 Now, before we move on, there's a concept we have to understand. If I have a particle in two-dimensional space, and as time goes by, I can plot its position. x-axis, y-axis. This is in meters. This is in meters. In other words, we can just say it's a unit of length and a unit of length. So the particle starts here. This is typical cannonball. Stuff goes up, comes down. It has an x-axis, which goes at constant velocity. There's not constant velocity there, but it moves in space. And I can get an equation for this. I can get an equation for this. And you might have seen this before. s equals s sub zero plus v sub zero t plus a half a t squared. So if we just look at the x-axis, so s is distance, so where is it? Well, that's the start position, that's the velocity, that's the time that passes, that's the acceleration. So if it goes at constant velocity, which is in the x-axis, velocity times time is going to give you a certain distance, so it's going to have that start distance plus another distance. If there is some acceleration to it, as there's negative acceleration and then positive acceleration, if positive is in the downward direction, or this I've drawn upwards, so negative acceleration, positive acceleration. Wow, let's draw it like this. Confusing the issue. In any space I can get a derivative. And you might tell me, well, all you really need is the position, an equation for position. And many times in physics we won't write s, we won't write x, we won't write y, but we write this q. It was a q's the position. You can say to me, well, if you give me an equation of q, I can get its first derivative, dq dt. I can get dq dt. Now anytime in physics, when we take a derivative with respect to time, we also write q dot, we put a dot on it. Dot on the top means the derivative. So dq dt, or in this instance would be dx dt, ds dt, dq dt for position, that is velocity. You'd agree with me. So you tell me that while all you only need, basically, is an equation that has position in it, and velocity is purely a derivative of that. And unfortunately that is not so. To describe the time evolution of a system, and that's the equations of motion. We have a time start time and we have an end time. How did the system evolve? And usually for us it would be a particle that moves. How does it move in time? We need at least two things. We need q and q dot. And although this is the derivative of that, we need both. And there's a simple explanation for this. Here I've already built the velocity into the equations, and then it's acceleration. So this equation here of position does not stand separate from knowing what the velocity is. I have two aeroplanes. They both have the same velocity. They're both travelling at the same velocity, same direction, same speed. But one starts at an airport here, the other one starts at an airport there. They both have the same velocity. Time equals zero. Time evolution. At the end they're going to be in different spots. I need to know both position and velocity to look at the start and the end. I cannot only know position. Take the first derivative. I have to have a very special function in which that knowledge is already built in. And that is where we get to this. This is called a phase space. Phase space. And these are two very special phase spaces. Here we just have length and length. It's two dimensional space. But I can also plot something different. I don't necessarily just have this line with the direction which is the evolution of time. I can have q on this axis and I can have q dot on this axis. So here I have length, and here I have length per unit of time. I have q and q dot. And you know any point in this space represents an instant time, and I can write coordinates for it. Here a coordinate for this would have been x and a y, values of length and length, dimensions of length and length. Here I'm going to have q and q dot, q and q dot. And just as a graph in this two-dimensional coordinate system gave me a function, I can also get a function from this. A very special kind of function and it's called the Lagrangian which will be a function of q and q dot. What am I trying to tell you here? Two things. One very exciting. Very quickly I'm going to tell you something about the Lagrangian and the Hamiltonian. You might have heard those words before. They are exquisitely interesting, fascinating things, and you need to know them before you can do physics. Why? Because force equals mass times acceleration is not the only things that we require to explain the system. I have a proton, I have electron in this probability cloud around it. I have an electrostatic force attracting each other. I also have a minute gravitational force. That electron stays there. That is not enough. Forces determine them as far as cause and effect. Those laws of nature are not enough to describe all systems in this universe. Certainly not that. We need something else and that is where the Lagrangian and Hamiltonian comes in. I need them to do q dot. Remember what q dot is? That's velocity. q dot is velocity. It's to that. Make them into vectors. That is velocity. But I can also just multiply this with mass just to make it into, if I multiply this with mass that just becomes, that just becomes p, the momentum. P, the momentum, so here I can have momentum which is mass times velocity. So here I'm going to have this mass factor, mass times velocity which is meters per second. So length per time unit. And on this axis I can have, on this axis I can have position which is this length. So any point here is going to have coordinates p comma q. And if I have, and if I set up the proper equation it's going to be equation called h and that's going to be of p, or p of q and p, position and momentum. Position and momentum. And I call that the Hamiltonian. That's a Hamiltonian function. Okay, but see what I'm plotting against each other. I'm not plotting two dimensional space. I'm plotting something very new. I needn't just do this. That's physics at college and physics at school. This is something completely different. I'm setting up a new coordinate system, points in this coordinate system. I'm calling that. And unfortunately for you to be able to do this you do have to do a multivariable calculus. We had a function in x and a multivariable calculus we have this function in mathematics we might write that as a vector. A function of a vector and that vector is a vector that contains both x and y. That is for x and y. In other words this is a function containing both x and y. In other words this might have been x squared but here I might have 2x plus xy. My function contains two variables. So here we're going to have a function that contains two variables. You have to know something about multivariable calculus. I do have an extended lecture series online for multivariable calculus. When we do use it though I'll give you a few hints maybe just to refresh your memory or to inspire you to go read up. One multivariable calculus. And if I have set up an equation and it has two variables in it position and momentum I call that a Hamiltonian. But unfortunately also not the full story you can also have this three independent variables where q is a function of time q dot is a function of time q is a function of time p is a function of time but there's an explicit function that function contains an explicit reference to time and here it's implicit because h depends on q just as f which is y depends on x h depends on q but q depends on t so implicitly h depends on t implicitly h depends on t explicitly f depends on x and y explicitly h depends on t but q and p might also be functions of t we call that implicitly dependent on h and we use h for Hamiltonian f for Lagrangian but it's nothing other than just a function f of x or f of x and y so we're going to set up these functions a Hamiltonian, a Lagrangian and a Hamiltonian depending on whether we deal with velocity or whether we deal with momentum and it's terribly exciting it's a brand new way of looking at nature as opposed to looking at the deterministic view of forces as causes with acceleration as effects.