Let's look at another example. It's basically a particle that's moving in a plane, but the radial motion is given through a spring, and see what happens here. In this case, Theta is the cyclic coordinate because the particle kinetic energy, you'll have your radial rates squared plus the tangential rates squared. You only have r. You have Theta dot and r dot, but no Theta in there, and the springiness is given through K over 2 r squared. That's the classic spring one. How much has it deflected, and 0 r was the zero deflection point. Theta, in this example, becomes the cyclic coordinate. Again, there's no other forces acting on it besides the spring. All the big cues are going to be zero, which makes sense because whenever oscillations are happening, you can solve for it, but if you oscillate in this way or you start oscillating in this way or maybe it has oscillations and lateral motion at the same time, hopefully, it's intuitive to you to think, "Well, what makes this direction special from this direction?" All that really matters is you started from some direction and how much have you moved since that original direction, that you could just shift everything 90 degrees, and it will give you the same motion. The result is just shifted 90 degrees. That's what it means to be a cyclic coordinate there. It does impact the actual where is the particle, but for the solving of the equations, you've got something that's going to be invariant, which is nice. In this case, Theta is cyclic, so p Theta is the partial of this with respect to Theta dot, and you're only going to get m over 2 times 2 times Theta dot times r squared, which gives you this. Mass times distance squared, that's basically the inertia of the particle about the point 0 times Theta dot. That's angular momentum. Again, that's why you can see conjugate momentum. That's where the name comes from. It's often momentum-like. But it's only that momentum along this one axis that we're claiming, and that has to be a constant, so that means when I solve Lagrangian dynamics, you get these differential equations, and there's no Theta appearing because it didn't appear to begin with, but because this is a constant, given initial conditions, I compute this constant, in this case, angular momentum, p Theta, and I can replace Theta dot with p Theta over mr squared. You plug that in, I get now a set of differential equations that are free of any Theta because any rates that appeared in there, I can replace through angular momentum conservation, and get rid of that condition completely, and now I only have a 1D problem, and that's going to be more stable if you can do this numerically because there's no integration errors to even occur. You've analytically solved for that part of the rate. It always has to be this over mr squared precisely, and you've imposed it precisely into the equations. That makes sense? Cyclic coordinates are really cool, they help you. Any questions on this one?