- This is a video about a single simple rule that underpins all of physics, every principle, from classical mechanics to electromagnetism, from quantum theory to general relativity, right down to the ultimate constituents of matter, the fundamental particles. All of it can be replaced by this single rule. It feels like we're approaching spooky territory.
- We're approaching spooky territory. I hear you and I agree. - [Derek] It may in fact explain the behavior of life itself.
- I think I'm stuck in some classical mindset where, to me, the local picture, the differential equation way of thinking about the universe, that that's really what's going on. But I fear that I have it exactly backwards. - And it all starts with a simple problem.
If you wanna slide a mass from point A to point B, what shape of ramp will get it there the fastest? This is known as the problem of fastest descent. - You know, common sense, you might say take the shortest path.
Straight line from A to B. - [Derek] But if you bend the ramp down a bit at the beginning, well, the mass accelerates to a higher speed earlier. So even though it travels slightly farther, it travels faster and beats the straight ramp.
So the question is, what shape provides the perfect balance of acceleration and path length to minimize travel time? According to Galileo, it was the arc of a circle. He showed that this is faster than any polygon.
But is it the fastest? Nearly 60 years later, in June of 1696, Johann Bernoulli set this problem as a challenge to the best mathematicians in the world. - Mainly because he's a big show off and he wants to show that he's better than all of them.
- [Derek] He gave everyone six months to come up with solutions, but none were submitted. So Gottfried Leibniz, a friend of Bernoulli's, persuaded him to extend the deadline to give foreigners a chance. - And I think Newton was probably the intended target because everybody thought of him as the best, and so Johann would've wanted to show that he was better than Newton.
He was no longer a really active mathematician or physicist. He was working as the warden of the Mint, like a big high government position. - [Derek] And on the 29th of January, 1697, Newton returned home after a long day at work to find Bernoulli's challenge in the mail.
Irritated, he wrote, "I do not love to be dunned and teased by foreigners about mathematical things. " But the problem was too enticing, so Newton spent the rest of the day and night on it, and by 4:00 a. m.
he had come up with a solution, something that took Bernoulli two weeks. Newton submitted his solution to the journal Philosophical Transactions. - He sent his solution there but didn't sign it.
And Johann Bernoulli, after seeing the solution, is alleged to have said, "I recognize the lion by his claw. " You know, that, "Okay, you don't need to sign it, Newton, I can tell who you are. Who else could give such a solution?
" - [Derek] And while overall Newton dominated Bernoulli, in this instance, Bernoulli's solution actually out-shone Newton's. - I could see why Johann Bernoulli wanted to challenge everybody, because he came up with a really clever, I would say like a truly creative, beautiful solution. - To do it, he took inspiration from a problem faced by ancient philosophers.
How does light travel from one place to another? This was contemplated by Hero of Alexandria in the 1st century AD. He realized that in a single medium, like air, light always follows the shortest path.
A consequence is that when light reflects, say, off a lake, the angle of incidence is always equal to the angle of reflection. Any other path between the start and end points would be longer. But when light goes from one medium into another, like from air into water, it bends in a peculiar way.
It refracts and it doesn't follow the shortest path. - If you've ever dropped something at the bottom of a swimming pool and you look for it through the water, if you put your hand in there, it's not necessarily where you think it is because the light has bent away from the flat surface. - So what is the guiding principle here?
Well, over the next 1,600 years people slowly figured out that the sine of the angle of incidence divided by the sine of the angle of refraction is equal to a constant, n, which depends on the nature of the two media. This came to be known as Snell's Law. But no one knew why it worked.
That is, until 1657. - So another great mathematician enters our story at this point. This is Pierre Fermat.
By day he was a judge, but at night he'd come home and hang out with his wife and kids and then do what he really loved the most, which was work on math for fun. And he mostly worked in pure math but at one point he got interested in the question of, why does light obey this principle of refraction? - [Derek] And he thought maybe Hero of Alexandria was on the right track, but it's not distance that is being minimized, but rather time.
But to see if this was true for refraction would be difficult. He would have to work out every possible path light could take by varying the point where it intersects the boundary and compute the time for each, and then show that light takes the path for which the total travel time is the shortest. - He doesn't know how to solve it and he worries that it's gonna come out complicated even if he could solve it, so he's not gonna do it.
I think it's 'cause he wasn't super interested in physics, actually. But anyway, years go by, he gets interested in it five years later and tries to solve it, and then he does solve it and he shows that Snell's Law actually pops out as the minimizing path for light, you know, under these conditions of moving from one medium with one speed of light to another with another. - [Derek] And that constant, n, well, that's just equal to the speed of light in the first medium divided by the speed of light in the second medium, which allows us to rewrite Snell's Law like this.
- And he says this thing, I just wanna read you a little quote 'cause I love it. He says that this is "the most extraordinary, the most unforeseen, and the happiest calculation" of his life. See, it's good to do physics.
(both laugh) If you use that principle of least time, you can explain everything that was known about light at the time of Fermat. It's the first time, as far as I know, that anyone shows that nature obeys an optimization principle, that nature does the best possible thing. In this case, that light takes the shortest possible time.
- [Derek] Now, Bernoulli knew about Fermat's principle of least time, and he thought he could use it to solve the problem of fastest descent. - He converted the problem from a mechanics problem about a particle sliding down a chute to a problem about optics. - [Derek] Instead of a mass that's accelerated by gravity, he imagined a ray of light that would go faster and faster as it went into layers of less and less dense media.
And if you make the layers thinner and thinner where Snell's Law is obeyed at each interface, you eventually get a continuous curve. Now the question is, how should the speed of light change from one layer to the next so that it accurately models a falling object? - [Steven] You could try to solve the problem by thinking, if the particle has to fall from A to B, it's gonna be picking up kinetic energy, it's going faster and faster as it slides down the chute and it's converting the loss in potential energy into this kinetic energy.
If you write down the conservation of energy for that relationship, you find that the velocity that the particle achieves at any time, having fallen a distance, let's say y, its velocity squared will be proportional to y, the height from the top. So velocity goes like the square root of y. And that'd be sort of like saying imagine light moving in a way where, instead of a constant speed of light, the speed of light is proportional to the distance from the top.
- [Derek] Now let's zoom in and look at a single interface. We can plug in our expression for the speed of light in each layer into Snell's Law. Then find that the sine of the first angle divided by the square root of y for the first layer is equal to the sine of the second angle divided by the square root of y for the second layer.
And now here's the key insight. Snell's Law also holds for the next layer, and there the ingoing angle is simply theta 2. So this is also equal to the sine of the third angle divided by the square root of y3.
And the same goes for the next layer and the next and so on. In other words, this ratio must be equal to some constant, call it k. And this equation, the story goes, Bernoulli immediately recognized as the equation of a cycloid.
That is the path traced out by a point attached to the rim of a rolling wheel. This is also known as a brachistochrone curve from the Greek for shortest time. - [Steven] And so the astonishing solution is that the fastest way to get from A to B is to follow an arc of a cycloid.
Not a circle, a shape called a cycloid. - Now, this curve also has another surprising property. No matter where I release the mass from, it always reaches the end at the same time.
For this reason, it's also known as the tautochrone curve, from the Greek for same time. Upon finding this solution, Bernoulli wrote, "In this way I have solved at one stroke two important problems, an optical and a mechanical one, and have achieved more than I demanded from others. I have shown that the two problems taken from entirely separate fields of mathematics have the same character.
" Little did Bernoulli know he was onto something much bigger. Around 40 years later, one of his students, Pierre Louis de Maupertuis, also studied the behavior of light and particles, and he noticed that there are cases where the two behave very similarly. This made him think, what if Fermat's principle of least time wasn't the most fundamental?
I mean, why should nature care about minimizing time? Maybe there is a more foundational quantity being minimized, one that doesn't only govern light, but also particles. So, in the 1740s, he proposed a new quantity, which he called the action.
It is mass times velocity times distance. His thinking went something like this. The farther something travels, the greater the action.
The faster it goes, the greater the action. And if it's a particle, then the more massive it is, the greater the action. If there are multiple segments to the journey, then the total action is just the sum of the mass times velocity times distance for each segment.
To see the principle in action, here is a super simple example with no friction or losses. Imagine a 0. 5 kilogram ball is rolled over the ground for 6 meters at 3 meters per second.
Then that would be 9 units of action. If the ball then bounces and travels another 6 meters at 3 meters per second, then the action for the whole trip is 9 plus 9, or 18 units of action. Now, what Maupertuis claimed is that out of all possible trajectories where the ball bounces off the wall, the path it will follow is the one that minimizes the action.
In 1744, he wrote, "This action is the true expense of Nature, which she manages to make as small as possible. " So what was the response to Maupertuis' revolutionary idea? He was attacked and ridiculed.
One of his longtime friends, a fellow physicist named Samuel Konig, wrote that "not only is your principal wrong, you also stole it from Leibniz. " Voltaire, who used to be a close friend of Maupertuis, accused him of plagiarism, bad physics, being stupid, and just about anything else he could think of. In fact, he wrote a 32-page pamphlet just to mock Maupertuis.
Of course, this may have been partly due to rumors Voltaire's lover had an affair with Maupertuis. But not everyone attacked him. Some just ignored him.
- Maupertuis, I've taken a lot of math and physics in my life. I think you're the first person I ever heard pronounce it. He doesn't get much mention.
- All of this was terribly stressful for Maupertuis who was nearing the end of his life, and more than anything, he thought the principle of least action would be what he was remembered for. That would be his legacy. But now he was attacked, mocked, ridiculed, and ignored.
Unfortunately, this treatment was at least somewhat justified because Maupertuis came up with this principle by kind of just picking it out of thin air. There was no obvious reason why nature should care about mass times velocity times distance, or even less, why that quantity should be minimized. And mathematically, the principle of least action wasn't rigorous either.
But there was one man who vehemently defended it, and that man was Leonhard Euler. The first thing Euler did was he replaced the sum with an integral so you could calculate the action while speed or direction changed continuously. And he used this to find the path of a particle around a central mass, like the orbit of a planet around a star.
Solving this meant that out of every possible path between two points, he would have to find the one for which the action was the smallest. This is similar to the problem Fermat tried to solve, only now, instead of changing one variable, he would have to vary every possible point along the path, which is infinitely many. Needless to say, this was an arduous task.
Math had not yet developed the tools required to handle such problems. Fortunately, Euler himself invented a new method. It was clunky and time consuming, but it worked.
Through this process, he realized that the principle of least action only works if the total energy is conserved, and it is the same for all paths considered. These were two conditions that Maupertuis hadn't realized were necessary. So Euler improved the mathematical rigor of the principle.
He found two extra conditions and he provided a specific example of it working. - Euler is an astonishingly powerful and not just great mathematician, but appears to be a good guy. As far as we could tell, he was very generous.
You can still read Euler and really understand it. He helps you, he's empathetic. You know, he was like you are, man.
He's trying to explain stuff. - But Euler was still far from a general proof. That would have to wait for another legendary mathematician, Joseph-Louis Lagrange.
Joseph-Louis Lagrange was a shy 19-year-old, mostly self-taught. But despite his age, he was already working at the forefront of mathematics, including with Euler's new method. In 1754, he shared his results with Euler, who replied that Lagrange had "extolled the theory to the highest summit of perfection," which caused him "the greatest joy.
" But besides being world class mathematicians, the two had another thing in common. They were both huge proponents of the principle of least action. And around five years later, just a year after Maupertuis' death, Lagrange succeeded in providing a general proof.
Is there any intuitive way to think about action? I feel like there's an intuitive way to think about force and there's kind of an intuitive way to think about energy, but is there an intuitive way to think about action? - I don't know.
I wanna watch your show to learn. I hope you'll come up with it because I don't have a good feeling about action. - [Derek] Now I wanna explain Lagrange's proof, but I don't want to do it the way he did.
So instead, we'll go through three steps. First we'll explain the general approach Euler and Lagrange came up with. Then we'll rewrite the principle into its modern form.
And finally we'll apply this math to a simple example to show you why it works. So first, the general approach. If there are infinite possible paths, how do you find the one with the least action?
Well, Euler and Lagrange realized you can do it in a similar way to how you find the minimum of a function. There you take the derivative and set it equal to 0. And where the slope is horizontal, that must be the minimum.
So if you took a tiny step to the left or the right, the value of the function basically doesn't change. And similarly, if you have the path of least action, then if you were to change it a little bit by, say, adding a tiny bump here or flattening it out there, imagine we're just adding a tiny function, eta, to our path of least action, well, then the action basically shouldn't change because we're at this really special point, the path of least action. You add a little bit to the minimal path but the action is still the same.
If that is the optimal path that has the least action, then any other path must have more action. - So the counter there is like all of this is the first order. So if you're looking at linear terms that are proportional to eta, the deviation, then the first order, the difference in action will be 0.
And the way you could imagine this is like let's say you're at the bottom of a bowl and you're at the minimum and you make some tiny step away and we call that step eta. If that change would be proportional to eta, you would maybe increase on this side but decrease on this side and then this would no longer be a minimum. So sort of the coefficient of eta has to be 0.
But since you're at a minimum, it kind of goes like a parabola, so it can be proportional to eta squared or potentially some higher order term. So there is a tiny deviation in the action, but it's not proportional to eta. So the first order, the change in action between the optimal path and some trial path is 0.
- So what you can write is that the action of this trial path minus the action of the true path is equal to 0 to first order. This is a compact way of writing the principle of least action, and it's the general approach you use to solve all these problems. So with that in hand, let's rewrite the principle into its modern form, starting with Maupertuis' action, which is the sum of mass times velocity times distance.
But Euler changed this into an integral, so it's the integral of mass times velocity integrated over distance. Now the velocity is equal to ds over dt, which we can rearrange to get ds equals vdt. And plugging this in, we have an integral of mv squared over time.
But wait, that's just twice the kinetic energy. And as Euler pointed out, the total energy must be conserved. Total energy is just kinetic plus potential.
So we can rewrite this as T equals E minus V. And filling this in for the second T gives us that the variation of T plus E minus V integrated over time is equal to 0. Now we can split this integral into two, and since the energy is constant, we can integrate this term over time to get this.
And we can simplify this even further. Just like with a normal derivative, we can write the variation of E times t as E times the variation of t plus t times the variation of E. But remember, as Euler found, the energy of different paths has to be the same, so the variation between them is 0 and this term drops out.
If we rearrange this like so, then we find that the variation of this integral is equal to minus the energy times the variation of time. This looks a lot like some other minimization principle, if only this was equal to 0. Well, we can make this 0 by only considering paths that have the same travel time.
If you do that, then there's no longer any variation in time and this term drops out. And what we find is that Maupertuis' principle has changed into another form where now the variation of kinetic energy minus potential energy integrated over time is equal to 0. - T minus V, kinetic energy minus potential energy, and then you integrate that along a path that you're traveling from A to B, and then integrate it with respect to time.
It's all very strange and yet that turns out to be the correct thing to integrate. - [Derek] Now this is a little weird. We started with mass times velocity integrated over distance, and now we have the kinetic energy minus the potential energy integrated over time.
And somehow both are ways to write the principle of least action. But that also means that this integral here, T minus V integrated over time, is another way to write the action. The first person to write the principle of least action like this was William Rowan Hamilton in 1834, and by doing so, he got the principle named after him.
- So the principle of least action that we write as integral of Ldt, L being being the Lagrangian, the T minus V, the kinetic minus potential energy, they don't call it Lagrange's Principle, they call it Hamilton's Principle. So I guess Hamilton is building on Lagrange in that way. - Hamilton's Principle is the modern way of writing the principle of least action and the way you'd find it in almost every physics book.
In part that's because Hamilton's Principle tells you how objects move from one place to another rather than just giving you the shape of the path. Two other important differences between both principles are that the action is now an integral over time instead of space. And a consequence is that with Hamilton's Principle, you now need a start and end point, and also a start and end time.
The third is that with Maupertuis' principle, you need to keep the energy of different paths the same, but the time can vary. While with Hamilton's Principle, the energies can differ but the time has to be the same between paths. So now we have our general approach and the modern way of writing the principle, so let's apply it to a simple example to see why it works.
Let's say I throw this ball straight up in the air, so it goes from some start point to say a different end point in a certain amount of time. Now if we call the height of the ball y of t, then we can plot these two points like so, and then we can imagine infinitely many possible trajectories that could connect these two points. Some go a little higher, some lower, some have wiggles, others don't.
The only condition is that all paths must have the same start and end point and the same amount of time elapsed between them. Now to find the real trajectory, we proceed as before. We imagine that this is the true path, y of t, the one with the least action.
And then we imagine making small variations to it by pushing it up a little here, down a little there, and so on, making tiny changes at every time step, which we'll call eta of t. So when you add y and eta together, you get this new trial path, let's call it q of t. And since the variations are small, the difference in action between these two paths is 0.
Our next task is to solve this equation. So we compute the action for each path. For this, we need the kinetic and potential energy for each, and we write them as a function of y and eta.
Plugging that in, you get the difference in actions is equal to this. But wait, this first integral is just the action of the true path, so these integrals cancel, and what you're left with is just that. m times dy over dt times d eta over dt minus eta times the derivative of the potential integrated over time is equal to 0.
We can rewrite this further by using integration by parts. That allows us to replace this term with this one. And if we plug that in, we now have some function that when multiplied by eta and integrated over time has to be equal to 0.
But since eta can be just about anything, this can only be true if this part is 0. So what we found is that the action is minimal for the path that satisfies this curious differential equation. Now it might look complicated, but it's not.
Minus the derivative of the potential is just the force. And the second derivative of height? Well, that's just acceleration.
So if we rearrange this, we find that the path that satisfies the principle of least action is the one that obeys F equals ma. In other words, the principle of least action is equivalent to Newton's Second Law, but it covers more than just mechanics. Fermat's principle of least time turned out to be nothing more than a special case of the principle of least action.
So with this single principle, you could suddenly describe everything, from light reflecting and refracting, to the swinging of a pendulum clock, to planets orbiting the sun and stars orbiting the core of the galaxy. What used to be viewed as entirely separate fields of physics were now all unified under a single simple rule. The variation of the action is 0.
After Euler found out about Lagrange's proof, he wrote to him, "How satisfied would not Mr Maupertuis be, were he still alive, if he could see his principle of least action applied to the highest degree of dignity to which it is susceptible. " With Lagrange's proof, we now have two ways to solve any mechanics problem. You can either use forces and vectors or you can use energies and scalars.
It seems like the principle of police action is just way too complicated. It is so unnecessary. Who would ever use this, you know, when you could just use Newton's Second Law?
Like that's a piece of cake. - Your options are either use all of this or just start with F equals ma, and they give you the same answer. Why ever use the principle of least action?
Well, that's because Euler and Lagrange came up with a way to make all of this like super, super simple. - [Derek] If this is the action, then T minus V is called the Lagrangian. Now let's replace everything we did before with the Lagrangian.
Then you see that the principle of least action works whenever this differential equation is satisfied. - So all you have to do now if you wanna solve any mechanics problem is you just write down the kinetic and potential energy and you plug it into this equation and you're done. And that becomes extremely powerful.
- I remember thinking, "Man, force is like hard to get the right answer. " You can do it if you're good, and people who are good at mechanics can do it. But with the Lagrangian approach, you have this machine crank out the principle of least action on it and you get the right equation to motion and you don't have to be a good physicist.
That was what I took from it, that as a math guy I can do physics thanks to Lagrange and Euler. - And it doesn't just work in one dimension. If you have more dimensions, then you just solve the Euler-Lagrange equation for each coordinate.
- Another thing that's great is you could use weird coordinate systems that might be better suited to the problem. Like if you were doing a problem with something rotating, you might want to use polar coordinates instead of Cartesian coordinates. And this will give you the correct equations of motion in polar coordinates, which again might be kind of tricky to do with vectors.
- [Derek] Like with the double pendulum. Trying to solve this using forces is extremely hard. - Because as one pendulum is swinging, it provides the attachment point for the pendulum hanging below it, and so that pendulum is in this moving reference frame as it's swinging.
It's a very nasty job to write down the correct F equals ma for a double pendulum. But if you write it down with kinetic and potential energy, it's pretty easy. - This is actually how we made this simulation.
Now, there is one little side note we should give about the principle of least action, because the name is a little misleading. - Although we often refer to the principle as the principle of least action, it's probably good to add a little caveat here, that sometimes it's not necessarily the minimum. Just as when you find in calculus when you set a derivative to 0, that doesn't guarantee you're getting the minimum of a function.
The principle of least action more properly stated should be the principle of stationary action. That the laws of motion come from demanding a stationary point, which is tantamount to this condition of setting a certain derivative to 0 and then getting the Euler-Lagrange equation from that. So very often it is a true minimum, but not always.
- But action is much more fundamental than just classical mechanics. Around the turn of the 20th century, action popped up as the key part of a solution to one of the biggest problems in atomic physics at the time. The UV catastrophe.
- It's kind of spooky that this breakthrough that starts the ball rolling toward quantum theory brings action in. Not energy and not force, action. It gives you a hint, yeah.
- But that and much more will have to wait for a separate video, so make sure you're subscribed to get notified when it comes out. The story of the principle of least action is the story of how knowledge compounds, growing through steady progress one step at a time until it changes how we see the world completely. And it's not just big scientific discoveries where this happens.
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