The hill that gives it back, and the forces that do not

Carrying a suitcase up four flights of stairs and carrying it up a ramp of the same height take the same work. Not approximately, and not because the numbers happen to come out that way — exactly, and for a reason that has nothing to do with suitcases.

That fact is so familiar that it takes an effort to see what an extraordinary claim it is. The two journeys have different lengths, different durations, different forces applied at different angles for different distances. Every quantity in the problem differs. One does not, and its refusal to differ is what makes energy the most powerful accounting device in mechanics.

Reading a motion off a picture of a hill #

The figure contains a complete description of a motion, and it is worth extracting it before saying where it comes from.

The curve is the potential energy at each position. The horizontal line is the total, which never changes. At any position the gap between the line and the curve is the kinetic energy, and kinetic energy cannot be negative — so the object simply cannot be anywhere the curve rises above the line.

That single restriction settles the qualitative question. The motion is trapped between the two crossings, it is fastest where the gap is widest, and it comes momentarily to rest at each crossing before returning. Those crossings are the turning points, and they are turning points because at each of them the object has spent all its kinetic energy and the slope of the curve is still pushing it back.

Nothing about the shape of the curve was needed for that argument, which is why the same reading applies to every landscape.

The pendulum’s landscape shows what a parabola conceals. Near the bottom the curve is very nearly a parabola, which is the small-angle approximation seen sideways: the reason a pendulum’s period is amplitude-independent is that a parabolic well has that property and a real pendulum’s well is nearly one. Further up, the curve flattens, the restoring slope weakens, and the swing takes longer.

Raise the energy line above the peak and both turning points leave the picture together. The pendulum goes over the top, and the transition between swinging and spinning — which the phase portrait shows as a separatrix — appears here as an energy line that has stopped intersecting anything.

Where the picture comes from #

The landscape exists because of a property that most forces do not have.

Work is force times distance along the path, added up. For a general force that sum depends on the route, so there is no such thing as “the work needed to get from here to there” — there is only the work along a particular journey. In that situation no potential energy function can be written, because a potential energy function assigns one number to each position, and one number cannot describe a quantity that depends on how the position was reached.

Some forces are different. Gravity near the ground does the same work $-mgh$ on anything climbing a height $h$, whatever route it takes, because only the vertical component of each step counts and the vertical components add up to the height however they are arranged. Such forces are called conservative, and for them the work between two points is a difference of two numbers.

That is what the suitcase argument was. The inclined plane trades force against distance and the product is fixed — and now the reason the product is fixed has a name. Every simple machine is a demonstration of path independence, and the reason no arrangement of ramps, levers and pulleys has ever produced free energy is that the potential difference between the top and the bottom does not care what machinery is in between.

The equivalent statements are worth having together, because different problems make different ones obvious. A force is conservative if the work it does around any closed loop is zero; if the work between two points is independent of the route; if it can be written as the slope of a scalar function; or if it has no curl. Those are four descriptions of one property, and a force with any of them has a landscape.

Energy as a bookkeeping that ignores the middle #

Given the landscape, an enormous class of questions is answered by subtraction.

How fast is the pendulum at the bottom? Take the height difference, convert it, done — no differential equation, no trigonometry, no need to know how long the swing took. How high does a projectile go? The vertical part of its kinetic energy becomes height. What speed does a roller coaster reach at the bottom of a drop? The drop, and nothing else about the track.

This is the same move as refusing to look inside a collision, and it has the same character: replace a question about a process with a question about its endpoints, and the process becomes irrelevant. The two methods are complementary rather than alternative. Momentum survives collisions that lose energy; energy handles smooth motion under forces that momentum cannot summarise.

There is a second payoff that is easy to miss. The landscape gives the force as well, since the force is the downhill slope of the curve. A steep part of the potential is a strong force; a flat part is a weak one; a minimum is where the force is zero. So a single curve carries both the dynamics and the statics, and the equilibria can be located by eye — stable ones at the bottoms of wells, unstable ones at the tops of hills, and the difference between them is the difference between a curve that turns up on both sides and one that turns down.

The double well is the most reused landscape in physics. A system in one of its wells stays there indefinitely if the energy is below the barrier, and the height of that barrier is what makes matter stable: a chemical bond is a well, a reaction is a passage over a barrier, and the fraction of molecules with enough energy to cross is the rate. Nothing in the picture is specific to a particle on a hill.

What the method costs #

The landscape is a compression, and it is worth being explicit about what has been thrown away, because the missing information is exactly what is usually wanted second.

There is no time in it. The picture states where the motion goes and how fast it is at each place, and says nothing whatever about when. Recovering the timing means integrating $dt = dx/v(x)$ along the path, which is a genuine calculation and, for the pendulum, the elliptic integral that has no elementary form. Energy answers “how fast, where” for free and charges full price for “how long”.

It is one-dimensional. A curve on a page describes motion along a line. Motion in a plane needs a surface, motion in space needs a function of three variables that cannot be drawn at all, and the intuition that an object rolls downhill on the potential surface is wrong in more than one dimension — a real object with sideways velocity orbits the minimum rather than falling into it, in the way a planet does. The landscape picture quietly assumes the motion has nowhere else to go.

It discards the direction of travel. Every point of the curve is visited twice per cycle, once in each direction, and the drawing cannot distinguish the two. The phase portrait restores this by plotting velocity against position, at the price of no longer being a picture of the force.

And the zero is arbitrary. Only differences in potential energy have meaning; the height at which the curve is called zero is a choice, made for convenience, and every prediction is unchanged if the whole curve is shifted. That sounds like a triviality and is the origin of a real result: a quantity that only ever appears as a difference is a quantity the physics has no way to measure absolutely, which is why the total energy of the universe is not a well-posed question and why the potential of a conductor is quoted relative to something.

Where the model stops #

The whole construction rests on the force being conservative, and the most common force in ordinary experience is not.

Friction has no potential. The work done against friction depends on the distance travelled, so a long route costs more than a short one, and the round trip does not return to zero. No function of position can describe it. The landscape picture simply does not apply, and the usual repair — draw the landscape anyway and let the object drift downward across the energy lines — is a picture of the failure of the method rather than an application of it.

That failure is not a technicality about a shabby force. It is where the arrow of time enters mechanics. Conservative forces are reversible: run the motion backwards and it is still a solution. Friction is not, and the energy it removes has gone into the disordered motion of enormous numbers of molecules, from which no arrangement of the original forces retrieves it.

Velocity-dependent forces are excluded too. Air resistance depends on speed, magnetic forces depend on velocity’s direction, and neither can be written as the slope of a function of position. The magnetic case is interesting because it does no work at all — the force is always perpendicular to the motion — so energy is conserved without any potential existing, which shows that the two properties are separate.

The energy has to be the whole energy. A landscape drawn for a block on a spring ignores the spring’s own mass, the sound it radiates, and the heat of its internal flexing. Each is small; none is zero; and each is an unlabelled leak out of a line that the drawing shows as perfectly horizontal.

And the system has to be closed. Energy is conserved because the laws do not change with time — Noether’s theorem again — and a system being pushed by something outside it has no such guarantee. A child pumping a swing raises the energy line, which the picture can show and cannot explain.

The landscape somewhere else #

The reason to spend this much attention on a curve is that almost nothing above was about mechanics.

The $-1/r$ well is the same picture with a different curve, and reading it the same way gives orbital mechanics. An energy line below zero crosses the curve twice and the motion is bound — a closed orbit. A line at zero has one crossing and the object escapes with nothing left over, which is the definition of escape velocity. A line above zero escapes with speed to spare. Three of the classic results of celestial mechanics are the three ways a horizontal line can meet a hyperbola.

The identical structure carries the electric potential, where the landscape is drawn for one unit of charge and the equipotential contours are the level sets of exactly this function. It carries molecular binding curves, where the well depth is the energy needed to break the bond. It carries the effective potential in a rotating frame, the free energy of a chemical mixture, and the loss surface of a fitted model. In every case the same three readings apply: minima are stable states, barriers are what has to be paid to leave them, and a horizontal line at the available energy says which regions are reachable.

That transfer is the actual argument for teaching energy at all. Forces are specific to the interaction; potentials all look alike, and a picture that describes a pendulum, a planet and a chemical bond with the same three sentences is doing something that no free-body diagram can.

Where the idea came from, and what it was resisted for #

Energy was not obvious and its arrival was slow, because for a century and a half two quantities competed for the role and neither side would concede.

Descartes’ followers held that the conserved quantity in a collision was the product of mass and speed; Leibniz argued from dropped weights that it was mass times speed squared, which he called vis viva — living force. The dispute ran from the 1680s into the 1720s, was conducted with some heat, and was settled only when d’Alembert pointed out in 1743 that both quantities are conserved and that they answer different questions. Momentum is a vector and survives collisions; energy is a scalar and survives smooth motion. There was never one right answer to be found.

The word energy in its modern sense arrived with Thomas Young in 1807, and the potential energy function with Lagrange, Green and Hamilton in the decades after — as a mathematical convenience, since it replaced three force components with one scalar and made the equations far easier to handle. The physical claim that the scalar is a substance being moved from place to place had to wait for Joule’s measurements of the mechanical equivalent of heat in the 1840s, which showed that the energy apparently destroyed by friction reappears, in the right amount, as warmth.

That is the sequence worth remembering. The mathematics of potential energy was in place, and being used, decades before anybody could say what happened to the energy that friction consumed. The bookkeeping was trusted first and justified afterwards, which is the opposite of the order the subject is usually presented in.

The ladder from here #

Later rungs on this anchor: the work–energy theorem derived rather than asserted, and why the factor of a half is there. The line integral, and the exact condition for path independence. Potential energy surfaces in two dimensions, and why an object with sideways velocity orbits a minimum rather than falling into it. The effective potential, which folds angular momentum into the landscape and turns the orbit problem back into a one-dimensional one. Escape velocity and the shape of the $-1/r$ well. Bound states, barriers and the rate at which a barrier is crossed. Lagrangian mechanics, where the difference of the two energies replaces the forces entirely. And tunnelling, where a particle is found on the far side of a barrier its energy line never crossed.

The next rung the reader is most likely to want is the electric potential, which is this same construction applied to a field rather than to an object, and which turns out to lose nothing at all in the translation.