 Now, let me give you your first tiny, tiny, tiny little example of a symmetry principle. Symmetry, symmetry principle. So I'm going to state a kind of symmetry. I'm going to invent one. And as I said, in any way or way, there's a symmetry principle, a conserved quantity high. So let's consider the system where all forces are radial. Can I give you an example of that? Yes, here's the sun. And there's a gravitational force in all the forces, no matter where you are. The force is going to be towards the center. It's going to be towards the center, no matter where you are. All the forces are radial. So they sit on this radius. So anyway, there is a symmetry here. And I call it a radial symmetry. Now, it's difficult for me really to put in words what is symmetrical here. But it should be plain to understand that if I create this system of radial, all the forces being radial, there's no force anywhere else here that points in another direction. All the forces in this area points towards the same spot. That's the symmetry. I have set a symmetry principle. Now you might remember angular momentum equals the cross product of a radius vector and a momentum vector. Angular momentum is this cross product. So if this is a vector and this is a vector, you should know or you have to know how to do the cross product of two vectors, a special kind of multiplication. It gives us another vector as an answer as opposed to a scalar which you get through a dot product. And you have to have done some linear algebra. You must know how to take a vector and do a cross product with another vector. It's very easy to get the direction difficult. You have to know how to do it to get the magnitude of this. The direction is easy. If you form a plane with these two vectors, one points in that direction, other one in that direction, I can make a plane of this and the resultant will always be orthogonal, perpendicular to that plane. First I'll point in the R direction with my fingers, then towards the direction of the P and my thumb with the right hand rule will point towards the positive direction for the resultant vector. Now I have a system of time evolution and I want to know what the LBT is. What is the rate of change of my angular momentum? So how do I do that? How do I take, once again you'll have to know multivariable calculus, how do I take this? Well it's the same as if I have the f of x and the g of x and they multiply with each other and I want to take the d dt of this. How do I do that? Or I can say the d dt, you might remember this, better u and v of the product rule. So it's going to be u prime v plus u v prime. Or f prime of x times the g of x plus the f of x times the g prime of x. So the product rule, the same applies here. So what you're going to have is this dr dt cross product with p plus we're going to have r cross product with dp dt. Now I'm holding this under the constraints of this symmetry principle. In my system, all forces are regular point. No matter where I am, I'm pointing towards the same spot and there is some symmetry in that situation. So what is rate of change of position? Well that's velocity times, what is momentum? Well that's mass times velocity. So this is a cross product of two collinear vectors, well actually the same vector, one is just multiplied by a scalar, the inertial mass. So it's the same vector, one is a different magnitude than the other. Remember if I do the cross product, if I have one vector, a mathematics cross product, another vector, let's make it vector r, that would be the norm of this vector times the norm of this vector times the norm of the other vector times the sign of the angle between it. So that's the same vector, the angle between the same vector that's of different sizes is zero, the sign of zero is zero. So here I have zero, the zero vector I should say, plus what do I have on this side? What do I have on this side? Well what is, we just saw in a previous video, what is dp dt? The rate of change of momentum is the force. That's force, isn't it? Now if I'm under this radial symmetry and I have any position here, the position vector is there, but from this spot all my forces point towards the same point from which this position vector came, so they are what do you call collinear vectors? Collinear, they point in the same line, they might point in opposite directions, but they're in the same line, so the angle between them is 180 degrees, the sine of pi radians, and what is the sine of pi radians or the sine of 180 degrees is also zero. So this cross product here is also a zero. So the rate of change d dt, d dt of L equals the zero vector, it's zero. And the only way that you can get a zero for first derivative is if that is a constant. I need the derivative of a constant, let's put it C, I need the derivative of a constant is zero. What is the dx of x squared? It's 2x, but what's the d dt, or d dx I should say of the constant 3? That's zero. That means angular momentum is a conserved quantity. Under the symmetry of a radial force, if I apply a radial force on a system with angular momentum, if I apply a radial force to that, I will not change, there will be no time evolution of this angular momentum. Angular momentum under this symmetry is a conserved quantity. So I have a symmetry principle and lo and behold, what hides a conserved quantity? A conserved quantity. We started with a symmetry principle. Wherever there is a symmetry principle, I will find a conserved quantity and I have to show this conservation of angular momentum. The ice skater is spinning around, arms outstretched, goes at a certain angular velocity, pulls the arms in, goes a lot faster. There has to be conservation of that angular momentum. Nature does not let it disappear somewhere, it can't be made, it can't be destroyed. If there is angular momentum in that system, that has to be conserved, it can't change. The cross product of this has to remain some constant for that system. So I have a star, it spins around, at the end of its life it's a massive star, becomes a neutron star, much smaller, lots of mass still, it has to go a lot faster. There's conservation of angular momentum, the skater pulls the arms in, goes around a lot faster because there's conservation of angular momentum. I have stated the symmetry principle as conserved quantity hides and I will show you later on how from this I can now derive a force, a law of nature. Law of nature. So instead of starting there, where we start usually in physics and work our way this side, at least to this point, we're now being proper physicists, we start here and we end there.