Our next last application is celestial mechanics - how the planets, and starts, and moons all move through space. You may recall from the very first unit where you first saw this picture, that 3 or more bodies operating under mutual gravitational forces can act chaotically, for the right masses and initial conditions, that is. This interaction of 3 stars is chaotic, as in the picture that I showed you of the spacecraft going to the moon. There is no dissipation here, by the way. At least not on short time scales, so there aren't any attractors. This is Hamiltonian chaos. Chaotic motion of stars and planets cannot happen if you have only 2 of them since the solutions to gmm/r-squared for that case can only be conic sections- ellipses, parabolas, and hyperbolas. Okay, so 2 bodies can't be chaotic, 3 bodies can. How many bodies are there in the Solar System? Certainly more than 2 and there's lots of chaos. I'm only going to talk about a few examples. Again, there are a lot more, all of which you are now fully equipped to look up and understand. People have been interested in the movements of the planets and stars for a very long time. Starting several thousand years ago, they even built special-purpose computers to simulate those orbits. These are what are called orrerys. They are mechanical computers. They have gears, and wheels, and cranks and when you turn the crank, the planets move in the correct way. Notice in the one on the right, that that builder of that one didn't know about Uranus, or Neptune, or Pluto. Here is Saturn and here is Jupiter. There is nothing outside of those. This I think from the early 18th century. The one on the left is called the Antikythera Mechanism and it's from maybe 100 BC. Here's a digital version of that mechanism. This particular one is now in the Smithsonian. It's called the digital orrery. It's about a cubic foot of electronics. Special-purpose computer could not browse the web and did not have a compiler. All it could do was integrate the differential equations that described the dynamics of the outer solar system. And it did not use fourth-order Runge-Kutta by the way, rather it used one of those high-power symplectic solvers that knows about and enforces the conservation of energy that is true in a Hamiltonian system where there is no dissipation. What Jack Wisdom and Jerry Sussman did with this was, among other things, establish that the orbit of Pluto is chaotic. When I give talks to lay audiences, this is where people get worried, but you folks know enough not to panic. Chaos does not mean Pluto is going to fly off and crash into the Earth. Pluto's orbit lives on a thinly-banded chaotic structure that doesn't intersect anything. So no need to worry. Wisdom and Sussman also studied the Kirkwood gaps - gaps in the asteroid belt between Mars and Jupiter. They showed that any asteroid in that gap would have a chaotic orbit that moves back and forth between an almost-circular shape at low eccentricity, which is the e on this axis. Down here things are very circular, then they become more and more long and thin ellipses as you go up. And this is the orbit for an asteroid in one of those gaps. And it moves chaotically back-and-forth between circular and elliptical. Why should this be a problem? Think about looking down at the Solar System from overhead. Long, thin elliptical orbits intersect other planets, especially the Earth. So any asteroid that was in a dynamical zone that had this kind of behavior is long gone because it whacked into us! This is a plot of the observed positions of the asteroids - which are the little symbols - and the edges of the region where that kind of Earth-crossing chaos arises. And I find this plot pretty compelling. That was all about planets, and moons, and stars and how they move through space. Chaos also turns up in how the orientation of an object evolved. That is, whether the object orbits nicely - like we do with our pole pointed generally at Polaris - or whether they tumble chaotically like that. The shape of a body, whether it's symmetric like a cube or a sphere, or axisymmetric like atop, or tri-axial like a book, dictates how it's orientation evolves. A symmetric body's initial rotation will not change. A top will spin and precess. So this is spinning. This is precession. And it will also nutate; It will nod like that. A triaxial body like this will tumble chaotically depending on how you throw it up. I don't know if you can see that, but there is the front of the book. I'm holding it like that. And there's the front of the book. Jack Wisdom and his students applied that reasoning to satellites in the solar system like Saturn's moon Hyperion and then predicted it would tumble chaotically. That prediction was borne out when spacecraft went by this moon somewhat later. Here is the movie made by the Cassini spacecraft that I showed you during the first unit. And Hyperion is not the only object in the Solar System that tumbles chaotically as you can see from this picture. But the really amazing thing about nonlinear dynamics and chaos in the solar system is the detective work that the math lets us do. We didn't have time to talk about Hamiltonian chaos or its crown jewel, the KAM theorem, or Kolmogorov–Arnold–Moser theorem. Among other things, that theorem tells us that every satellite, wherever it is, unless it is perfectly spherical and in a perfectly circular orbit, tumbled chaotically at some point in the past on its way to its current equilibrium position. And that includes the Earth. We are not perfectly spherical - we are somewhat oblate (we're short and fat) - and our orbit around the sun is slightly elliptical. So that slide that I just showed you applies to us. Imagine what the sky would look like if we were tumbling chaotically. The sun would be zigzagging all over the place and, at night, the stars and the planets wouldn't be doing their beautiful, slow rotation across the sky. Life would be very very different and nonlinear dynamics tells us that's how it was a couple billion years ago. Of course, there were only bacteria around to observe it at that point.