How physics becomes advanced

In today's video, we will start with the most basic of Newtonian mechanics and then build up and up and up, overcomplicating things more and more until we reach the Lagrangian in order to understand what is an action. So, in other words, we'll go from basic physics to a setup for advanced.

[Music]

It's okay. First things first, we need to start with the Newtonian mechanics. Yes, I know it's basic, but we're trying to showcase how physics changes as it becomes more advanced. We need to start with the basics. So, let's start with the laws of motion.

So, the first law of motion, what is it? Well, it states that a body remains at rest or in motion at a constant speed in a straight line unless it is acted upon by a force. Wow, that that Newton guy really was a genius. He effectively just said that if an object isn't moving, then if you don't move it, it won't move. Yeah, almost as good as when Einstein said that light travels at the speed of light. No, but jokes aside, this is important because this introduces the concept of a force. So basically if you have something like a cube and you'd like to move it, you need to apply a force at which point it will keep on moving forever and ever and ever, stopping because of the or in motion at a constant speed in a straight line. So this cube will never slow down, right? Well, no. That's because of friction or rather the force of friction. That's how we do things in Newtonian mechanics. If something is moving or rather if we want to change that movement, we do it using the concept of a force.

But that's only one-third of Newtonian mechanics because next we have the second law, which states that at any instant of time, the net force on a body is equal to the body's acceleration multiplied by its mass, or equivalently, the rate at which the body's momentum is changing with time. So basically, if we apply some kind of force onto our cube, then if we know the cube's mass, we can figure out by how much it will accelerate. That's simple enough. Elementary school physics level.

But more importantly, let's ask the question of what is acceleration. Well, thanks to Lenet, don't tell big N I said that, we have this cool tool called the derivative, which basically works like this. Imagine that you have a function of position in relation to time. In this case, we can see that the position of our object would start at zero, then go up to one, and then fall down to zero. But what about velocity? Well, if you think about it, velocity is just the change of position per unit time. Or in other words, if you have some kind of point on our graph, then we can set another point like this. Then what would be the velocity between these points? Well, it's meters per second or distance per time. Or in other words, this link right here by this link right here, which mathematically we can say is just the slope. So basically, the higher the slope, the higher the velocity. The lower the slope, the lower the velocity. If it's flat, there is no average velocity. And if it goes down, well, negative velocity.

But there is something tricky about it. As you've probably noticed, right here between these two points, we are changing position. Most notably, our cube goes up and down. But the average velocity between these two points is zero. Why is that? Well, that's because we're talking about average velocity. What we want instead is instantaneous velocity. And that's why we need derivatives. Because derivatives basically just bring these two points together to create a slope at a point. I am glossing over it quickly, but the point is this slope right here represents our velocity. The higher the slope, the higher the velocity.

And another way for us to represent it, we can just make another graph. So first we have the graph of position. Second is the derivative in relation to time, so velocity. And then we can take a further derivative on velocity and get acceleration. Even more, we can keep going forever. Derivative of acceleration to get jerk and so on so forth.

But how does that relate to the second law? Well, it relates to the second law because F=ma can also be written as F=m times the second derivative of position in relation to time, or alternatively, m times the derivative of velocity in relation to time. Or finally, if we just multiply the velocity by mass, momentum in relation to time. So just to tie it all together, momentum change is equal to the force applied. So if you want to change the momentum of an object, you need to apply a force on it. Which once again is why when our cube is traveling forward, it won't ever stop unless we apply a force because in order for it to stop, its momentum would have to change.

And so there we go, only one law left. The third law: if two bodies exert forces on each other, these forces have the same magnitude but opposite directions. This law is effectively the whole "every action has an equal opposite reaction." So basically, if we really quickly go to space and have a make it with an apple, then we can have make it throw the apple. Make it will apply a force onto the apple. And so according to the third law, the apple will apply a force onto the make it. So they will both push themselves away. This is fairly obvious, I would say.

But the more interesting thing here is what happens when we take those forces and we calculate the total force that happens in our interaction. The total force would just be the make it to apple force plus apple to make it force. And since apple to make it force is just equal and opposite, we can effectively note it down as force minus itself, which equals zero. And since we've just learned that force is equal to change in momentum, we know that this means that the total change in momentum is always equal to zero. Or in other words, momentum stays conserved.

There we go. Newtonian mechanics. These are the basics we'll start with. But let's solidify our understanding a bit. So first, the simplest one, a free particle. Free particle just means a particle without any kind of potential force, which means that there is no force acting on it effectively. And since we know from before that force is just change in momentum, that effectively means our momentum doesn't change. So it could either be our particle standing still or moving forward at a constant pace. Either way, the momentum isn't changing.

If we were to graph the energies of our free particle, we'd set the potential at zero and kinetic at, well, whatever the particle is at. Now, since there is nothing happening to our particle, this just keeps going forever. Yep, that's the entire particle. Since the kinetic energy of a particle is equal to one-half mv squared, I won't get into why that is, left as an exercise to the reader. We can say that the total energy of a free particle would be equal to one-half mv squared plus zero, where this is the kinetic and this is the potential.

Then for the slightly more complicated example, let's talk about the harmonic oscillator. Despite sounding way more advanced, it is still relatively simple. It is effectively just like the free particle, but this time in our equations of total energy, we'll have the kinetic plus the potential. What would the potential be this time? Well, in the harmonic oscillator, we'd have the force acting on it be equal to minus kx. So, basically, whenever it moves away from the middle, a force directly proportional to how far it went will act on it. So, move right, it gets pushed left. Move left, it gets pushed right. It's called an oscillator because if we let those forces act on our object, it will oscillate. Yep.

Now, there are a lot of things that can be approximated with a harmonic oscillator, like an awful lot. But for now, let's keep it intuitive. So, I'll just give you some examples, like a spring, where if you move it to the right, it wants to stretch, moving the ball further into the middle. But if you squish it too much, it wants to spread, pushing it back. Another example could be a ball on a parabola. In that case, we have our ball. And if we put it somewhere on the right, it will want to roll down and to the left. And once it rolls to the left, it will want to roll back to the right. But the most common example of a harmonic oscillator would have to be the pendulum. Pendulum is one of the most classic harmonic oscillators. So, it should be a pretty fine example to continue on.

And that's because it's finally time to get back to our equation and ask, what is the question mark? Well, that's a good question. In order to get that energy, all we need to do is just integrate the force. And congrats, this is our kinetic energy. I won't explain how integration works because it's boring if you're just talking about some equations. So, instead, let's go back to our pendulum and let's graph its kinetic energy and potential energy. As you can see, we have three lines. The yellow-greenish, which is kinetic, the purple, which is potential, and blue, which is the total. As you can see, when the pendulum swings to the extreme, the kinetic energy reaches zero because it's no longer moving, but the potential reaches the maximum because at this point, it's pulled back with the strongest force. Once it gets back into the middle, the potential vanishes because there is no longer anything pulling on it, but the kinetic is maximum this time because it's traveling the quickest it could. Then it swings back and so on it goes. And as you can see, our blue line, the total energy, stays constant because of the whole "energy cannot be created nor destroyed" thingy.

So there we go. That's how it works in classical Newtonian mechanics. If we wanted to create a more general case, we'd have our ball and our total energy would be equal to the kinetic plus potential. While kinetic is relatively easy to get, potential is way more complicated in a real world because potential would mean gravity, friction, electromagnetism, air resistance, all of those things. That's why we usually note it down as V, so just a function, because actually expanding it would be a bit much.

And so there we've done it. The basic background of Newtonian physics took us long enough. Now we can finally up the ante by talking about the Lagrangian and the Hamiltonian, I suppose. So basically, there are two ways you can combine the energies: either adding them or subtracting them. If you add them, that's a Hamiltonian, which will become important once we get to quantum field theory, but isn't really right now. So let's talk about the Lagrangian. The Lagrangian will effectively be this: we plug in some kind of coordinates and what we get is the kinetic energy at those coordinates minus the potential at these coordinates. Except these coordinates might not be exactly what you think.

Okay, so going back to our pendulum, if we were to apply this Lagrangian idea, then we need to plug in our coordinates. And so you might be tempted to think that this Lagrangian will be dependent on the x-coordinate and the y-coordinate. But in reality, we don't actually need both of these coordinates. Like, yeah, theoretically we could define each point based on these coordinates, but that's a completely unnecessary thing to do considering how both kinetic and the potential can be described by a single coordinate of the angle. This is a generalized coordinate because you effectively just use the constraints placed on our system, the fact that the ball is always a constant distance from the top of the pendulum, to reduce the amount of coordinates.

And okay, the concept of a generalized coordinate is a bit annoying to explain because usually, when a dry mathematical concept becomes too complicated, you can just bring it into real life and as such, it would just be way simpler to understand. But when it comes to generalized coordinates, however, the exact opposite is true. Effectively, this just means the minimum amount of variables we need to describe a system. So, for example, if we go back to our ball on a parabola just oscillating back and forth, in this case, how would we describe the position of this ball? Well, we could do it with Cartesian coordinates like so, but that's complicated, as you can see, confusing. How else can we represent it? Well, we can also represent it with polar coordinates, where this blue line will represent the angle, the yellow the distance. And this is what we get there. But that's still not generalized. Why? Well, it's because there's another way for us to describe it, and that's with just the position of the ball on the line. Now, the ball is oscillating from minus one to one. And we can just describe it as that coordinate. So, here's the ball described with the generalized coordinate. Here is it described with the Cartesian. And here is with the polar. And the whole point is that if you want to describe the system, you can do it with these two, these two, or this one. And if we can describe it with one, that means the doubles carry redundant information. So we only need this one.

So basically, once again, just to make it as clear as possible, in order to find the generalized coordinates for any system, you need to find the minimum amount of values to describe that system. So if you have a train running on some train tracks, XYZ can describe it in three coordinates, but the distance along the track can describe it in one. So that's the generalized coordinate. Simple.

And with that in mind, let's get to our phase space. Phase space is just effectively the combination of our generalized coordinates. So, for example, Q and Q dot. Q in this case is the generalized coordinate. Q dot is the change in the generalized coordinate in relation to time. Now, that really is it. But let's go over an example to really solidify our understanding. Back to the train. If we wanted to truly describe the train with Cartesian coordinates, we need six: position x, position y, position z, velocity x, velocity y, and velocity z. Now, these are all Cartesian coordinates, but we're dealing with generalized. So, position along the track and velocity along the track. And now, if we look at the phase space, we can place a point on it. And this point will perfectly describe our system. So, if we move it left and right, we have the train move along the track. If we move it up and down, we'll have the velocity of our train change. That is the phase space. And right here, I will move it around some more. That's because I feel like it would really help if you actually focus on this animation, realize how generalized coordinates work, and just appreciate the amount of effort and time I have spent on this single animation that I'm just going to show once in this video and then throw it away to never see the light of day again.

Okay, now let's do something festive. Go to our phase space and plot de Broglie on it. And with this plot, we can have some fun. Fun with actions. So let's imagine a point A and point B and draw a line between them. Over the course of this line, it will go up, it will go down, it will go all around. But the point is, it tells us how the Lagrangian changes as we move on this line. But we don't need to go in a straight line. We can curve it a bit. And if we curve it, we get a slightly different path. Not only is it longer, but its height changes as well. So right here on the left, I'll draw seven different paths and on the right, I'll represent their heights. Now, what action effectively does is it just ranks them. You can think of it as basically assigning them a score of "annoyance," which says just how annoying a path is to follow. This straight path isn't annoying to follow at all. It's short. It doesn't change the height much. It's simple. This elaborate one with loop-de-loops is really annoying to follow. The height changes, it goes up. It's long. It's awful. The ones in the middle are slightly more passable, but they have their own problems. But the point is, more passable, more passable.

So now, let's ask another question. How likely is it even to travel from A to B? Well, it would try to keep it at least annoying. So, we would prefer to travel with the least annoying ones. So what we can do is we can add all of these paths together, weighted by the action, and what we get is like a general trend of least annoyance. And what this gives us is a kind of like a sum of annoyance. So basically, kind of like a score of how annoying it is to go from A to B. And this right here is the path integral of A to B. So basically, once again, we have some kind of two points. We draw a path in between them and we rate it using the action described by this integral. This integral effectively just says the total Lagrangian over the path. The more Lagrangian there is, the higher the path action. So a path like this has a high action because it's long and high, and this integral ends up being high. A path like this has a smaller action because it's shorter, because it's slower, and this is its action. And as such, this path will be preferable to this path.

And with that in mind, let's ask another question. How does the action change? But first, a brief disclaimer. That's because you might have noticed that one week ago, me and Remco have done a live stream covering the exact same thing as we've covered in this video. And there's a reason for it. Here's the whiteboard. And you can see that going left to right, top to bottom, it is exactly chapter by chapter what we've covered today. And that's because this episode is actually a part of a series that we'll be starting where we'll start with basic physics and eventually reach quantum field theory. And when I say quantum field theory, I don't mean basic quantum field theory, which is what I've covered on this channel so far. No, I mean proper quantum field theory. But it's going to take a really long time to get there. Hence why we will actually revisit the path integral in the future. So if you don't understand completely, don't worry. I just wanted to mention it because the last chapter is actually Remco's chapter from the live stream. This will be just an edit of a live stream. Remco doing the explanation with all matter's art in the corner. Hope you enjoy.

Let me actually just write the action first. So, in case we forgot it, I'm going to use a bit of abstract notation. The action is the integral of the Lagrangian with respect to time. What we are going to do now is we are going to make a change in a path. This is because this is a path integral, it represents a path in this phase space. So instead of having to deal with a normal function, you have to deal with x, we have we now have to deal with paths, and this is slightly different, but it's not that intuitively impossible to understand. Let's just say we have a path. This is our x of t. This is a path that takes place in this phase space. What we now want to do is we want to vary it a little bit. Like what we usually do with integrals is we have a point, we go slightly away from the point and see how it changes, how the function changes with respect to that point. We're now doing the same, but with a path. So what we're doing is we're slightly going away from this function. So we are, it doesn't have to have the same shape. It is just the path needs to be close to it, significantly close. This is our original path. We are going to use one of these white paths, and I indicate this by the normal path, but we add a little bit, and this could either be positive or negative. Doesn't matter. The main important thing here is that at the end points, the paths are the same, which means that the variation, as it is called, the deviation from the path at the end points, it's going to be zero, holds for both end points.

Now we can actually see how our action is going to change when we take one of these other paths instead of the normal one. In this case, for simplicity, let's just entirely forget about time. It's a function of x and v, which means we are changing the path. We are also implicitly changing the, well, we don't even care what happens with the v. We just know it's a function that depends on these x and these v. So if you want to see its change, we will need to see how it changes with respect to x and we will need to see how it changes with respect to v. And we all want this to be zero because that's what it means to be an extremal value. That's what it means to be in a minimum or a maximum.

Okay, pause because what Remco just said is pretty important, but it's kind of hard to understand maybe without the proper visualization. So, basically, if we go back to our environment of hills and valleys, well, these extreme points, what are we effectively looking for? Well, we're basically looking for a point where the gradient is zero. So, basically like a flat point, which would either mean a peak, it would be in a valley, or a saddle point like this. So, here's our environment, right? And you can consider it as some kind of just abstract space. The exact space doesn't really matter. What matters is that in this case, I have it colored in so that the height is represented also with color, where the brighter it is, the higher the height. That's all pretty obvious. But what I can do with a slight tweak is I can change it so that instead of it being colored by the height, it can be colored by the gradient like so. And right now, all of these black lines are basically the areas that are flat in one way or another. And so effectively, what we're looking for with the zero is all of the areas where it's pitch black, where if you can, if you imagine like an abstract ball rolling down a hill, then this would be these would be the places where the ball would come to a rest because the area is flat and so it's no longer rolling down. I just thought that this kind of brief visualization could help to visualize it a bit better. So if we look at the mountain, you can see the top of the mountain. It's pitch black because it's flat over there. Right here, it's really steep, and right here in the valley, it's black again.

Anyway, back to the explanation. So what we can now do is we can rewrite this a bit, where the variation of the velocity, we can just write this as the time derivative of the variation of x. We are assuming that variation commutes with the time derivative. This is a fair assumption. We are now going to use integration by parts, but effectively what you're doing is you are changing the integral of the derivative on this variation. You're putting it instead on the term before it. So the derivative of the Lagrangian with respect to the velocity, and you still got this one, and you can do this at the cost of your boundary conditions. So you will need to compensate for these. What we know is, so this term needs to be evaluated at A and B. But what we know is that the variation at the end points, they are they're glued to the point. So this entire term will be zero. So we don't have to worry about this. Very convenient.

What we can then do is write one more step and then we're done. So we're almost there. And what we end up with is this. Now comes the main crux of the problem, which is I have not specified which variation. So it could be any variation. So this expression needs to be zero because that's what it means to be an extremal value. It needs to be zero for every single variation. Now, if you have an integral that needs to hold for every single value, then this needs to be zero inside the integral. The integrand needs to be zero. This is the result we were after, namely that the derivative of the Lagrangian with respect to x minus the time derivative of the derivative of the Lagrangian with respect to the velocity. This needs to be zero. This is the very famous Euler-Lagrange equation.

>> Oh man.

>> Oh, you did. Oh, you didn't. Oh, so I went for a lunch break, mate. I don't know what you're >>

And I guess if you, you don't have to understand where this equation is coming from. It, as you can see, it required some intuition. Does that matter? The main important thing is we have this equation now. So we got the Euler-Lagrange equations. Let's see what we can do with them. We have seen two examples already in the Newtonian case. The first one is the free particle. So let's just go back to our free particle. We know that the Lagrangian is the sum of, is the kinetic energy minus potential energy. I'm going to write this as one-half mv squared. Don't worry. Yeah, exactly. Minus zero.

So, let's actually check this equation now. We can do this now. We can check the, we can just, we have the Lagrangian now. It was very easy to find it. We just needed a kinetic term and a potential zero. You can argue we knew this already. Yes, but that's because we're fact-checking right now. So, let's just use this Lagrangian. Let's first of all, partially differentiate. We see that this means setting all the v to a constant, which means this is just zero. Now, the other one with respect to v, this is going to give us mv. This factor of two is going to negate this factor. So we're just getting m from there. So what the Euler-Lagrange equation is telling us is that minus the time derivative of the second part equals zero. Now, what do we know about the derivative of velocity with respect to time? That appears to be the acceleration. So what we get is mass times acceleration equals zero. So what we saw is exactly the same equation. So until now, our formalism works out perfectly. That is lovely. So once again, we get that the force equals zero, even though we didn't even notice, we didn't even use the notion of a force equal to zero. And we can solve this equation and just find the same linear dependence of position, once again, on time.

So now let's make it a bit harder and go for the >> In other words, as you're noting down, basically we just figured out a thing about the free particle that we assumed on the very beginning, but now just from the fact that we knew the energy of a free particle, we were able to figure it out without the notion of force. >>

Who remembers the energy, kinetic or potential, from the harmonic oscillator? The potential energy, exactly. It's going to be one-half kx squared. We can fact-check this with the previous one. We are going to use the Euler-Lagrange equations again. I'm going to highlight them because that's how important they are. So let's just go right back to it. Derivative with respect to x is giving us, in this case, not zero. Well, we get minus kx. The two is going to cancel the half. The derivative with respect to v equals, once again, just mv, which means that the time derivative of this is going to be ma again. So that changes. So what you can already see is that from the kinetic energy, we get most of the time the mass times acceleration. So Newton's law and the potential energy sort of determines the force in this case, exactly determines the force. So let's combine them. We got minus kx from our derivative with respect to x. We then get a minus and then we get ma, and this needs to equal zero. So if we rewrite this, we get exactly ma equals minus kx, and this is exactly the force that we found before. Once again, perhaps we can show the potential of that.

>> So some like like like a hat is what you want, like a very specific kind of hat.

>> Yeah, I was thinking Mexican.

>> Well, in that case, say no more.

>> And where should I place this this point then, Ramco? Would it be, would the center be fine?

>> Perhaps let's just do it in the center.

>> And perhaps perhaps let me do some improve right now.

>> Yeah, we did just go from we did just go from classical mechanics to creation of the universe in the space of three-ish minutes.

>> Yeah, this was also not planned. So >> kinetic energy like this.

>> Oh,

>> just some potential energy. I you can fill in your e to the x here. You can fill in your e to the ix here. You that's a phase. Why would you even put a phase in the potential energy? You can fill your x to the pi in here. It it's it's all left to interpretation. And this is why this formalism is so powerful. I can make it as global as I want. So let's just solve it. We we know this one. We can simply, we can simply just write it. It's minus the derivative of this. Now I can't go further because I don't know what this means. But in case of the e to the x, we would have e to the x again. This is once again mv, which means that if I am already, I'm doing it a bit faster because we have seen this a couple times. What we get is ma equals minus Vx, and this is potential energy, or well rather, the V is potential energy. So we're taking the potential energy with with the derivative with respect to x. And if you just go back to our harmonic oscillator. So what we did is starting with potential energy and we just defined force as Pazan was mentioning. And this is deviously why it didn't matter which functions you were going to say because I was going to generalize it anyways. I'm sorry. What you can do with this formalism is any function of x can just be plopped down. You will get a force equal to the minus the derivative with respect to x. But what you could do is indeed, as was mentioned in the chat, you could think of potential energy in the form x squared, v to the fourth. So you could couple, well, velocity and position in physical terms. And you just need to do some integrations, well, no differentiations, and you will find your equation of motion. But usually they are difficult to solve or impossible to solve in that, like if you want to find x of t equals something, it will not be analytically possible.

For now, I think it's really, let me give like a short, like, let's just say a short cliffhanger for next time. So if we have this Lagrangian formalism, kinetic energy minus potential energy, or actually, let's just consider the Hamiltonian formalism in this case, which means the plus. What we then could do, let's just first write it in terms of, actually, let me write it in terms of momentum in this case, and we also have some, we also have some potential energy. We have talking about Hamiltonian, that's why there's a plus sign. What we could do, and this is perhaps a really important step to take, is if we have this Hamiltonian, and we just by some sign of divine intervention >> put some little hats. My goalie, the mad lad, has put hats onto these variables. What does it mean? Well, I suppose you'll have to wait around two weeks to find out because in around two weeks, we'll do the next live stream, and in around three, the video should come out. Terms and conditions pending. The exact timing might change. We're still scheduling it. But there we go. That was it.

Special thanks to Remco for, it doesn't even feel right to say helping me with this video. I would say more like co-making this video. Remco did all the research. I did the script writing and video making. So, thank you so much, Remco. And everyone, you really should tune to the next live stream because it's a lot of fun. Thanks to all matter for this lovely art commission. It's, it's honestly incredible, and I do adore it. And if you'd like more, I have other physics videos or chemistry videos or computer science videos, or I have a whole other channel where I just do like this. Exactly. This guy is my favorite YouTuber. What a curious time for you to join back in, Remco. I was just, um, working. So, you'd like to know how my mask works, or if you'd like to know how I can use this to crack my cards eventually, then I would recommend my second channel where I do more technical applied science. Also, I have a Patreon in case you'd like to support me and my lavish lifestyle of living below the Polish minimum wage. Thank you all so much, my beloved patrons. Without you, I really wouldn't be able to even keep the lights on, let alone make these videos. And also, I have a Discord server in case you'd like to chat with me or the man himself. He has a whole Discord channel over there where people ask him questions all the time, and he answers. And for now, that'll be it. Thank you so much for watching and have a great day. Bye. That actually, next chapter will be all about hats. So, you know, I need to get myself ready.