Now, we've been talking about a whole range of different coordinates, and we've seen hints that not all coordinates are equally important. There was one problem where there's a particle on a plane it's attached to a string in a whole and somebody's pulling at a certain force on that string, so that it starts to spiral quicker and quicker, and what happens to that system? That was one of them. In the end, if you look at the equations of motion, not all the coordinates actually appear, and there was some interesting things happening in there. Let's talk about cyclic coordinates quickly. What does that mean? This is your classic Lagrangian. I'm just going back to the very basic form. We're not doing constraints. Constraints would expand it a little bit. But here, What happens with the cyclic coordinates? You'd have N differential equations. But what if your coordinate itself did not appear? We've had stuff like that where maybe the angle doesn't appear, or the position, we'll have some examples we go through, doesn't appear in your kinetic energy expressions. Here too, we're assuming the virtual work acting on this coordinate is zero, so that's why Qj is zero. Maybe there is a force acting on the system, but for this particular coordinate that force dotted with the partial velocity corresponding to that coordinate was zero or, orthogonal to the force, non-working. In that case, we've had cases where Qj then became zero all of a sudden, and we got invariants of motion out of it. That's what we're talking about here. Because if this is zero and this is zero, then you end up with the time derivative of something scalar equal to zero. Well, that means that scalar must be a constant. That's what it means that in that case, if that scalar, I'm just calling it p, and p is typically called the conjugate momentum in these Systems. Also, if you go to Hamiltonians and stuff, you'll see when you see people talk about conjugate momentum. It's basically once you have your Lagrangian, you take the partial with respect to generalize coordinate rates and whatever comes out is like a rigid body moving. You would have your linear momentum or your angular momentum is actually coming out of that. That's what's your measure is, so your p. But this must be a constant, so on that string where you're pulling it in turns out it starts spinning faster and faster because while you're putting force on it, that force is always in a way that's not working and momentum ends up having to be preserved as you're pulling in that little particle, and it spins faster and faster as it comes in. With this system like Lagrangian formulation, sometimes total momentum is not preserved, but momentum along one axis might be preserved while the other two may not be constant. This will tell you right away, there's something preserved, and that's nice because then we have an integration constant. This partial will give you an expression that it's a first-order differential equation because it could have rates of the coordinates in there, but it's not a second-order one and this must be constant. That means Q_j is said to be a cyclic variable or also called ignorable variable because it's an integration constant. The solution you get for the rest of the system, and we'll see this in examples next, don't depend actually on this one. Think of like a bouncing ball. You can solve this with a bouncing ball, and I'm solving it right here over this location. But if my x coordinates are here, and I move half a meter over and do the same bouncing ball, I will get the same height motion happening over and over and yes there's an x, but it turns out x is basically a fixed one, or if you had some initial velocity then you would just keep moving at that fixed velocity as you're bouncing. The vertical bouncing has nothing to do with the x in the end, so x would be an ignorable coordinate. In fact, because there's an invariant thing that you can solve for, you can explicitly express. Where is it? Oh, the time. I don't have to solve a second-order differential equation. That's what it means to be an ignorable coordinate or a cyclic coordinate or the system. We need these two conditions. This one goes to zero and this one goes to zero.
