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PROFESSOR: Here are the
differential equations

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for the angular momentum
of a tumbling box.

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Try throwing a book, or a box,
or any rectilinear object whose

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three dimensions are all
different, into the air

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with a twist, to make a tumble.

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You could go to rotate
about its longest axis,

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or about its shortest axis.

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But you can't get to rotate
about its middle axis.

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Let's examine that
phenomena numerically.

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Here's the anonymous
function defining

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those system of three first
order differential equations.

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Now I'm going to start with
an initial condition that's

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near the first critical point.

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1, 0, 0 is a critical point.

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And I'm going to take 0.2
times a random number,

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to sort of be near the critical
point, and then normalize it,

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so that it has length 1.

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So the largest component
is the first component.

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And the other two are
small, but not too small.

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This is an easy
problem numerically.

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There's no stiffness
involved here.

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So I'm going to use ODE
23, integrate from 0 to 10,

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and here's the solution.

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The blue component is the
first one, and it stays near 1.

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And the other two are periodic,
rotating around the 0.

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Let's go back and take
another starting condition.

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Here it is again.

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Now the other two
components are really small.

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And when we integrate
that, the blue component

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stays flat near 1.

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And the other two guys
hardly move at all.

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Now I'm going to go to the
third critical point, 0, 0, 1.

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Do the same thing.

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Take a random number near there.

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Use ODE 23.

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Now the yellow component
stays near one.

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And the other two move
periodically around 0.

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Run that again.

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The third component is near 1.

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The other two are not too big.

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And run the ODE 23.

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The other component
stays near 1.

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And the other two rotate
periodically around 0.

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Now we're going to go to
the middle critical point.

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We're going to try and get
the box to rotate around

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its middle axis.

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The second component
is the one near 1.

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And now we see completely
different behavior.

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This sienna component
doesn't stay near 1.

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It goes down near -1,
and comes back up.

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Let's integrate over
a longer period,

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so we can see that behavior.

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So it's periodic.

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But it goes down to -1
and comes back to 1.

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And the other two move in
large amplitudes around 0.

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So this is the instability of
that middle critical value.

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Let's doing again.

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Same thing.

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1 down to -1, and back up.

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It's periodic.

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These solutions
are all periodic.

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But that middle critical
point is unstable.

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Now I want to view these in
a different way, graphically.

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The differential equations have
these three critical points.

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Any solutions to start exactly
in these initial conditions

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stay there.

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But what happens if you start
near these initial conditions?

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Well it turns out, that x and
z are stable critical points.

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But y is an unstable
critical point.

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If the angular momentum
is near x or near z,

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it stays near there.

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But if it starts near y,
it moves away quickly.

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You can think of x as the short
axis, and z is the long axis.

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Rotation near the
short axis is stable.

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And rotation near the
long axis a stable.

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But rotation near the
middle axis is unstable.

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We can see that in
the following graphic.

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It turns out that
if a solution starts

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with an initial condition
that has norm 1,

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it stays with norm 1.

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So the solution lives
on the unit sphere.

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Here's our unit share with our
three critical points, x, y,

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and z.

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If this were the Earth,
z will be the North Pole.

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Axis where the 0-th meridian
crosses the equator.

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That's in the east Atlantic,
a little off of West Africa.

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y would be where the 90th
meridian crosses the equator.

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That's in the Indian
Ocean, west of Sumatra.

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If we start with an
initial condition near x,

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the solution orbits around x.

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That's stable rotation
around the short axis.

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If we start with an
initial condition near z,

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the solution orbits around z.

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That's stable rotation
around the long axis.

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But if we start
near y, the solution

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takes off, goes over to
near -y, turns around,

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and comes back to y.

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Periodic, but goes
clear around the globe.

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That turns out, that's a circle
actually, an orbit around x.

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If we come up a little above
y, we get an orbit around z.

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Go down a little below y,
we get an orbit around -z.

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Go to the right of y, we
get an orbit around -x.

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Let's zoom in a little bit.

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And we can see that y is a
classic unstable critical

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point.

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Let's conclude by
drawing a few orbits.