…a different inertia for moving in the x direction and the y direction and the z direction. It's rather an interesting generalization of Newton. In a real crystal, if you actually put an electron on an atom, well, first, if you took an electron on an…

…So I don't want to waste it, so I will give it to you. They are lectures on curved space. According to Newton, everything attracts everything else. by force inversely proportional to the square of the distance between them. And they respond by accelerating…

…When we write the relativistic form, we don't change them. We just rewrite them. This formula is not the original law of Newton as Newton stated it, but contains, of course, all those square roots of 1 minus v squared over c squared, which were…

…we write the relativistic form, we don't change them. We just rewrite them. This formula is not the original law of Newton as Newton stated it, but contains, of course, all those square roots of 1 minus v squared over c squared, which were discovered by…

…equations is sort of the way it was discovered, what the transformation is. They would not, if you took F equals ma directly from Newton, look nice. We'll have to modify that one in order to be able to write it this way. When we do it with Newton's…

…the time, you'll find that that number you'll get is bigger than it is for the actual motion. In other words, the laws of Newton could be stated not as a form F equals ma, but in the form that the average kinetic energy less the average potential…

…for any number of particles. I said Newton's laws, it isn't quite true, because Newton's laws include non-conservative forces like friction. Newton said, just put F over here. It only works for conservative systems. We also know that on a microscopic level,…

…the same at the surface of the sphere as if all the matter were concentrated at the center. That was a theorem that took Newton a while to convince himself it was true in order to publish his theory, if you remember. And we did that by actually doing…

…sufficiently smart, you'd be able to figure out everything else in the rest of the course. But as you remember, the law of Newton was very simple to write down, but it had an awful lot of complicated consequences that took an awful long time to learn.…

…with density at the normal pressure. And if we can find that derivative, then we have the speed of sound. First time Newton did it, he did it wrong. And it lasted for several hundred years before somebody figured out what was wrong. What's important…

…it was found that it behaved in many respects like a wave. On the other hand, light at first, well, at Newton's, after Newton, people thought it was a particle. Then they discovered, as we have here, that it looks like a wave. Then in the beginning…

…kinematics. It's definitions. Rather, it's the definition of the torque and the angular momentum. But the dynamical law, the law of Newton, F equals ma, for three-dimensional space rotations has as a consequence this great law, which you see here. That's it.…

…the same line, which is what he said or should have said if he didn't. I haven't been able to find it in Newton, actually. But he somewhere tacitly assumes it. In other words, if between two particles the forces are not only equal but opposite, they'll…

…we can figure out how much thrust we're going to get per ampere. When you multiply that together, you get 3 quarters of a Newton. That's very poor, lousy, low. And that's a, well, an ampere isn't a hell of a lot of current, but 100 amperes or 1,000…

…And the quantum mechanics is not a study of quanta in that sense. It's the laws of mechanics in a different form than Newton, and that's all they are. And it includes the mechanics of electrons and protons and so on, and of photons. But technically, for…

…the result isn't a result of experiment and not a result of plain thought, as we can illustrate as follows. In the first place, Newton believed that it was true that you couldn't tell how fast you were moving if you were going in a uniform velocity or…

…you couldn't tell how fast you were moving if you were going in a uniform velocity or straight line. In other words, Newton first stated the principle of relativity. In fact, the one quotation I made last lecture was a statement of Newton. Therefore, why…

…words, Newton first stated the principle of relativity. In fact, the one quotation I made last lecture was a statement of Newton. Therefore, why didn't the philosophers make all the excitement about all is relative or whatever it is? in Newton's time. Just…

…define without measuring something outside, that is an elementary point of view at a low level of understanding, a point of view that Newton, in fact, had. But with an appreciation of the fact that the velocity of light is a whole 186,000 miles a second,…

…uniform velocity in a straight line. Now, the main difference, of course, between the relativity of Einstein and the relativity of Newton is that the law of transformation connecting the coordinates since times in one system and in the moving system are…

…Lecture 15, Relativity. November 17, 1961, Lecture 15, Relativity. Some classes Nobody's left. Oh here they come. The equations of motion of Newton, which we've been studying so far, were believed to be right for over 200 years. And the first time that an…

…to be right for over 200 years. And the first time that an error was discovered, correctly discovered, that is, in the laws of Newton was by Einstein in 1905. The equation of Newton, which is the d dt of mv, is equal to the force. Newton added the…

…first time that an error was discovered, correctly discovered, that is, in the laws of Newton was by Einstein in 1905. The equation of Newton, which is the d dt of mv, is equal to the force. Newton added the hypothesis that m is a constant. But it's…

…in the laws of Newton was by Einstein in 1905. The equation of Newton, which is the d dt of mv, is equal to the force. Newton added the hypothesis that m is a constant. But it's been found that it isn't true that the correct formula is that m is a…

…transformation of coordinates into them. And then these laws transform to a new set with all primes. In other words, the laws of Newton are the same in form in the moving system as well as in the stationary one. And therefore, you can't tell by making…

…nature appear to be conservative. This is not a consequence of Newton's laws. From Newton's point of view, as far as Newton knew, in fact, the forces could be, like friction apparently is, non-conservative. But when I say friction apparently is, I'm taking…

…it's, of course, of some interest to see how it comes about that the energy is conserved in accordance with the laws of Newton. And in order to explain how that's done, we'll take the simplest possible example and analyze it and then develop harder and…

…Newton's laws. In other words, I'm going to take the derivative with respect to time of the kinetic energy and then use Newton law to understand what's going on here. This is equal, when you differentiate the square here, since this is only one…

…energy is the same as it would have been if that whole shell were collapsed at the center as one piece. And that's, yeah. Newton had a lot of trouble proving this. He found it very difficult. And he worried about it a long time, because he wasn't sure…

…a deep problems in philosophy. So it always is interesting to ask, what does it mean? Let's ask, what do the physical laws of Newton mean, in which we say F equals ma? What is the meaning of the mass? And, well, we can intuitively feel that. And we…

…have independent properties. Now the feature is that the specific independent properties that the force have were not completely specified by Newton or by anybody else. And therefore, the physical law F equals ma is an incomplete law. It says, If you study…

…a suggestion that the forces will be simple. Example of such forces was the complete law of gravitation, which was given by Newton. And thus, he answered the question, what is the force? If there were nothing but gravitation, then the combination of this…

…important characteristics is that force has an origin. And it's not just the definition. It's that some other thing is making the force. Newton also gave one rule about the force, that the force on the action on one object with the forces were equal and…

…In what? In meters and so on? OK, 8.85 times 10 to the minus 11 coulombs square meter square per unit force in Newton or something. At any rate, that's the unit. That's the constant that belongs in here. Of course, in nature, the most important charge…

…Well, I need more room. Move it over. x is measured, say, from some origin this way. Where do we put the origin? All that Newton would tell us at first is that there is some place that you can measure from. So this would be the center of the…

…some kind of a scale. so that one pound of force, or whatever you measure forces in, will be equal Newtons, I guess. One Newton would be, say, 2 and 1 half inches or something on the scale. But once you've decided upon that, all the forces can be…

…any lecture Tuesday. Hey, where are the notes? Today's lecture is on the conservation of momentum. If you're given the laws of Newton, which relate the force to the acceleration, then any problem in mechanics can be solved in principle. For instance, for a…

…conservation of momentum. As I tried to explain, Newton's laws were the kind of program that said, pay attention to the forces. And Newton told us two things only about the forces. In the first place, in the case of gravitation, he gave us the complete…

…discovered one rule, one general property that they have, which is called his third law. And that's the total knowledge that Newton had about forces, the law of gravitation and this principle, but no other details. And this principle is that action equals…

…this way between the two. Equal and opposite and oppositely directed on the same line. That was an hypothesis or a law that Newton proposed. And it seems to be quite accurate, but not exact. And we'll discuss the errors later. But at the moment, we'll…

…see, this is in the theory of relativity again. It's an effect of relativity, if you wish. One of the propositions of Newton was that the interactions at a distance were instantaneous. It turns out that's not the case, that in electrical forces, for…

…called dynamics. having taken care of kinematics. This lecture today is on the laws of dynamics, which were degenerated first by Newton, and we'll take the approximation of Newton. Later, we'll study Einstein's laws of dynamics. The discovery of the laws of…

…kinematics. This lecture today is on the laws of dynamics, which were degenerated first by Newton, and we'll take the approximation of Newton. Later, we'll study Einstein's laws of dynamics. The discovery of the laws of dynamics or the laws of motion was a…

…Saturn on Uranus and so on, which alters the motion of all of them a little bit from the ellipses. But after Newton, there was a complete understanding. Even the slight variations away from Kepler's laws, the perturbations of one planet on another, were…

…figure out the other remark. How does it change its speed if something is affecting it? And that is the contribution of Newton. Newton wrote down three laws. The first law was a mere statement of the Galilean principle of inertia, which I just mentioned.…

…out the other remark. How does it change its speed if something is affecting it? And that is the contribution of Newton. Newton wrote down three laws. The first law was a mere statement of the Galilean principle of inertia, which I just mentioned. The…

…by the time it took as the time it takes gets smaller and smaller. infinitum. That is an idea which was invented by Newton and by Leibniz independently, and is the beginning of a new branch of mathematics called the calculus. So calculus was invented in…

…in some other division, to learn. However, you've got to learn it in order to do physics. Incidentally, it was invented by Newton because he couldn't figure out motion without it. I'm very sorry. We can't wait for the mathematicians to teach it to you.…

…looking out by any effects that you're moving relative to all the stars, if you want. This proposition was first stated by Newton. Let's take his law of gravitation, for instance. It said that the forces are inversely as a square, let's see, what else,…

…no effect of such a drift through space on the motion of the planets around the sun, according to Newton's law, so that Newton said, the motion of bodies among themselves is the same in a space, whether that space is itself at rest relative to the…

…moving at a uniform velocity in a straight line. Now it turns out that as time went on, new laws were discovered after Newton, and those were the laws of electricity by Maxwell. And one of the consequences of the laws of electricity were that there…

…an extra mass because of the relation of mass and energy. So things get heavier when they move. It was first believed by Newton that this wasn't the case, that the masses stayed constant. And so when it was discovered that that was false, everybody say…

…no explanation of the machinery of gravitation. So you might want to look further. And various people have tried to look further. Newton was originally asked. It doesn't mean anything. It doesn't tell us anything. He says it tells you how it moves. It…

…invent a better one. And maybe you can't, because nobody knows the ultimate. But up to today, from the time of Newton, no one has invented another theoretical description of the mathematical machinery behind this law, which does anything else but say the…

…fact that the force is toward the sun and that the areas are equal. Not ingenious, no? This was, I borrowed this from Newton. It comes right out of the Principia, diagram and all. The letters are different, that's all, because he wrote in Latin. These…

…Principia, diagram and all. The letters are different, that's all, because he wrote in Latin. These are Arabic numerals. But, incidentally, Newton made his proof geometrical like this and made all his proofs in his book geometrical of this type. Today we…

…area is the component of force at right angles to the radius. But if the force is in the direction of the radius, as Newton said, then there's no force at right angles to the radius, and that means that the rate of change of area doesn't change. I…

…as the friction on the floor and so on, the thing would go at a uniform speed forever. The next point was made by Newton, who discussed the next question, which is, when it doesn't go in a straight line, then what? And he answered this way, that a…

…the masses are different. This is a way of measuring the masses by how much, how hard it is to change the speed. Now, then Newton saw, and from this, that for instance, to take a simple example, if a planet is going in a circle around the sun, no…

…do need a force. So it became apparent that the origin and that the force was toward the sun. As a matter of fact, Newton was able to demonstrate that the statement that equal areas are swept in equal times was a direct consequence of the simple idea…

…had a different conclusion. Their theory was that it was the earth that was pulled by the moon away from the water. So actually, Newton was the first one to realize what actually was going on, that the force of the moon on the earth and on the water…

…of the way they went around over a long period of time, one could be very careful to check that everything was according to Newton. Turned out not to be the case. The moons of Jupiter appeared to be first to get sometimes to eight minutes ahead of…