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.