 Now, what I want to show you in this video quickly, well, I'll say quickly, but a very simple way for you, again, an example to start thinking like a physicist, can I derive Newton's third law from a symmetry principle? So we're going to have a symmetry principle, we're going to do a thought experiment, so that's exciting as well, Einstein is famous for his thought experiment, we're going to do a thought experiment, it's called virtual displacement by D'Alembert, then we're going to see that it's a conserved quantity and from that we're going to get a law of nature. So this is the proper way of thinking. Since the time of Einstein, since 1905, we know that we don't start with the laws of nature, start with a symmetry principle. We are going to accept two laws of nature though, we're going to accept them as being true and that's the second law and once we accept the second law, we have to invariably accept the first law as well, you cannot have the second law without the existence of the first law, equilibrium in a translational space, you can't have that. So the symmetry principle we're going to have is, let's look translational, translational invariance, translational invariance in homogeneous space, homogenous, homogeneous space, so that's my symmetry principle. What I'm saying is the following, translation means I move an object, a system of particles from one location to another location, I translocate them, translational, you get a rotational, you get all sorts, here we're going to translate them and if I have a system of n particles, I want that also to be a solid crystalline structure. So the inter particle distances and relative positions also remain intact, it's not gas atoms or molecules that are moving around, invariance, invariance is just a fancy word meaning there's no change, there's some aspect of it that doesn't, of something that doesn't change, okay so there's no change happens, okay, of something that I am referring to. Homogeneous space that just means wherever you are as far as position or concern, from a macroscopic to the tiniest submicroscopic properties of that space does not differ, so that's not really true of time space, space time I should say because you do get a warping due to large mass, et cetera, et cetera but we're going to have homogeneous space in other words all properties, all properties of the space is exactly the same aspect of a position. So I'm going to have this translational invariance in homogeneous space that is a symmetry, there's some symmetry to it, there's a beauty in it, it's the same no matter what you look, where you look from or whatever, it's a symmetry principle, there's no other words to use for it. So I'm going to have translational invariance, so it means something doesn't change because of this movement in space, so that's my symmetry principle. Now I'm going to make use of a thought experiment and this is, now I can say I can spell, so we're going to have that, we're going to have that, it's called virtual displacement, virtual displacement. So what I'm going to do, I'm going to imagine the system of n particles, n particle system, n particle system and I'm going to move it from point A to point B to there and there is my vector R, that's my position vector and I'm going to move it from there and the length of this is some distance delta S, some distance delta S and let's make that a vector as well but it's a distance and a direction. Okay but something is not going to change in doing this, that is the invariance, the translational means I actually have it there, then I have it there, the space is homogeneous, so all the properties are exactly the same as all the properties there and the virtual displacement is I'm going to consider a thought experiment whereby I have the object there, 3D letters so the atoms don't change position of the relative to each other and I just translate it from there to there but it's instantaneously, no time, not a limit as time goes to zero, no time. So it's a thought experiment, I can't ever do this in reality. If that happens, the other thing I should say is an n particle system in a closed system, n particle system in a closed system so there's no external forces other than the forces in between the particles. So if I let this thought experiment, that just means that the work done should be zero, no work is done, instantaneously it's there, then it's there, so this delta W is zero and what is work? It's the scalar product or the dot product of force and distance. What are the forces involved? Now my little thought experiment is a closed system so the only force is involved is the sum of all these internal forces, k equals 1 to infinity and that's a dot product with delta s. So I have that dot product, now I can write that summation a bit better, I can really write this as the sum of k equals 1 to n of the sum of, so for instance i equals 1 to n of f, let's make it ki, yeah I have i that's not equal to k, okay I rewrite it like that, doesn't matter how you write it to me but you can say this is the force on the kth particle by the ith particle, just think about it as n particle so every particle will have other forces on it, n minus one other forces, there are n particles looking just at one so n minus one are all there on the left, they all have an influence on this particle. So I'm just summing them, so I start with the first particle and I can't have i equals k so it can't be 1, 1, that one particle is not going to have a force on itself. So particle 2 on 1, 3 on 1, 4 on 1, 5 on 1 and then start with particle 2, 1 on 2, 2 on 2 can't happen then 3 on 2, 4 on 2, 5 on 2, so this is the rewrite of that just to be proper. But that's a dot product and it's going to equal 0, okay, through my thought experiment. And the surgery principle that I've got on there, now how can that be 0? One thing, one method for it to be 0, we remember that the dot product, the dot product between two vectors, if I have vector a, dot product vector b, what is that? I can also write it as the norm of vector a and the norm of vector b, in other words the length times the cosine of the smallest angle between them, okay. So for cosine for that to be at 0, I have to take the cosine of pi over 90, so the angle between those two are 90, but I don't want that, I want invariance, translational invariance in homogeneous space, in other words, I want to arbitrarily be able to make this move, this b needn't be there, b might be there or there or there, so I don't want to constrain myself to this pi over 2 radians or 90 degree angle between the two. So I can't consider that angle to be the cause of this dot product being 0. I also want to actually move it, there must be some translation, I have to translate it from one position to another position. So this can't be 0, I don't want that to be 0. So the only other option available to me to have this little algebra being 0 is to have the sum of all the forces, the sum of all the forces 3 equals 0. That's the only way I can do this, that's the only way I can do this. Now, think about it, we are now going to accept Newton's second law and then biopsically we accept Newton's first law as well where we have the fact that f equals dp dt, rate of change, rate of change of momentum, Newton's second law. I'm going to have that and I'm going to plug this into my equation. So instead of that, I'm going to have the sum of k equals 1 to n of dp dt, dp dt of k there equal to 0. And if I sum all of those pk's, I can call it one, probably one big p, capital P, you've seen us do that before. And the only way that a function like that p can be 0, its first derivative at least can be 0, if p is a constant. If p is a constant and they suddenly jumps out my conserved quantity, momentum is conserved. Now let's consider this to be n particles, let's consider this n to be 2, n to be 2. And so I know p, if p is then equal to p sub 1 plus p sub 2, then if I add those two together, it's going to be a conservation of linear momentum. So number one is from the symmetry principle, translational invariance and homogeneous space. I can show also that there's a quantity in that conserved quantity as linear momentum. Okay, so if there's only a two particle system, remember now if I have dp, which is all my k's all put together, the whole sum, dT. That is now dp1 dT plus dp2 dT. I'm just taking the first derivative with respect to time on both sides. I'm using the property of derivatives that I can just take, take it on both sides of that plus sign. So, and that's going to equal 0. We just saw that dp dT is going to equal 0. So this is the 0, in other words dp1 dT equals negative dp2 dT. Or if particle 1 equals negative if particle 2 or the equal and opposite, the force on one is equal and opposite the force from the other. Apologies for that. The recording stopped. We're not quite sure where it stopped. What I did show was we started with a symmetry principle. We used a virtual displacement thought experiment. We got our conserved quantity because the only way that we accepted Newton's second law and thereby implicitly Newton's first law, but not the third law. We're deriving the third law. We showed that because of the symmetry principle, wherever there's a symmetry principle, hiding somewhere is a conserved quantity, a conservation law. We showed that to be linear momentum. So from the symmetry principle, we see that linear momentum is a conserved quantity. And we used Newton's second law then implicitly Newton's first law as well. And we showed that we used that to rewrite this force. And for that to be 0, for the rate of change of linear momentum to be 0, that means momentum has got to be constant because the derivative of a constant is 0. And if I just consider a two-particle system now and I take d dt of both sides, I'm going to end up with that equaling 0. If I take it to the other side, bring it back to f, I have Newton's third law.