Everything, Everywhere, All According to the Principle of Least Action

On Physics, Light, Lifeguards, and the Deepest Mysteries of the Universe

What need is there for toil and strife?
The path of least resistance is the way of life.
For energy to find its way,
It must take the path that requires the least to pay.
This law of Nature, Least Action, is true
For Nature loves to find the path that's least to do.

(This original poem was written by the latest OpenAI (Artificial Intelligence) model, GPT-3, unveiled last week, which will likely change all of our lives . . . but more on that another time.)

The Principle of Least Action (PLA) has to be one of the most interesting ideas in the universe. Everyone should know a little bit about it. That’s because everything moving around you, and everything moving inside your own body, down to the tiniest electrons and quarks, is following this basic principle, even as we speak. Physicists regard it as more fundamental even than Newton’s famous laws.

Below is an explanation I wrote for myself. Given that I’ve named this blog “Least Action,” it seems appropriate to share. By the way, I’m not a physicist or mathematician, so if you see any errors, please do let me know. (Also, if sciency stuff like this makes your head hurt or just bores you to tears, you can choose to receive updates only about particular topics - or no topics!)

Anyway, the PLA holds that physical objects take an optimal route through space, given the forces acting on them. To oversimplify, objects always and everywhere travel along the most time-and-energy-efficient route. Thus, the PLA has been described as evidence of a “Lazy Universe.”

To understand this fascinating concept, it helps to break down the name.

“Principle.” A “principle” is a general concept that explains how more specific things work. PLA explains how objects in the world move - their dynamics or mechanics. In fact: “Essentially all the laws of physics, describing everything from the smallest elementary particle to the motion of galaxies in the expanding universe can be understood using some version of this principle.”

This includes Newton’s laws, relativity, and even quantum mechanics.

“Action.” “Action” is a term of art, so throw your everyday definitions out the window. In physics, the Action is a quantity. Its units are expressed as either “energy-time” (commonly joule-seconds) or momentum-length, depending on what kind of object we’re talking about. The fact that the units can be expressed in different ways hints at the enormous complexity lurking beneath the surface, which we’ll only scratch. Note, too, that energy-time is not the same as energy/time (energy per unit time), which is the unit of power.

Action (uppercase “S”) is expressed as the integral of the difference between kinetic energy and potential energy over time:

Again, that’s:

The integral
     Of the kinetic energy
          Minus the potential energy
               Over time

This is often stated in technical jargon as “time integral of the Lagragian:”

“S” is the Action. The integral symbol ∫ means the sum under a smooth mathematical curve (a “function”) from Time 0 to Time 1. “L” is the Lagrangian function that gives us that curve. “dt” means that the function is “with respect to changes in time.”

If you took calculus, recall that an “integral” expresses the size of an area under a mathematical curve (think of the squiggly line that appears on a graph). The integral is expressed by this funny ∫ shape.¹ You probably know how to find the area of a two-dimensional square space, like a box: length time width. But finding the area under an any-dimensional curve requires calculus. Calculus is a method of chopping up the space into infinitely many little boxes and adding them all together.

If this simplified illustration were describing the Action, the curve would represent the quantity of the Lagrangian changing over time (time being the horizontal axis) and the sum of the little boxes would be the Action

The Action is a time integral because the curve we want to sum underneath tracks points in time rather than space. So rather than moving from Point A to Point B, say, in a room, our curve is an abstraction that chart’s the Action’s change from Time A to Time B - say, from 11:58 a.m. to noon on a Tuesday.

The “Lagragian” is the kinetic energy minus potential energy. It’s named after the 18th century mathematician Joseph-Louis Lagrange. The Lagragian forms the basis of a whole branch of physics that is an alternative to Newtonian mechanics. Remember: mechanics is the study of how things move through the universe. Also, by “Newtonian,” I just mean the basic high school physics of objects in the everyday world where, for example, objects aren’t moving near the speed of light. Lagrangian and Newtonian mechanics are consistent, but sometimes it’s easier to think in terms of one or the other. In fact, all the essentials of Newtonian mechanics can be derived from the PLA.

Kinetic energy is the energy the object has due to its motion. In Newtonian terms, the Kinetic energy (uppercase K) equals one-half the object’s mass times its velocity squared: K = 1/2mv^2. 

Potential energy (uppercase V) is more complicated. It’s the energy due to position - generally within a force field, like an electric or gravitational field. A simplified way of thinking about potential energy is the object’s inherent energy. For example, atoms have a lot of inherent energy due to the strong forces holding their centers, or nuclei, together. It’s this potential energy that gets released - that is, converted to kinetic energy - in a nuclear explosion. However, the most familiar example of potential energy is that of a large object due to gravity. This is equal to its mass times its height (in the field) times a constant number h that depends on the particular gravitational field. So, V = mgh.

The total energy of a system is the Kinetic + Potential Energies (K + V). But the Kinetic minus Potential energies (K - V) is what is called the Lagrangian. And the sum of the area under the curve representing the change of the Lagrangian over time is the Action.

“Least.” Finally, the path of “Least Action” means the path that will minimize the Action over a length of time. To minimize the Action, it helps to have a smaller Lagrangian (K - V). You make the Lagranian smaller by either having less kinetic energy (to keep L low by keeping K low) or more potential energy (to keep L low by subtracting more from K). This path of minimized Action can be considered more efficient, because we get to our destination in time and space while trying to use as little kinetic energy as possible and hoarding as much potential energy. Finding the Least Action typically requires a particular type of calculus, called “the calculus of variations,” that helps us optimize the sum of the area under the Lagrangian curve, or the “integral.” What it tells us is which of the many different paths (infinitely many, in fact) will produce the Least Action:

Here, “q” is a position, and “t” is a time. Given particular q’s and t’s, the PLA tells us what path the object actually took to get from one to the other.

Actually, “Least Action” is a bit of a misnomer. What we really want is the path that produces a stationary Action at every infinitesimally small moment in time along the path. “Stationary” means “not changing.” Of course, the Action does change, because an “infinitesimally small moment in time” is just a mathematical fiction - an approximation. But the calculus of variations allows us to find the motion that will produce zero change in the Action as the distance traveled through time approaches zero. So the PLA is sometimes called the Principle of Stationary Action. But “Least Action” sounds cooler and is easier for non-physicists to grasp. 

Zooming out to the real world, the PLA would explain why throwing a baseball from one glove to another on earth makes a parabola shape. Not just any parabola, but the very parabola that you observe. The main force acting on the ball after it’s released is gravity. (We’ll ignore things like wind and air resistance). If instead the ball zoomed up in the air at the speed of light, moved in a straight line, and then dropped back down at the speed of light, the Potential Energy would be higher, which would help bring the Action down. But: the Kinetic Energy needed for the whole “speed of light” bit would be way higher, and would produce a much higher Action overall. 

We’d have the same basic problem (too much Kinetic Energy) if the parabola was any taller than we observed. On the other hand, if the baseball formed a kind of “convex parabola,” dipping down and rising up again to hit the second glove: then we would be needlessly losing out on Potential Energy. Same, too, if we had a regular (concave) “gravity’s rainbow”-style parabola that was any less high than what we actually observed. 

In this way, the gravity’s rainbow you observe is always exactly the one that minimizes the total Action over the path through time. The Kinetic and Potential Energies are perfectly “optimized” in a finely calibrated balance at every moment. 

What’s more amazing is that the PLA appears to work everywhere in the universe. It doesn’t just work with baseballs and balloons here on earth. It works, with some highly “mathy” tweaks, at super high speeds, with massive objects like planets and galaxies, and with super tiny objects like electrons or quarks. Also, unlike less fundamental parts of so-called “classical” mechanics, like Newton’s laws or aspects of Einstein’s Relativity, the PLA works in the mysterious and counterintuitive “quantum” world (i.e., the world of teeny-tiny objects).

It even works for beams of light. Because light essentially consists only of kinetic energy (due to its momentum), and has no potential energy, the way to minimize the Action for light is simply . . . to get from Point A to Point B as quickly as possible. Which in turn usually means moving in a straight line.²

But the path of Least Action for light is not always a straight line! Here’s a really cool and profound point: If light has to choose between the path of the least distance or the path of the least time, it will always choose the path of least time, because that path will produce the Least Action. This is why you get weird optical effects when light passes through two mediums, like air and water, before hitting your eye. The light you see appears refracted, or bent, because it travels faster through air than through the denser medium.³ 

So, just as with the baseball example, the light going through two mediums will follow a minutely calibrated path to minimize the Action at each point in time. But, instead of a parabola or a straight line, the path is basically two straight lines - a steeper one that allows it to get out of the slow water “faster”, and then a shallower direct line from the water’s surface to your eye. Any other path than the one you see - including a single straight line - would take more time. Thus, as the early 20th-century physicist Max Plank put it, photons seem to “behave like rational beings.”

One of the many natural phenomena explained by the PLA. Refraction by a prism, which slows down light, creates different “paths of Least Action” (which is also least time) for photons (tiny “packets” of light) of different wavelengths.

The physicist Richard Feynman compares this optimized path to the one you might take if you were a lifeguard standing on the beach who had to rescue a swimmer in the water at a diagonal several hundred yards from you. You wouldn’t just take a straight line to the swimmer, because you move slower in the water, so you want to spend less time moving through water. Nor would you entirely minimize your time in the water by running horizontally along the beach and then making a sharp right turn to the water. Instead, you would run along the beach at a diagonal covering most of the horizontal distance to the drowning person, and then swim directly to the person at a different angle once you hit the water.

The same basic principle explains why lenses in microscopes and glasses work - by “focusing” paths of Least Action from the object to your eyeball. The PLA explains, in fact, pretty much everything in the physical world.

Given its vast explanatory power, it’s remarkable that the PLA is not more widely known. In physics circles it has attained quasi-mythical status. According to Planck, it “arouses the impression as if nature be governed by a rational and purposeful will.”

“​​During the last three centuries, no other principle has nourished hopes into a universal theory, has constantly been plagued by mathematical challenges, and has ignited metaphysical controversies about causality and teleology.” - Philosopher Michael Stöltzner.

For while we know that everything in our universe seems to follow the PLA, we don’t really know why. For example, the fact that subtracting the potential energy from the kinetic yields anything interesting and testable in the world (the Lagrangian), rather than some random number, is not really explainable. It just seems to work out that way. This is surely one of the deepest mysteries of physical reality.

Further reading:

Richard Feynman’s classic lecture on the PLA

Leonard Susskind’s “theoretical minimum for doing physics” (more technical)

Sean Carroll’s “The Biggest Ideas in the Universe: Space, Time, and Motion” (still complex, but more accessible)

1.  We won’t be detained by how calculus works, but here’s a cool little book from 1912 (!) that explains it more simply than anything else I’ve come across.

2. “Straight” describes the motion from light’s perspective - light can appear “bent” if the space is warped by gravity from our frame of reference, such as near the sun. This bending by gravity is also explained by a PLA, albeit a more complex version.

3. “Wait, I thought light only moved at one speed: the, um, speed of light?” Well, yes, in a vacuum, but things get more complicated when light moves through a denser medium where it can run into other stuff like electrons.