 Now, let's try and set up the Hamiltonian, Hamilton's equation. Now, we know the ground is equation del L del q equals the first time derivative of the partial derivative of L with respect to velocity. Now, once again, just some basic calculus if I have f and f is a function of x, y and t, the f dt is partial derivative of f with respect to x times x dot, partial derivative of f with respect to y x double dot plus the partial derivative of f with respect to t. So, if I have l, and I remember and if I take l to be a function of q, q dot and t, now that it's x, y and z, or x, y and t there, the l dt is going to be del L del q, q dot, del L del q dot, q double dot plus the l del t, del L del t. Okay, now, I know what del L del q is though. I can just replace that. So, we have the l dt is going to equal that, the d dt of del L del q dot, q dot plus del L del q dot, q double dot plus del L del t. So, that is where we stand. Now, if we look carefully though at these two, if we look carefully at these two, I can rewrite that. Look at that. The l dt equals the d dt of by l del q dot, q dot plus the del L del t. Because if I take the first derivative of this product of two functions, q dot, q dot, I'm going to get the d dt of this times that plus this times the d dt of q dot, which is q double dot. So, if I take the first derivative of this product, I end up with those two. So, I have to simplify things. Now, let's try to simplify things even more. I'm going to get all the d dt's on one side. So, let's bring this over to this side minus del dt. That is going to equal the d dt of, if I take this to that side and I then have the d dt of something minus the d dt of something, I can put that with each other again. So, d L del L del q dot q dot minus L. That is what I'm going to have. Now, I want to say, we want to say one thing. Let's set this negative del L del t to zero. The only way I can put that to zero is if this L is a constant. We know that. If L is a constant or at least not a constant, it can be a constant with something else, but it has no, it has no dependence on t. So, if L is a function with no dependence on t, it's only L of q and q dot with no dependence on t. There's got to be no dependence on t whatsoever. If that is true, in other words, I have no hidden degrees of freedom. I'm setting that as a prerequisite for setting up the simultaneous. The Lagrangian does, should have no dependence on time. If that is zero, the only way that that can be is if this is a constant, because the d dt of a constant would be equal to zero. In other words, del L del q dot q dot minus L is going to be some constant and we call that the Hamiltonian. It's equal to some constant. It is equal to some constant. Now, let's think about this, let's just think about this or let me clear some space and then we'll carry on from this to get the two equations for the Hamiltonian of a system. Good. Some space on the board, let's carry on. I said this was equal to a constant. It's only because del L del t was equal, it was a negative, that the matter was equal to zero some dependence on t. I just want to remind you what the Lagrangian was. It was t minus v. In other words, it was a half in q dot squared minus v of q. So it was that. So if I took, that was L, if I took del L del q dot q dot was there, that would have just been m q dot and that's mv, mv and that is momentum, momentum p. Okay, so this part I know is p momentum q dot minus L equals this Hamilton's functional values, our Hamiltonian. Now, just think about what we are saying here. We're saying momentum times velocity, momentum is mass times velocity times another velocity minus that is t minus v equals h. So this is mv squared. What is that? Well, that's twice half mv squared, isn't it? Two times a half is this one. Minus t plus v equals h. This is two times t, two times the kinetic energy minus t plus v equals h. And what is that v for h? h equals t plus v, the total energy. So the Hamiltonian of our system is the total energy, total energy if there are no hidden degrees of freedom as far as time is concerned. If there is no, if the del L del t equals is zero, if only if that is true, then I have that the Hamiltonian equals t plus v. In other words, it's going to be a half m q dot squared minus plus v of q. And so they're very simple terms. We have the Hamiltonian of a system.