Is ACTION The Most Fundamental Property in Physics?

Thank you to NordVPN for supporting PBS. It's about time we discussed an obscure concept in physics that may be more fundamental than energy and entropy, and perhaps time itself. Ready?

Action! In order to wrench the laws of physics from the reluctant clutches of nature, we need to watch for clues, patterns in its behavior that can reveal deeper truths. Here's an example.

In 1940 AD, Heron of Alexandria noticed that when light moves between two points, and of all possible paths between them, it follows the shortest one. He proposed what you might call a principle of least distance to define the path of light. Of course we now know that light doesn't always travel in a straight line, like when

it's refracted by glass or traveling through a gravitational field. It took a millennium and a half following Heron for Pierre de Fermat to propose a solution for the case of refraction. What if light didn't travel the path of least distance, but rather of least time?

If light changes speed between media, then the fastest trajectory is no longer a straight line. With this principle of least time, Fermat was able to explain all of the field of optics. It seemed as if we'd stumbled upon a powerful guiding pattern in nature, minimizing principles.

What if there was a similar property that could be minimized to determine the trajectories of matter as well as light. There were some efforts. For example, Euler considered properties involving momentum and kinetic energy, but to no avail.

In the meantime, Isaac Newton came up with his laws of motion, which gave us the instant and miraculous power to explain the motion of all particles in the universe. To do this, all you needed to do was know the exact vector forces on each body at each time which lets you calculate that body's exact vector position, velocity, and accelerations

at each time. That's doable for say a ball flying through the air, but it gets frankly pretty hellish for complicated systems. So maybe we were still in the market for a simple guiding principle after all.

It was Joseph Louis-Legrange who found that principle. Consider again the ball flying through the air. We can describe that motion without the painful vectors if we use energy. The ball starts out moving fast, it has a lot of kinetic energy, which it trades for

potential energy as it rises, and then back to kinetic as it falls. If you need a refresher on the energy stuff, check out our previous video when you're done here. You can figure out a lot just using energy like the maximum height the ball will reach,

but because energy doesn't include directional information, vectors, it seems that you shouldn't be able to determine the exact path. Except you can. Lagrange realized that a particular combination of kinetic and potential energy had the same

minimizing property as does time for the path of light. He found that the ball will always follow the path that minimizes the time-average difference between these energies. In other words, add up the difference between kinetic and potential energy at all time steps

during the flight and you get a number. Then try the same for any other path that you can imagine. The result will come out higher for every path other than the path that the ball actually travels.

Lagrange called this new quantity the action. Formally, it's the integral over time of the kinetic minus potential energies. And that difference in energies itself got the name Lagrangian, which itself turns out to be an enormously important and useful quantity.

Lagrange proposed that everything in nature happens in such a way that action is always minimized. And so was discovered the principle of least action. A century later, William Hamilton made the important correction that action is always minimized or maximized. Here's an imaginary graph of how the action can change between paths.

You get paths where it's at its lowest and at its highest. Moving objects tend to choose paths where the variation of the action between nearby paths doesn't change much. In other words, where the slope of this curve is zero. And that's true at both the minima and maxima.

Most of the time it's the minimum, which is why people still call this the principle of least action, but others call it the stationary action principle. If the principle of least action is true, then it should be possible to figure out the path that any object will take between two points as long as you can find a way to determine

which has the minimum action. That sounds hard, but thankfully Lagrange and Euler already did most of the work for us by finding the Euler-Lagrange equations, allowing both physics students and nature to minimize their action.

The principle of least action in combination with the Euler-Lagrange equations leads to Lagrangian mechanics. When you study Newton's mechanics and then move on to Lagrangian mechanics, it seems almost like a miracle.

The hellishly complex equations needed to solve problems using pure Newton seem to just melt away into much simpler forms. This simplicity comes from the fact that with Lagrangian mechanics we can dispense with forces and vectors in general and only consider the energies.

The principle of least action is so powerful that it really seems like it must be telling us something deep about the universe. But what? It's hard to interpret what this action property really signifies? If it really is so fundamental, then it should apply well beyond classical mechanics and into

our modern theories, in particular general relativity and quantum mechanics. And perhaps in these we'll find clues as to what the action is really about. Let's start with Einstein's general theory of relativity, which as we've discussed before describes the force of gravity in terms of the bending of the fabric of space and

time. One of Einstein's motivations in developing general relativity was the fact that Newtonian mechanics failed to correctly predict the orbit of Mercury. If you calculate that orbit using the Einstein equations in their full glory and full gory