 Hi, I'm Zor. Welcome to Inizor Education. We will talk today about inertial systems, about Galilean transformations, and how Galilean transformations are invariant relative to certain laws of physics. Now, let me first remind you that previous lectures were related to certain concepts which we call inertial reference frames, inertial systems. And what's very important is we basically took as an axiom the fact that physical laws are supposed to be the same in all inertial systems. Now, let me just remind you that the inertial system is the one where an object which does not experience any kind of a force, just a completely free object, is supposed to move along the straight line with uniform speed. So, there are many different inertial systems, and if you found one, then there are infinite number of other systems, because any system which moves along a straight line with a constant speed to the first inertial system would be inertial, obviously. And we have actually introduced the way how coordinates in one inertial system can be transformed into coordinates of another inertial system. This is called Galilean transformation. Let me just remind that let's say you have one inertial system, x, y, and z. And you have another inertial system, lowercase x, y, and z. If the origin of this inertial system, let's call it beta, and this is alpha, if origin of beta reference frame is moving along a straight line with a constant speed relative to this inertial system, then we can actually transform coordinates. Now, the way how we did it before was let's just assume that at time t is equal to t is equal to 0, lowercase t is time in this system, uppercase t is the time in this system, and we assume that the time is absolute and universal and is the same everywhere. So, if at moment time z, at moment time 0, the centers, the origins are coinciding and axis are coinciding as well. And then, as the time grows, this system goes only along the x-axis, which means only x-axis is changing. Then the transformation would be, that's supposed to be uppercase t. They are the same, but I still prefer to maintain different letters. So, this is the coordinate, x coordinate at time t. This is in beta system, this is x coordinate in alpha system at time t, minus v times t, where v is speed, this beta system is moving along a, alpha system along the x-axis, while other coordinates are without change. So, this is just a reminder of what is Galilean transformation if the beta reference frame is moving relative to alpha just along the x-axis. From the beginning, and at the beginning they completely coincide. Okay, that's simple thing. Now, my issue is that we have agreed and took it as an axiom that all the laws of physics are supposed to be the same in all inertial systems. Now, why we took it as an axiom? Well, it corresponds to our experience, our common sense, and physical experiments until certain level where it started not actually to confirm. And we will discuss this separately. I mean, that was the beginning of theory of relativity. Right now I'm just trying to approach this moment when everything was great. So, laws of physics are supposed to be the same. These are transformations which are called Galilean transformations, which physicists actually took as reasonable and they checked. It looks like it was working at the time. So, let me just demonstrate basically what kind of different physical laws were checked against this particular transformation and really prove that it does make sense to consider the laws of physics exactly the same in all inertial systems and taken into the law of transformation of coordinates. This Galilean transformation. So, the first thing which I would like to check is the first Newton's law. The first Newton's law should look the same in both inertial systems. So, it looks in one way, in one system, then it should look exactly the same in other systems. So, let's just check it out. Now, what is the first Newton's law? Let's say you have an object which moves without any kind of external force applied to it. So, it moves freely. It moves along the straight line with a constant speed in alpha system. Now, if our presumptions about Galilean transformation are correct, if we will apply Galilean transformation to its coordinates, we should also have in another beta reference frame that its movement is a straight line with a constant speed. Let's check it out. First of all, what exactly is the movement in case our object is moving in the alpha system, in the alpha reference frame, with constant speed along the straight line trajectory? Well, that's basically a three-dimensional straight line parameterized by the time. So, what is it? Well, x of t is supposed to be equal to x0 plus vx of t, not to be of multiplied. So, x0 is x-coordinate at time 0, and then as the time grows, it linearly increasing with the time. So, that's what makes a straight line. Same thing with y. That would be a y-component of its velocity, and z would be z0 plus z-component of its velocity. So, I assume that velocity vector, it has coordinates vx, vy should be capital Y. Capital Y and vz. So, if this is velocity, constant velocity, we are talking about constant, the first law of Newton, no forces. So, it's a constant velocity. So, if this is the velocity vector, then this is basically how the coordinates of this particular object are changing with the time. This is a description of the movement of the object in the alpha reference frame. Now, let's just think about the beta reference frame, what happens in beta. All right, let's just apply the transformation. So, we don't need this anymore. And let's apply Gaussian transformation. Let's just assume for now that the beta system is moving along the x-axis. So, it's only the x coordinate is affected. So, the x of t, as we know, should be equal to x of t minus v times t. Okay? Now, y should be equal to y and z should be equal to z. Okay. So, in this case, it would be equal to x0 plus vx times t minus v times t equals to x0 plus vx minus lowercase v times t equals, and I don't know if it equals. I can just replace capital C with a lowercase t because they are the same. Now, what is this? Well, this is, again, a straight line. Now, y of t would be equal to the same thing, and z of t should be equal to the same thing. Just change capital T to lowercase t because the time is absolute and the same. Now, these three equations, this, this, and this, define the straight line movement with the components of the vector of velocity vx minus v comma vy, again, capital Nvz. Now, these are all constants. These are three components of the vector in the alpha reference frame. It's a constant speed, constant velocity, speed along the trajectory. Lowercase v is the speed of the beta system moving along the x-axis relative to alpha system. Now, if it, by the way, I will address this a little bit later in this lecture, but it's obviously that if it's just any kind of movement along the straight line of beta relative alpha, not necessarily along x-axis, there will just be different coefficients here. It will be vx, lowercase vx, vy, and vz. But later on I'll talk about this again. So the fact that I'm moving along the x-axis is not really any kind of a simplification. It's just easier to talk about. Okay, so that's the first law of Newton. Okay, let's talk about the second law. The second law is related to force. So there is some kind of a force. It's one force or it's a sum of many different forces which are acting on the object. And again, let's right now consider a constant force. Constant force has in the alpha system certain coordinates fx, fy, and fz. This is a vector of force. Let's call it f. Obviously, vector f. f is equal to m times a, where a is acceleration vector. f is the force vector, m is a mass of the object. So this is the second law of Newton. So let's see how it will be transformed from alpha system. So let's consider this is alpha system. Let's think about how it will be transformed into beta reference frame. Now, first of all, these coordinates fx, fy, and fz. These are coordinates of a constant vector in alpha system. Basically projections of the vector to x, y, and z coordinate axes. Now, the beta reference frame is moving in such a way that its axes are always parallel to the alpha reference frame. Now, if axes are parallel, projections of this vector will be exactly the same. So I can say that the same fx, fy, and fz are coordinates of this vector in the beta system. So f is basically the same vector. So left part is not really changing whenever we are going from one system to another. How about right? Well, acceleration definitely is different. Something must be different because what is acceleration? Acceleration is the second derivative of coordinates. And since my beta reference frame is moving relative to alpha reference frame, coordinates are changing. And we know basically how they are changing. They are changing according to the other land transformation. That we know. Great. Now, obviously since time is universal, I can put lowercase. So the same functions x, y, and z can be with lowercase t. And let's take the second derivative from these coordinates, from these functions, and see what happens. Now, the first derivative is derivative of v times t would be v, right? v is a constant. Now, the second derivative will be what? Second derivative will be second derivative of this. And second derivative of constant is zero, right? So what do we see? We see that acceleration in the beta system is exactly the same as acceleration in the alpha system. I can now return it back to capital T. So it will be definitely acceleration in the alpha system. So this remains the same. And that's the proof that the second Newton's law looks exactly the same in both alpha and beta reference frames. Okay? All right. So what else can we do? So the third thing which I wanted to do today is talk about how speeds are basically changing. So we know that acceleration is not changing. And we know the position is changing according to Galilean transformation. Between acceleration and the position, there is a speed. Because speed is the first derivative and acceleration is the second derivative. All right. So what happens with the speed? So let's consider our object is moving with certain speed, with certain velocity vector to tell a little bit more precisely. So what happens with the same object? What happens with its velocity vector in the beta system? So given velocity vector in the alpha, and I actually have it as a V with alpha superscript. So it's variable. Let's just consider it's a different kind of a movement, more complicated. Maybe it's acceleration based on force or whatever else. Some kind of variable velocity vector, which is basically the first derivative of covariance. And it's what? It's V alpha x of t, V alpha y of t, capital T, and V alpha z of capital T. That's my vector, velocity vector. Now, it's obviously equal to x first derivative of t, y derivative of t, y t and z derivative. Now, derivative is derivative by time of this. Okay, this is my velocity vector. Now, how can it change in the beta system? And here, I would like actually to a little bit more complicated. It's not really complicated way of moving beta system relative to alpha system. Before, I was talking about movement only along the x-axis. And the constant V was my movement, right? But let's consider it a little bit more complicated. Let's say there is a vector V and I will also call it alpha because it's all how beta is moving relative to alpha, which means in alpha coordinates, right? So let's consider there are three constants along the three different axes. So that would be V alpha x, V alpha y and V alpha z. So these are coordinates of the vector of movement of the entire beta system. And these are constants. Now, before we had this and this zero, basically, right? But the movement is very, very similar and Galilean transformation would be exactly the same. Just slightly different, which means x of t would be equal to x of t minus V alpha x times t, y of t, which is the same as before. Now it would be y of t minus y component times t. And z coordinate in a beta system would be z component in the alpha system minus V z times t. So that's basically exactly, again, a very simple continuation of moving only along the x-axis. This is moving along all three axes in some direction. But it's a constant. The x, the y and the z are constants. That's what makes it movement of the beta system along the straight line with constant speed relative to relative to. Relative to x system. Okay. So that we have done now. Let's consider now vector of movement in the beta system of t. Now this is basically very similar to this. It has beta x component, beta y component and beta z component, which are derivatives of corresponding coordinates, right? So this is x derivative by time, y derivative by time and z derivative by time. That's what it is. Similar to this one. Now, knowing the transformation, let's just check it out. So if I will take derivative of this, again, lower case and capital case t are exactly the same. So I can put lower case t and differentiate. So it will be x of t minus vx alpha y derivative of t minus v alpha y comma and v z. So derivative of z of t minus v alpha z, which let me convert it into vector form. It's vector x t, y t, y of t and z of t, which is this one, right? So it's alpha minus. And these are three components of my vector of movement of the entire beta system. So this is, compare this to this, this is the law of addition of velocities. If you have some kind of a movement of an object in the alpha system and v represents, basically v can be variable, v of t, not necessarily along the straight line. So any kind of a movement in the alpha system, which is described by this vector of velocity. And this is description of the same object's movement, but in the beta reference frame. And we know that the beta frame is moving along alpha frame with constant velocity vector v alpha, whatever the components are. This is the constant vector, which is given because beta system is given as moving relative to alpha system. And this is basically the law of addition of velocities. So you just have to subtract the vector of velocity, we subtract from it the vector of movement of an entire system. And this is basically the law of addition of velocities. Let me just write it down again. So the beta vector of velocity is equal to alpha vector of velocity minus vector of vector, which describes the velocity of an entire beta system in alpha coordinates. Okay, so that's basically it. I would like you to look into the description of this lecture, there are notes. So everything is presented on Unisor.com. You have to go to Relativity part of this website, Relativity course. And if you will open the Relativity course, you will have it in the first part, which basically describes the principles of relativity. So this is Galilean invariance and addition of velocities. So thank you very much and good luck.