What other insight do we have, if we assume, as we did with the pole hold plots? So without loss in generality, remember, we can always flip the principal axis such that the largest inertia is axis one, intermediate inertia is axis two and the least inertia axis is axis number three, right? So, I assume this ordering. Then, we found earlier the B's for one and three had to be positive, meaning they have very strong stability for large departures. Here, that's negative, something we have to consider. But intermediate axis, we know, this one is always unstable, right? That was the separatrix stuff, so not too surprising. It's actually linearly stable, nonlinearly unstable and the other two's you don't have it defined. It depends on the situation, on how you're doing that and how you tumbling it for that, you know, the energy levels to get the right sign. So, it's different ways we can do this. So, when you're looking at these systems for small departures, we said, well, you know, k times x is always going to be bigger close to x's around zero, than another a times x is going to be bigger than b times x cubed if x gets small enough. As x gets very very small, x cubed is even smaller, right, whereas x is going to be bigger in that sense. At some point, this magnitudes always crisscross and as x gets large, at some point, x cubed will dominate. So, you can use these coefficients to argue if you have a particular sign. If A1 is positive, that means I've got a positive spring stiffness. That omega one motion should stay nicely bounded and stable, nice oscillatory part. If you have big departures and B is negative, that means once you go too far from the origin, it doesn't just oscillate at some point, it's going to just take off to infinity. That's what you find out in this analysis. Or vice versa, if a is negative, that means, well, it won't be stable. Small departures drive you away from the origin, but if B happens to be positive, in that case, remember you can't have your x's then go to infinity because at some point x cubed is going to dominate and it will try to bring it back. And, in nonlinear system you get things called limit cycles, and it doesn't converge to zero, but these two terms fight each other and you get these cyclic motions that happen in the end. That's what it converges to. So, the take off, this whole class is on nonlinear systems that talk about these things. So yeah, B1 can always be restoring if it's positive, and we showed that. Phase plane plots. Let's talk about that. Many of you have seen them, but not everybody, not everybody's had quite an engineering background. So, if I have a simple mx double dot plus kx system, all right, what is a phase space plot? What's on the horizontal axis? Gamma x. x. And what's on the vertical axis? x-dot, right? So you solve this. So, after some math or numerically, you find what x is going to be, and you have also a solution what x-dot is going to be. If it's numerical, you could, you know, you're always getting your positions and your rates and you're integrating, so you could keep track of all of that. The phase space plot is nothing but a visualization of my position and rates. So, I'm not plotting it versus time, but I'm plotting added positions and velocities compared themselves directly. So, if you have an oscillator equation I give it an initial deflection here, that's where I'm starting, and I have a sinusoidal response, right? Initially, I take the spring. I deflect it and then I just gently let go. The system would just oscillate, right? What happens to the phase space plot? What does that curve look like, for spring mass system? Andrei. Circle. It's a circle, for this case, right? So, at some point, this potential energy goes into kinetic energy right and then it slows down again. And then, the net positions become negative and then it becomes positive again and you're just going to circle around, right? If I give it less deflection, you have a curve that does this. If you give it more deflection, you have bigger curves, right? Do these curves ever intersect? If you start here, could this curve do this and go through here? Then it's a different system. If it's the same dynamical system, can you have phase space lines intersect? And the answer is really no. The challenge with that, if they intercept, if you had a different phase space plots, right? It means if some curve does this and some other curve does this, when you hit this intersection points which way do you go? It's a second order dynamical system. If I'm giving it a position velocity, the accelerations are dictated by equations of motion. You can't be going in two different directions. Maybe unless you go on quantum stuff or something, you know. But for classic mechanics systems, if given a position velocity, there is a very distinct acceleration and that velocity will evolve in a certain direction. So, you don't typically see these lines intersect, but you get these flow diagrams that you can draw, and they will show you how the stuff evolves. OK? So, That's just a phase space, there should be a dot here. That's what a phase space plot is. With a phase space plot, let's still look at x double dot + kx, just x double dot + the natural frequency squared x equal to zero. What is an equilibria, no, actually that's not the one I'm looking for. Sin(x) equal to zero. This is basically your planar and normalized version of a planar pendulum. That was this problem we're talking about earlier. With this system, what are my equilibriums going to be? When is x doubled dot and x dot gonna go to zero? If what? When is x double dot gonna be zero? That's the easy one. Zero and 180. Thank you. So, if, x is equal to zero or pi. So, on a phase space plot, plus or minus pi, right, because you can go upside down this way or this way, and if it hangs straight down, let's call that zero. Right? We know if we linearize this, it's just x double dot plus x, and you guys were arguing earlier, in that case, around the equilibrium, that's at the zero equilibrium I would have something where my phase space plots look like circles. Alright? And they have a certain direction. But if you go here, and you'd linearize around that. Here, you'd end up with x double dot equal to minus x, x double dot minus x is equal to zero, which gives you negative stiffness, and it's actually hyperbolically unstable. So there's these asymptotes you can draw, and that's good for both of them. So as we have larger departures, these phase space plots can all look like this. And, if you're on these lines, you would do weird stuff, where you could tumble from one with enough speed and hit the, you know, you can go from here and give it enough initial speed to where you would hit over here again, and do different kinds of motions, right? But, that's what these phase space look like, so you can quickly identify equilibriums. If you do this and there's programs that will give you this flow, the phase space flow, and you will see the dead spots. Those are the equilibriums and around it, the flows visually very quickly tell you this looks stable neighboring stuff or no neighboring. Some of it comes in, other stuff goes crazy, right? So, that's my quick review, if you haven't seen phase space plots. So, these kind of things with linear and cubic part can be used for that, as well. So this is one of the phase space plots we could talk about, where they flow. Omega's nearby, those omega two's look good, but then there's a limiting case, as well. And it turns out, with this omega two this was the separatrix case, but that was the intermediate axis case. There's a limit. If I'm doing a pure spin about omega two, that's one of the conditions, and then anything inside is a non-pure spin about the intermediate axis, and you can, it turns out, you can show, this paper showed, I'm not doing that in class. But there's different limiting conditions that they can come up with, physical motions can never be outside here. You have to be within this arc. You can never actually exceed initial conditions. It will give you nonphysical initial states that you'd have to have to get there.
