 Hi, I'm Zor. Welcome to Unizor Education. Today is an interesting lecture, which is, I would say, more philosophical than physical or mathematical, but it actually lays the very important foundation to entire physics, if you wish. Let me start from something which I was trying to basically convey throughout the whole course, that we are trying to build a course of mathematics or physics very logically, which means that I'm trying not to say something which I cannot prove, based on something which I have conveyed before. And whatever I was talking about before was based on even earlier and earlier and earlier items which were offered within the same logical sequence, down to basically certain statements which I could not prove, because there is nothing before that, which we call axioms. So this lecture is more or less within the same kind of vein. I will not try to prove anything. I'll just explain how certain concepts might be based on something earlier in physics. Now, this lecture is part of the course called Relativity for All presented on Unizor.com, together with two other courses, Maths for Teens and Physics for Teens, which contain basically material absolutely necessary to understand this one. The website is totally free. There are no advertisements. You can use it any way you would like. So I do suggest you to at least pay attention to the previous courses, Maths for Teens and Physics for Teens, at least to whatever items are suggested there, because otherwise you will not really understand the relativity part. Okay, what else? Yeah, the site contains a lot of problems to solve exercises, etc. Also, every video lecture is complemented with textual presentation of exactly the same material like a textbook. So you have basically completely free visual and textual presentation. All right, so back to this particular lecture. It's called Conservation Laws and Notar's Theorem. Okay, we will start with whatever I just talked about, that every statement is supposed to be based on something else, logically derived from it, and whatever was something else, before that it should be derived from before and before down to axioms. Now, one of the most important properties which we in physics accept as given are conservation laws, conservation of energy, conservation of linear momentum, and conservation of angular momentum. That's basically the properties which were used in all the courses, physics which was presented here, physics for teens, and obviously in relativity as well. Where did we get it from? Well, from experience, that's as much as we can say. So it looks like if we take these axioms as given, I mean the statements as axioms, so to speak, then whatever the consequences, logical consequences, theoretical consequences we are coming up with really correspond to our experience, our practice, etc. So we kind of used to have, okay, yes, of course, conservation of energy or conservation of linear momentum. Yes, we do have these laws. Now, what's the problem about this? The problem is that these conservation laws are really quite complicated. Maybe our experience is not a sufficient foundation for just staging that these are the laws of the universe we can take as axioms. What I would like to say is that it's not easy to take as an axiom something as complicated as a conservation law. When Euclid presented five axioms of geometry, they were not taken as something unusual. I mean it was kind of a natural that you have two points, you have only one line which goes through them. It's kind of a natural thing. Conservation law, well, personally I wouldn't say that this is an obvious and intuitively obvious thing. So we accept these as axioms throughout the whole classical physics, but we always kind of have some kind of a thought in our mind. Maybe these conservation laws are real laws of the universe only to a certain level, to a certain precision of our instruments, the way how we can measure it. Because energy can be transformed from one type to another, from mechanical to heat, from heat to electricity or something, electromagnetic oscillations. How can we really follow these transformations of energy and state that, okay, the energy is conserved in whatever form it is? It's not easy. I mean it's too complicated to take as an axiom. Intuitively we would like to take as an axiom something much simpler which intuitively obvious. Okay, here comes a very important person. Her name is Emy Noter. She is German mathematician, not even a physicist at that particular time, mathematician. However, many mathematicians are involved in aspects of physics. So what she did, and she published this in 1918, if I'm not mistaken, yes, in 1918, she published a very important article which was a proof, basically. These articles contain mathematical proof that our laws of conservation can be derived from something much more fundamental about our universe. Now that was so important. These kind of theoretical derivations that Albert Einstein said that Emy Noter is the most important woman in mathematics. So what exactly is much more intuitive and fundamental statements which we really can take as axioms much easier that Emy Noter derived the conservation laws. And here it is. And it really sounds extremely simple. Obviously it's not simple to derive this derivation from these simple statements to laws of conservation, but she did it. And it is just a little bit complicated for me to offer it right now as a lecture. So that's why this lecture would be called kind of exception when I'm staging something without proving. So what I'm staging is what Emy Noter proved. And here it is. Well, let's consider our three laws of conservation, conservation most important, conservation of energy, conservation of linear momentum, and conservation of angular momentum. And what she did, she derived that the conservation of energy is a consequence of uniformity of time, which means if we are making an experiment right now at a certain place in certain conditions, whatever the conditions are, and we will do exactly the same experiments, a certain amount of time afterwards. In exactly the same place, with exactly the same conditions, we will have exactly the same results. That's what uniformity of time actually is. That whatever we are doing is the same, would be the same, today as tomorrow if we repeat exactly the same thing. The results will be exactly the same. So uniformity of time, which personally I can accept much easier as an axiom than conservation of energy. But conservation of energy can be derived from uniformity of time, and that's what she did. That's number one. Number two, conservation of linear momentum. Again, conservation of linear momentum can be derived from uniformity, linear uniformity of space, which means if I'm doing something at this point in space, and on that point in space, and I'm doing exactly the same thing, results should be exactly the same. So life, universe, the laws of physics, whatever, should be exactly the same in this place as in that place. And finally, angular momentum, conservation of angular momentum, then the rotation is involved. Again, she has derived that this is following from the directional uniformity of the space. So regardless of where we will turn our experiment, the result should be exactly the same. So time uniformity, linear space uniformity, and angular space uniformity are accepted as axioms and accepted much easier as an axiom. And if we do accept them as axioms, the conservation laws will follow. I think it's not only important, but it's also a beautiful result, quite frankly. I mean, when I first learned about this dependency of, let's say, uniformity of time and the conservation of energy, I was really in awe. I mean, this seems to be kind of obvious that the time is uniform. And okay, maybe time is not uniform, but under the consequence, under whatever existing situation in our life in physics, I think it's very natural and intuitively obvious to assume that as the time goes by universe basically is the same. And if conditions are the same, the results should be the same for any experiment. So time is uniform. One second today and one second tomorrow will be exactly the same second. And the same thing about linear and angular uniformity of space. These statements seem to be natural. And whenever you are witnessing that something really complicated, like the laws of conservation, can actually be derived from something as simple as uniformity of time and space, I think it's just absolutely astonishingly beautiful result. So I did spend some time actually to follow the logic how she derived one from another. And it's very brilliant. I do suggest you, if you are comfortable with everything else, try to and you do need some mathematics, of course, in this. I do suggest you to just go to internet and find this article. It's very easy and try to follow the logic. But again, this is absolutely beautiful result. And it was regarded very highly by basically all physicists. I just quoted Albert Einstein's quote. And many other physicists established basically such a pride that we have made such a great derivation of really complicated things from something simple, which we can really accept as axioms. So basically, I would say that whatever Eminotir did is probably in physics, in parallel with whatever Euclid did for geometry many, many, many years ago. He put geometry on a good solid axiomatic foundation. And that's what basically Eminotir did. And again, I'm completely in awe about whatever was done by her. So that's basically it. I think that's all I wanted to talk about. And again, you can read about the whole thing which I was talking about on the website and Unisort.com. Just go to Relativity for all course. And in the menu item called Conservation, you will find the first lecture about Eminotir's theorem. So thank you very much and good luck.