Force That ClicksBonus: Force and Mathematics ① (trilogy, part 1)

Behind force lies a quieter principle ── force was never the lead role

The Principle of Least Action Why does a ball fly along the path it does? "Pushed by force, one step at a time" was Episode 1's answer.
But nature has another face ── of all possible paths, it selects the one that makes the "action" stationary. F=ma was a derivative of that.

Tools you'll need: F=ma of Episode 1, the geodesic of Episode 5, a feel for integrals Nature makes S=∫L dt stationary

In the main series, force was peeled back all the way to "relationship, geometry, gauge." In the bonus cluster "Force and Mathematics," we shine a light on what lies even further behind. The first installment is one of the most beautiful ideas in physics ── the principle of least action. Throw a ball and it traces a parabola. In Episode 1 we explained it as "at each instant, pushed by gravity, it bends a little by little via \(F=ma\)." But nature has an entirely different way of telling the story ── of the countless paths connecting the start and end points, only the one that makes a single quantity called the "action" stationary is realized. And from this telling, \(F=ma\) can be derived. In other words, force was a derivative of a quieter "stationarity principle."

01Action ── one number determined per path

To a given path (a record of position over time), we assign a single number, the action \(S\). How to build it ── take, at each instant, "the kinetic energy \(T\) minus the potential energy \(V\)," and add it up (integrate it) over time along the path.

Definition of the action
$$S=\int_{t_1}^{t_2}\!\!\big(T-V\big)\,dt \qquad (T:\text{kinetic energy},\ V:\text{potential energy})$$

Fix the start and end points, and as you change how it passes in between (the path), the value of \(S\) changes. What nature selects is the path for which \(S\) is stationary (unchanged to first order under a small shift) ── this is the principle of least action (Hamilton's principle).

"Stationary" means that shifting that path ever so slightly leaves \(S\) unchanged to first order (zero slope, like the bottom of a valley or a mountain pass). In many cases it's the minimum of \(S\), which is why it's called "least action." In the figure below, shift away from the true path (the parabola) and confirm that the action \(S\) always increases. The true path sits at the bottom of the valley of \(S\).

Figure: the path of a thrown ball. Shift from the true path (indigo) to a trial path (orange) and the action S increases (the bar at lower right). The true path minimizes S ── which is why nature selects it
true path (S minimum) trial path (shifted)

02F=ma comes out of it ── force is a derivative

Write down the condition "the path that makes \(S\) stationary" mathematically (the Euler–Lagrange equation), and what comes out is ── none other than \(F=ma\). Precisely, \(m\ddot{x}=-\dfrac{dV}{dx}\), that is, the \(F=-\nabla V\) of Episode 6 itself. The principle of least action and \(F=ma\) are completely equivalent restatements of the same content. So which is "more fundamental" is a matter of viewpoint, but many physicists place the action in the leading role. The reason is next.

The heart of this installment ── force is a face of the stationarity principle

"At each instant, moved by being pushed by force" (\(F=ma\); local; Episode 1)
= "as a whole, selecting the path that makes the action \(S\) stationary" (least action; global).
Two ways of telling the same motion. Force can also be seen as a derived quantity, derived from a single quantity called the action.

The advantages of putting the action in the lead role are immense. Its form doesn't change when you change coordinates (it gets along well with relativity), the same framework works for light and for fields, and ── next time's star, symmetry and conservation laws (Noether's theorem), can be told most naturally in the language of the action. In Episode 9 we saw "the origin of force is gauge symmetry," but that symmetry means a symmetry of the action. The action is the soil in which this series' backbone puts down its deepest roots.

A connecting voice ── light, and free fall, follow the same principle The stationarity principle is not for mechanics alone. Light travels along "the path of least time" (Fermat's principle) ── this too is a stationarity principle. Free fall (Episode 5) travels, in curved spacetime, along "the path where proper time is stationary (the geodesic)." A thrown ball, a ray of light, a spacecraft in free fall ── all select "the path that makes something stationary." Nature follows a stationarity principle everywhere. Force was one manifestation of it.

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03What peeling it back revealed ── the stationarity principle behind force

The conclusion of Force and Mathematics ①. Force was not the principle at the very root of nature. There is a quieter, more global principle ── "nature selects the path that makes the action \(S\) stationary" ── and \(F=ma\) is derived from it. In the main series we peeled force back to "force is not a noun but a relationship," yet even that relationship was a manifestation of a single variational principle, "make the action stationary." Force yields its leading role to the action.

The honest line

"Least action" is the common name; precisely it's the stationary action (not necessarily a minimum ── it can be an extremum or a saddle point). Also, the action principle and \(F=ma\) are equivalent, and "which is fundamental" is a matter of interpretation (the action is preferred for the practical benefit that it connects naturally to symmetry, fields, and quantum theory). Some forces, such as forces with friction, cannot be written with a simple action (dissipative systems). In quantum mechanics, a particle passes through not one path but every path, and the action sets the weight of each path (Feynman's path integral) ── the classical "single stationary path" is a special case of this.

The action \(S\) in the figure is a value computed numerically for a simple parabolic motion, with its units and magnitude normalized for display. The way of shifting is also just one example (a single hump with both ends fixed).

Practice problems
  1. How do you build the action \(S\)? Which path does nature select?
    See the answer
    The action S=∫(T−V)dt is (the kinetic energy T − the potential energy V) at each instant, integrated over time along the path. With the two ends fixed, nature selects the path for which S is stationary (unchanged under a small shift).
  2. What is the relationship between the principle of least action and \(F=ma\)?
    See the answer
    They're equivalent. From the condition that makes S stationary (the Euler–Lagrange equation) you can derive \(m\ddot x=-dV/dx\) = F=ma (the F=−∇V of Episode 6). Force can be seen as a derived quantity derived from the stationarity of the action.
  3. Give one example, outside mechanics, that follows a stationarity principle.
    See the answer
    The way light travels (Fermat's principle = the path of least time). Or free fall = a geodesic of spacetime (the path where proper time is stationary; Episode 5). Nature follows a stationarity principle everywhere.

SummaryForce was a derivative of the action

The path of a thrown ball can be told as "pushed by force, one step at a time," or as "selecting the path that makes the action \(S=\int(T-V)dt\) stationary." The two are equivalent, and from the stationarity condition \(F=ma\) (= F=−∇V) is derived. So force can also be seen as a derivative of a quieter, more global stationarity principle. Light (Fermat) and free fall (the geodesic; Episode 5) were both selecting the same "path that makes something stationary."

Behind the "force is a relationship" that the main series peeled back, there was one more sheet: "nature makes the action stationary." And this action is precisely the soil in which next time's Noether's theorem (symmetry → conservation law) and the gauge symmetry of Episode 9 can be told most naturally. Force yields the leading role to the action ── that's where "Force and Mathematics" begins.

This document is installment ① of "Force and Mathematics," a bonus of the "Force That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. That Hamilton's principle (the stationarity of the action \(S=\int(T-V)\,dt=\int L\,dt\)) and the Euler–Lagrange equation derive \(m\ddot x=-dV/dx\) (Newton's equation of motion); that this is equivalent to Newtonian mechanics; that Fermat's principle and general relativity's geodesic principle are also variational principles; and that in quantum mechanics Feynman's path integral contains the classical stationary path as a special case ── all of this is established content. "Least" action is precisely the stationary (including extremum and saddle point) action. There are cases, such as dissipative systems, that cannot be represented by a simple action. The action values in the figure are a conceptual model, a numerical integration of parabolic motion normalized for display. ── To print, use your browser's "Print" and choose "Save as PDF" (in the print version the slider and answers are static and hidden).

Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider lets you see the action increase as you stray from the true path. "See the answer" opens each solution.