What makes the principle of least action so important? Who came up with it? Is it all just a bunch of math or does it actually mean something? I discuss these questions and more in today’s Ask a Spaceman!

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EPISODE TRANSCRIPTION (AUTO-GENERATED)

Before I dig into today's topic, I want to recommend a book to you. You are always asking for recommendations. Well, here it is. And I've started this amazing partnership with Chirp, an audio book retailer known for their great deals without any commitments or subscription. And we've got a book club going on that you can join. And my pick for this month is the Copernicus Complex by Caleb Sharp. You know, my book are placed in the universe. The original title was You are Not Special and this book is a perfect companion piece to it. It's about the discovery of our cosmic insignificance, how important that revelation was and, and, and how that's something that drives scientists even today, including me and my men, late night existential crises. It really digs into the facets of this complex topic and it's, it's a really, really fun read. So to get it, you go to chirp books dot com slash spaceman and get the Copernicus Complex on sale from $19 to only 2 99 for a limited time and be sure to press follow, to join my club to stay in the loop on future picks and other exclusive content from me.

That's chirp books dot com slash Spaceman going in there really helps to support the show and I appreciate it. There are laws of physics and then there are laws of physics folks today. I gotta warn you, it's an especially nerdy topic today, but I want to do it because a bunch of you asked and the punch line is worth it. It's gonna blow your mind in a surprising way. We're, we're used to having our minds blown by supernovae or the beginnings of the universe or the intricacies of string theory. This is a very subtle mind blowing that I think as a dedicated listener of this show, you are going to appreciate it and, and it's gonna take a little bit to get to the punch line, you gotta hold on, you gotta be patient here. But trust me, I think it's gonna be worth it because today we're talking about perhaps the most important idea in all the physics. And I mean, all the physics, not the most important idea in cosmology or electromagnetism or relativity or condensed matter physics or thermodynamics.

I mean all of physics today, I'm talking about the principle of least action, which is more than a law of physics. It's a principle that enables us to create laws of physics. It's a machine that turns simple descriptions of nature into powerful predictive equations. In fact, once you write down a few basic terms, you can use the least action principle to derive all of physics, all of it. Every single corner, every single physicist working today on the planet is really just investigating one tiny aspect of this meta theory given to us from the gods themselves. This principle of least action and the weirdest thing, it's a secret. Sometimes I feel like scientists are a little too uh exclusive. They use jargon and dense math to cut themselves off from the rest of society. And you're right, I'm a little different.

I've never minded being a heretic. I feel like I'm in good company in the least action principle or the principle of least action is a prime example. If you've had any sort of stem i education, you probably took some intro to physics class in college or at least took some version of physics in high school and you probably learned something like Newton's laws, you know, mass force action, reaction, acceleration, et cetera, et cetera. The three Newton's laws with these three laws. It's, it's a backbone of classical physics. Good old fashioned Newtonian mechanics from Newton's laws. You get the physics of movement and motion and the physics that governs the vast majority of what one encounters in everyday life. A car crashes, bowling tournaments, cheese acquisition, airplane flights, the works. Your life is described in large part by Newton's laws.

That's why they're a massive insight into the way the world works. Uh For example, one of those laws is force equals mass times acceleration. If you want to know how an object will move, you add up all the forces acting on it. Uh There's the gravitational force, maybe there's some friction, maybe some air resistance, et cetera, et cetera. Maybe it feels some electromagnetic force, you add it all up. And if you know the object's mass, you know its acceleration and from there, you can calculate its future path. That is a powerful statement of nature, that is a powerful law or theory of nature. And it all rests on New Newton's conception of how the world works, which is based on forces and masses and acceleration. This is how Newton looked at the world. This is the lens that Newton used to look at the world and this is a very powerful lens and you are probably at least somewhat familiar with these rules or I've encountered them before, but you probably didn't learn about the version of mechanics developed 100 50 years after Newton, uh completely different ways of approaching the same problems and the same questions.

But with a much cleaner, much more elegant mathematical structure, there are two of them that survive today in some sort of useful form. They're called as opposed to Newtonian mechanics. There's Lagrangian mechanics named after Joseph Louis Lagrange, who we met before with the Lagrange Points in Hamiltonian mechanics named after sir William Roe in Hamilton who deserves his own episode. Feel free to ask Now, this episode is not going to be about Lagrangian and Hamiltonian mechanics and these different approaches to solving everyday problems. Uh But I need to introduce it in order to get to the promised land of this punch line that will blow your mind and these approaches to classical physics. So we're leaving aside quantum physics, we're leaving aside general relativity. We'll get to those later, we're talking about classical physics. There are different ways to look at the world than the Newtonian forces and masses and accelerations. And we get these from Lagrange in Hamilton.

But the vast majority of people are not familiar with Lagrangian or Hamiltonian mechanics. These topics are typically only reserved for undergraduate acolytes in the arcane arts of physics itself. And even then you don't learn about their true power until well into grad school. When, when one day the professor stops herself, mid lecture turns around, takes off her glasses, looks you straight in the eye and says we've been lying to you and it's time to tell the truth and then she tells you about the principle of least action. And now it's time for me to let you in on the secret. Why am I going on about these different versions of mechanics? Hamiltonian, Lagrangian. I'm not gonna get into the details of how the those mechanical systems work. But the general idea here is that physicists are always playing with concepts and descriptions of nature, some of those descriptions are useful, some are less. So uh we look around the world and we see objects doing things and interacting and all of these objects have properties.

Some of these properties are useful and we can build physical theories out of them and they respond to other things like the classic example is force and mass and acceleration, mass is a property of an object. Acceleration is a property of an object. It's describing what it's doing and force is a, a good description of its environment of what it's experiencing. And it turns out these three random quantities out of all the ways I could possibly describe an object in its environment. It turns out that force and mass and acceleration are very, very useful other concepts, other qualities, other quantities are not as useful. One of my favorite examples is uh position is useful knowing where an object is at any one point in time is pretty useful. It's velocity, it's change in position over time. That is also a very useful quantity. It's acceleration is also very useful. How does the velocity change with time that is called acceleration? And that's a very useful thing. It correct connects directly to the force that an object experiences.

You could also measure if you wanted to something called the jerk. I'm not making that up. The jerk is the rate of change of acceleration. It's a real thing. It's a measure every object has some amount of jerk in it, but it's not really useful because it doesn't connect to much of anything else. And so it doesn't appear much in our equations in our physical theories. But here are two other very useful quantities. The kinetic and potential energies of an object, the kinetic energy is just the energy an object has through its own motion, just one half of the mass times the velocity squared. So if you know an object's mass and velocity, you know it's kinetic energy. This is the amount of energy it has through motion. If, if I throw a ball at you really hard, it has a certain amount of kinetic energy and then if it hits you, it transfers that energy into you. And so you it, it's good to know that energy. The other useful quantity is the potential energy. This is uh in the case of the earth's gravitational field, uh how high an object is above the surface of the earth.

So a ball that's sitting on the ground does not have a very high amount of potential energy. A ball that's sitting on the top of the shelf does have a lot of potential energy. It has energy just by virtue of its placement in a gravitational field or different fields give you different kinds of potential energies. There can be electric, potential energy, et cetera, et cetera. But for the case of the earth's gravitational field, then the higher up a ball is the more energy it has just by virtue of its placement and you care about it because if it were to roll off that high shelf and hit you on the head, that's how much energy it has available to hit you with. So you care about it. And when it comes to physics and physical situations, I can look at the same situation through multiple lenses and use different mathematical languages and get different insights. Like some things sound prettier in French than in English. For example, I can use Newtonian mechanics. I can also use Lagrangian or Hamiltonian mechanics. I I can focus on forces in mass and acceleration.

Like for example, if I take a ball and throw it up in the air and it falls to the ground, I can use Newtonian physics which uses forces and masses and acceleration F equals ma all that good stuff to describe and predict how that ball will behave. In this case F equals ma is what's called an equation of motion. It tells me how that object will change its position over time. An equation of motion. This is a fundamental thing we use in physics all the time. We want to know how things will move because we want to predict where they'll go. And we need an equation of motion for that F equals MA force equals mass times acceleration is great and grand and wonderful. And it can be applied easily to a bunch of situations. Uh Technically it can be applied anywhere everywhere where classical physics applies. But sometimes it can get a little bit unwieldy because adding up all the forces in a complex uh situation. Yeah, you can use F equals ma to describe an airplane in flight. But that's like really complicated.

And so even though it can apply, it's not always the best tool for the job. Alternatively, we can look it through these alternative approaches, these Lagrangian and Hamiltonian mechanics. We don't have to look at a situation in terms of forces and masses and accelerations. We can look at potential in kinetic energies, for example. And we can build a theory of physics that allows us to describe the future motion of an object by just by analyzing its potential in kinetic energy. And we can write down a specific combination of terms in this case, the kinetic energy minus the potential energy and use that. And that's called the Lagrangian because it's the central mathematical core of Lagrangian mechanics. The central mathematical core of Newtonian physics is f equals ma the central core of this Lagrangian picture where we're gonna take a different combination, a different set of, of qualities and attributes to every object and use that to build our physical theory.

And in this case, it's kinetic energy minus potential energy at first blush. That seems pretty random and arbitrary. Why those two, why do you take the difference and not add them? What if you take kinetic energy minus twice the potential energy or potential energy minus the kinetic energy. Like why that order, why that specific arrangement, like I said for centuries, physicists have played with various quantities and measures and combinations of those quantities. And it just so happens that the Lagrangian which is the difference between the kinetic and potential energy of an object is really super duper important. It, it, it works because it does the same way F equals ma works because it does because that was Newton's insight by playing with all these concepts he's found, oh force and acceleration are connected this way. And then boom, he was able to unlock classical physics. If you take the difference between kinetic and potential energies, you get Lagrangian mechanics in boom, you, you can explain all of classical physics.

If, if Lagrange had lived first or you know, a different evolutionary pathway, we may have came, come up with Lagrangian physics first. Before we came up with Newtonian physics, they're, they're equivalent, they're just different ways of describing the same situation, but just focusing on different properties. So in one case, in in one way, Lagrangian physics isn't all that special. By the way, if you add Connecticut potential energy, you get Hamiltonian mechanics and for what it's worth, that's very incredibly useful for other things unrelated to this episode. And at first glance, Lagrangian, it's just like, OK, like just translating different languages. If, if a situation is a little too tricky for Newtonian physics to to be wieldy and efficient. Well, then I can just switch to Lagrangian physics. This alternate picture that is exactly equivalent mathematically proved to be equivalent by the way. So you don't have to worry about it in, in some physical scenarios are easier to deal with. But little grand has one other superpower and hence I can finally introduce to you the star of our show today.

It's a quantity called The Action. But before I continue, I need to take a quick break for a word from our sponsor. Better help. Mental health is so important. And you know, I'm a firm, firm advocate for mental health. You, you take care of your body, you go to the doctor when things are a little off or just do regular checkups with your doctor. You should also take care of your mind. I know a lot of you tune in to this show to just escape and relax and have your mind blown. Well, maybe you should have your mind helped a little too. I've gotten a lot out of therapy and I'm not ashamed to admit it. I think it is a powerful tool for everyday life and that's where better help comes in. Better help is online therapy. It's like a podcast where you get to do a lot of the talking. That's pretty cool. And someone's there to listen up real professional over the video, over phone, even live chat only sessions. It's more affordable than in person therapy.

You can be matched with the therapist in under 48 hours. This is a powerful tool for your everyday life and I seriously encourage you even if you don't think anything is wrong, you will be surprised at how much therapy can help. Ask a Spaceman listeners get 10% off of their first month at better help dot com slash Spaceman. That's better. He LP dot com slash Spaceman. If I throw a ball at you, it starts out leaving my hand with a certain amount of kinetic energy and a certain amount of potential energy as that certain amount of kinetic energy because I pushed it, I threw it and it has a certain amount of potential energy because it's a certain height off the ground. It will go on its path, it will arc up, reach the top of its flight path and there its potential energy will be highest and its kinetic energy will be lowest and then it will fall back down before reaching your hand as the ball travels, as the ball flies from me to you, its kinetic and potential energies are always changing.

And so the difference between the kinetic and potential energies are is always changing that difference. That value, the difference between the kinetic and potential is always changing. The action is the total difference between kinetic and potential potential energies across the entire path. So at least my hand, I measure kinetic and potential. Take the difference. Boom, it it advances one tiny little step. I have a new value for the Connecticut potential energies. I take that difference. Boom. And I and I keep adding it as this ball is traveling, I keep adding up these differences between the Connecticut potential energies as it's traveling. And at the end of the whole thing, at the end of the whole trajectory, I have some number that represents the total difference between the Connectican potential energies. And that is the action for you stem nerds. It's the integral of the Lagrange and it's the integral of the difference between the kinetic and potential energies across the path before you skip to the next episode and before you fall asleep, let me blow your mind.

If I toss a ball at you and you've had a little bit of practice, you catch it as soon as you see the ball in flight, you're ab your brain is able to predict where the ball will be and you put your hands in roughly the right spot. Some of you are probably better at this than others. But you get the idea. How did your brain know that? How did your brain know that when I threw the ball, once you got a sense of its motion, its velocity, its direction, you could roughly predict where the ball will land. Yeah, you won't catch it every time. But you have a rough idea of what the ball is gonna do. Your brain knows that from learning, you've, you've watched lots and lots and lots of balls being thrown at you and, and when you throw a ball, you see how it behaves, it follows a regular and predictable path. And we could predict that path from Newtonian physics. That path is called a parabola as the name for the mathematical equation that describes the path of that ball. It's a parabola. You know, when I throw a ball at you, it's gonna be roughly a parabola and you can roughly guess where the ball is gonna be because you've seen this all the time.

But why that path? Why not another path? Why didn't the ball zoom way up and hang out between us and then wiggle around a bit and then come to you. Why didn't it fly past you and then stop and turn around and come back? Why didn't it like spiral around me like a tornado and then roll along the ground for a little bit and then pop up into your hands? Why didn't, why any random path? Why this path all the time? In fact, if I take a ball throwing machine and it throws the ball with the exact same precise initial kinetic energy and angle and everything at the beginning, it will follow the exact same path every single time for now until eternity, you know, we'll do it all across the universe every single time Y now, you might say Paul, it didn't follow those paths because that would be stupid and it would violate the laws of physics. We know our laws of physics. We know Newtonian mechanics. We know F equals ma, the ball has to obey force equals mass times acceleration and force F equals ma gives that path every time.

So of course, the ball has to do that. Anything else would violate F equals MA? it would violate the equation of motion. It would violate the laws of physics. Well, my response to you is Francis, I'm assuming your name is Francis. In this example, why do the laws of physics give that path? And not another one? Why did the universe choose a parabola for the motion of a throne ball instead of literally any other path? The universe had an infinity of paths to choose from. And it's the parabola that comes out every single time Newtonian physics tells it's a, it's a parabola. But why is Newtonian physics correct? F equals MA works. But why doesn't work? We're not talking about laws of physics and equations of motions here, we're talking about why the laws of physics and why the equations of motions are the way they are. And the reason that we have the laws of physics that we do in the equations of motion that we do is the principle of least action if I throw the ball to you.

And it follows that parabolic path, I can calculate the action, the total difference between Connecticut potential energies along that path. And that's a number, it's just one number five. If it were to take some other path like the cork screw path around me or it zigzag goes out to the moon and hangs out for a while and then comes back that I can calculate the action for that path and the other path and the path where it twists around and it like uh uh you know, it describes your sky, writes your name in the space between us. Before reaching, I can calculate the action for all those different paths. The Parabola is the one that has the least amount of action. So if I want to know which path the ball will take when I throw it to you, I find the path that has the least amount of action, the minimal action. And that is the path the ball will take every single time it gets wilder. You don't even need a particular scenario for this particular ball.

And OK, calculate the action and, and calculate all these flight paths. I can just write down a general formula for the potential energy of an object in earth's gravitational. Well, which is just its height times the mass times g the gravitational constant. And I can write down a general formula for its kinetic energy. Like I said, it's just half the mass times the velocity squared. And without knowing anything else anything about the exact setup of the system? Just knowing that this is how we write down kinetic and potential energies that these are just the properties of an object. And I can write down symbolically any object has this kinetic energy. And this potential energy, I can calculate a formula for the least least action. I can write down the kinetic and potential energy. I can write down their difference. That gives me the Lagrangian, I can minimize this quantity using a mathematical technique called the calculus of variations. And when I do that in that general sense, not just a narrow specific sense of when I throw this ball with this mass with this velocity. And I calculate all the numbers for the action I get the parabolic path.

This is just a broad like any ball thrown anywhere or any classical motion happening at any time in the universe now and forever. This is the kinetic and potential energy. I can use some mathematical tricks to just write a general like what is gonna happen? What do I get? I get Newton's laws, all three of them. I write down a single Lagrangian kinetic energy minus potential energy. I apply the least action principle once. And what comes out of that is F equals ma and the rest of Newton's laws, I can derive Newton's loss from the principle of least action I can create Newton's laws. Physicists go nuts for this kind of stuff. Because it looks like we've uncovered something more fundamental than Newton's laws. Newton's laws tell you how objects behave. The least action principle tells you why Newton's laws are correct.

Why Newton's laws are the way they are. Why is it F equals ma and not F equals a half ma or MA squared or M minus a, the least action principle tells you why that works. The least action principle generates Newton's laws. You can use the least action principle to create from thin air entire laws of physics. All you need is the kinetic and potential energies and you're done. So when I throw a ball at you and you ask, why does the ball take that path? And not, not another, a high schooler or a college grad might say well because it's obeying Newton's loss. But now you can say something better because that is the path that minimizes the action. I know why F equals MA is correct because F equals ma is the only formula that satisfies the least action principle. And folks, we are just getting warmed up because the least action principle doesn't just give you Newton's laws. As long as you can write down the kinetic and potential energies of any system. In any scenario, you can get any equation of motion, you can get any law of physics.

You write down the kinetic and potential energies of a charge part particle moving through electric and magnetic fields. You use that to write your Lagrangian and apply the least action principle. You get all of Maxwell's equations of electrodynamics. You write down the kinetic and potential energies of quantum particles interacting by the strong and weak nuclear forces. You apply the least action principle. You get all of the standard model of particle physics. You write down the kinetic and potential energies of mass and energy moving in a spacetime metric that can deform. You apply the least action principle. You get all of the general theory of relativity. Ladies and gentlemen, you can write down a Lagrangian an expression, a mathematical expression. That's about half a dozen terms long. You can print it on a t-shirt and it perfectly encompasses all of known physics. You apply that least action principle to the Lagrangian that's half a dozen terms long.

And you can explain the entirety of the natural world as viewed through physics. All of it, it all rests on the least action principle. If we were to discover some new force of nature or new particles, if we wanted to know how objects behave according to this new force or with these new particles, we just write down the Lagrangian and apply the lease action principle. And now we know how it works. It's that easy. I mean, actually it's incredibly complicated but, but conceptually, it's easy. The least action principle is a machine. It allows you to derive equations of motion, it tells you why natural laws have the form and structure that they do. Why do electric charges behave that way? Why does the strong force do that? Why does General relativity work the way it does? Because they are all obeying the least action principle according to their Lagrangian, the least action principle is the mother principle behind all the physics. And it gets weirder. I can see you asking, I hear you, Paul.

It's pretty fun and wild. I get it. Let's go back to throwing the ball a simple example. I understand how the least action principle generates Newton's laws which gives the Parabola, which is the path the ball takes. But how does the ball know that baseball doesn't understand la Grand jas and the calculus of variations in kinetic and potential energies. It's, it's just a dumb ball. It doesn't even contribute to Patreon. That's Patreon dot com slash PM Sutter where all smart objects in the universe go to support the show. And I truly do appreciate it. How does the ball know to take the correct path? Well, for the ball, it just feels the force of gravity and has its initial momentum and at any point it's just reacting to the force of gravity. This whole Ranian and least action principle business works because well, because it does the kinetic minus the potential, the added up difference between that the calculation of the action and minimizing the action. Well, it just works because other combinations don't. But the least action principle seems more powerful because we can use it to explain why other forces of nature, like why are Newton's laws correct?

Well, because the least action principle is correct. But what about, you know, this whole quantum stuff, you know where things have fuzzy motion and everything's a probability like the least action principle. This this sounds great for classical motion. But we know that classical physics doesn't apply to quantum mechanics. So for example, a firm his least time principle he was examining light rays and how they reflect and how they refract as they go in different mediums. And he discovered that the beam of light always minimizes the travel time, it will choose the path so that it can get to its destination as quickly as possible. But how does it know that like a beam of light is a quantum object? How does it know it or the famous quantum two slit experiment? How does a quantum particle know which path to take when it's going one at a time? Like just how does it know these path? How do we translate this knowledge of the least action principle into the quantum regime which is full of fuzzy probabilities? The answer is Richard Feynman's insight into quantum mechanics, which we call the path integral approach or the sum over histories approach.

It deserves its full episode probably as a series in quantum mechanics that I swear I'm gonna get to one of these days if I let's skip throwing a ball because we gotta go into the quantum world, let's say, shoot an electron at you. When we're talking about these different actions and the different paths, we can apply the same language to an electron. Uh One path is it just shoots right at you. Another path is it dances around you for, for a little bit flirts with you. It tickles you right behind your ear and then lands in your hand. Uh The one path is it goes up to the Andromeda galaxy hangs out there for a million years and then comes back, we can envision all possible paths that the electron can take. And in quantum mechanics, everything's got a probability, everything has a chance that it there's nothing outlawing the electron from going to the Andromeda galaxy before coming back. There's nothing that says it can't, especially in quantum mechanics. But the probabilities of these paths you actually have to calculate these probabilities. And the probability of the electron taking one of these alternate path is given by a particular quantum mechanical quantity which I'm not going to discuss yet because I'm gonna save it for a series on quantum mechanics.

And Feynman discovered that one way to approach well, why does the electron take this path instead of any other path is once you assign and calculate the probabilities of all these other paths, all the other probabilities, all the other paths cancel each other out and what's left is the path that's given by the least action principle. This is a quantum version of the least action principle where you do envision the electron taking every single possible path. But then all those little paths have probabilities associated with them, chances of them happening and then they end up canceling just washing out. And all that's left is the path that the electron takes that you would expect from the least action principle. This is how the quantum world connects to a classical one. And how fuzzy probabilities transform into motion quantum mechanics is happening all the time. But far from the classical motion path given by the least action principle, all the other pastures cancel each other out and you're left with the traditional motion that you're used to and where you start to get all that interesting juicy quantum stuff is that in very delicate quantum situations, the probabilities for some alternate path start to increase.

And so electrons start to get to the freedom to travel some alternate paths because now the probabilities outside of the classical least action principle path start to become allowed starts to get some new ideas. But again, that's more for a quantum mechanics episode. Now Richard Feynman took this all the way and he said when I literally throw a baseball at you, it is taking every single possible path to get from me to you. But all those other paths quantum mechanically cancel each other out and you're left with just the classical motion. Not everyone agrees that that's a valid way to look at it, but it's a fun thing to think about, but no matter what, without the least action principle. Well, you wouldn't catch the ball. Thank you to Kevin O on email at Ken Richards at Twitter and Scott M on email for the questions that led to today's episode. Thank you to my top Patreon contributors this month. That's Justin G Chris L Barbara K Duncan M Corey D, Justin ZN Age, Andrew Eia, Aaron S Scott M Rob H, Justin Lewis, Paul G, John W Aaron J, Jennifer M Gilbert M Joshua Bob H John S Thomas Dee and Michael R.

Seriously, I, I can't thank you enough for all the contributions. If, if you're wondering, what can I do to support this show Patreon dot com slash pm, Sutter and a small contribution every month. That is the best way to support this show. Also, you can support the show by giving reviews on itunes or your favorite podcast downloader. I really do appreciate it. Helps other people find the show and just sending me questions, really, the more questions, the better I love, love hearing what all of you are curious about. I would have never imagined in a million years that I get to do an episode on the least action principle. But there you go, you asked and you received, you can send those questions to ask us spaceman at gmail dot com. The website is ask us spaceman dot com. Hit me up on social media. I'm at Paul Mats Sutter on all channels and I will see you next time for more complete knowledge of time and space.

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