $e^{\gamma t}(\tfrac12 m\dot x^2 - \tfrac12 kx^2)$ is the Lagrangian of an undamped oscillator whose mass and spring constant both grow exponentially with time. Such an oscillator’s momentum, $e^{\gamma t}m\dot x$, is conserved in the absence of the spring, and its velocity must therefore fall as the mass grows: an object gaining mass while its momentum stays fixed slows down, which looks exactly like friction from outside. The model has swapped a constant mass losing energy for a growing mass keeping it.

There is a second reading in which nothing grows. Write the position as $x = e^{-\gamma t/2}y$, scaling the coordinate by the decay of the motion itself. The weighted Lagrangian becomes, up to a term that does not affect the motion, the Lagrangian of an ordinary undamped oscillator in $y$, with its frequency lowered slightly to $\sqrt{\omega^2 - \gamma^2/4}$. The damped oscillator is an undamped oscillator watched through a lens whose magnification shrinks exponentially. The force a coordinate cannot see found how much a change of coordinates can hide or reveal; here a time-dependent change of coordinates turns friction into no friction at all, which is a mathematical truth about the equation and a warning about reading physical meaning into the variables of a Lagrangian chosen to produce it.

What is conserved instead of the energy #

Noether’s theorem still applies to the weighted Lagrangian, and it is instructive to ask what it conserves. The Lagrangian is not unchanged by a shift of time, because of the explicit $e^{\gamma t}$, so energy is not conserved — as it should not be. But it is unchanged by a shift of time combined with a rescaling of the coordinate by $e^{-\gamma\,\delta t/2}$, and that combined symmetry has a conserved quantity of its own:

$$I = e^{\gamma t}\left(\tfrac12 m\dot x^2 + \tfrac12 kx^2 + \tfrac{\gamma}{2}m x\dot x\right).$$

It is the oscillator’s energy, plus a small correction, multiplied by the exponential. As the energy decays, the exponential grows to compensate, and $I$ stays fixed. In the $y$ variable of the previous section, $I$ is simply the undamped oscillator’s energy. The conservation law a symmetry hands over found that every symmetry hands over something conserved; here the something is a bookkeeping quantity that grows the record of the decaying energy back to its original size. It is conserved exactly along the damped motion the figures draw, and it carries no information a measurement of the oscillator alone could use. A conservation law obtained by hiding the loss is not a law about where the energy went.

Quantised, the oscillator knows too much #

The damage becomes unmistakable when the system is quantised. Quantisation imposes the commutation rule $[x, p] = i\hbar$ on a coordinate and its canonical momentum. For the Caldirola–Kanai Lagrangian the canonical momentum is $e^{\gamma t}m\dot x$, so the rule for the physical momentum becomes $[x, m\dot x] = i\hbar e^{-\gamma t}$: the uncertainty principle for position and physical momentum weakens exponentially with time. The fourth figure follows a quantum oscillator that starts in a thermal state with an uncertainty product five times the minimum. Under the Caldirola–Kanai evolution the product falls exponentially, as the classical phase-space area did, and after about eight time units it passes below $\hbar/2$. From then on the model describes an oscillator whose position and velocity are both known more sharply than any real quantum system permits — and in the limit it is a particle at rest at a known point, which quantum mechanics forbids. The motion that cannot be stopped found that every oscillator retains a zero-point motion; this one loses it.

A real damped oscillator does nothing of the kind. Its energy goes somewhere: into the molecules of the air, the vibrations of its support, the electromagnetic field it radiates into. Model that environment as a large reservoir of other oscillators, each weakly coupled to the system, and let the energy flow into it. At zero temperature, the oscillator’s motion decays exactly as friction requires, but its uncertainty product decays only to $\hbar/2$ and stops: the oscillator settles into its quantum ground state, as the dotted curve shows. The reservoir does two things at once. It takes energy away, which is the friction, and it kicks back with random fluctuations, which keep the uncertainty principle intact. That pairing is the fluctuation–dissipation theorem, which half a kT in a piece of wire found in the thermal noise of every resistor: any channel through which a system loses energy is also a channel through which noise comes in, in a fixed proportion. A Lagrangian for the oscillator alone has the dissipation and not the fluctuations, and a quantum theory with dissipation and no fluctuations is inconsistent.

Bateman’s mirror #

The other classic way round the difficulty is to give the damped oscillator somewhere to put its energy. Harry Bateman showed in 1931 that the damped oscillator can be paired with a mirror image — an oscillator with the same frequency and the friction reversed, so that its motion grows as fast as the first one’s decays — and that the pair has a perfectly ordinary Lagrangian, with no explicit time dependence and a conserved Hamiltonian. The last figure integrates it: the first oscillator rings down, its mirror rings up, and the Hamiltonian of the pair stays constant to the precision of the arithmetic.

The physically faithful version replaces the mirror with a crowd. Amir Caldeira and Anthony Leggett showed in 1983 that a particle coupled linearly to a large set of oscillators, with their frequencies and couplings chosen so that the set absorbs energy equally well at every frequency, obeys exactly the classical damped equation of motion on average — and, quantised, carries exactly the fluctuations the fluctuation–dissipation theorem requires. Their model, built to ask whether a superconducting circuit damped by its resistance could still tunnel quantum mechanically, is the standard account of quantum friction, and it is Bateman’s idea taken to its honest limit: the environment is put back, not as one mirror, but as the infinitely many degrees of freedom friction actually feeds.

The mirror is the environment reduced to one degree of freedom. It is not a physical object anyone builds, and a single growing oscillator is a poor model of an environment, which is many degrees of freedom that never give the energy back. But Bateman’s construction makes the logical point cleanly: the principle of least action can describe a damped system perfectly well, provided the system’s surroundings are part of what is varied. Modern versions of the idea, which double every variable and treat the forward and backward evolution together, are used to write variational principles for dissipative systems from first principles, including the radiation reaction of orbiting black holes, where the “friction” is the energy carried off by gravitational waves.

Rayleigh’s way, and why it is not a principle #

Engineers met the problem long ago and solved it practically. Lord Rayleigh introduced a dissipation function, half the friction coefficient times the velocity squared, and added its derivative to the Euler–Lagrange equations as an extra generalised force. That gives the right equations of motion for any system with friction proportional to speed, in any coordinates, and it is how damped mechanical and electrical systems are routinely analysed. But it is not derived from a stationary action; it is an addition to the equations that the action produces, a statement that some forces lie outside the principle. Friction that does not depend simply on speed, such as the dry friction that the swing that dies in a straight line found bringing a swing to rest in a straight-line decay rather than an exponential one, does not fit even Rayleigh’s form.

That is the honest summary. Least action describes closed systems, whose energy goes nowhere. Friction describes open ones, whose energy goes somewhere the description has left out. Either the somewhere is put back in, as a reservoir or a mirror, or friction is added to the equations by hand, or the Lagrangian is tilted in time and pays for it with variables that no longer mean what they appear to.

Where the weighted Lagrangian is still useful #

None of this makes the Caldirola–Kanai Lagrangian worthless. Classically, its equation of motion is correct, and as long as only the motion is wanted, not the phase-space structure, it gives the right answer. It has a real use in another setting entirely: in an expanding universe, a field’s equation of motion picks up a friction term proportional to the expansion rate, the Hubble friction, and the field’s Lagrangian really does carry a weight that grows with time, the volume factor of the expanding space. There the “canonical momentum” that grows is a momentum per comoving volume, and the uncertainty relations are applied to the correctly normalised variables. The exponential weight is legitimate when it describes something real, like an expanding volume, and misleading when it is used to fake the loss of energy to an environment that is not in the model.

The bath the oscillator’s energy goes into #

The figures compute the motion of a single damped oscillator and its quantum moments. They do not show the environment the energy goes into, except in the dotted curve of the fourth figure, which uses the standard equations for an oscillator coupled to a large reservoir rather than simulating the reservoir’s own oscillators. The phase-space clouds are classical ensembles; the quantum figure tracks only the variances and covariance, which for these linear systems is all that changes. And the first figure varies paths along a single family of bulges; the stationarity it shows holds along every family, which the calculus of variations guarantees and one family can only illustrate.

Still open: a variational principle for the second law #

Whether every dissipative process can be derived from a principle of stationary action, in the way every conservative one can, is not settled. There are partial answers — Onsager’s principle of least dissipation for slow, near-equilibrium processes, the doubled-variable actions for open systems, and the path-integral formulations of stochastic dynamics, which assign probabilities to whole trajectories and recover irreversibility as the overwhelming likelihood of entropy-producing paths over their reverses. The ring that forgets and then remembers found how delicate the distinction between reversible and irreversible can be in practice. Whether these variational descriptions of irreversibility are aspects of one principle, and what that principle’s version of Noether’s theorem would conserve, is an open question at the boundary of mechanics and thermodynamics.

The habit worth carrying away is to ask where a lost quantity went. Friction cannot come from a Lagrangian that does not depend on time, and the one that does — $e^{\gamma t}(T - V)$ — produces the damped motion only by making the canonical momentum grow as $e^{\gamma t}$ while the real one decays, so the phase-space area it conserves is not the physical one, which shrinks as $e^{-\gamma t}$, and its quantum version drives the uncertainty product below $\hbar/2$. A damped oscillator is half of a closed system; the action that describes it honestly includes the other half, and the other half kicks back.