Now, here, I ran through a simulation

to kind of show you how these things evolve,

and how we can use these Omega's phase-space plots.

So I got Omega 1 versus Omega 1 dot.

So my state versus a rate, state versus rate,

state versus rate, the same phase-space plot.

I'm starting out with a minimum energy case actually,

which is here. So, H overI2,

if T becomes my intermediate

one, this value goes to one,

this is kind of a nice normalized way to look at it.

But here T is less, so this value goes bigger, so here

on this pole hold plot and spinning right here purely about B1,

in the phase-space plots so you can see I have a B1

and no other velocity, it stays there.

If it weren't in equilibrium that point

would be moving through that plot.

And the other two are perfectly zero.

So that's a pure spin about B1.

That's a minimum energy case.

Now I'm holding momentum constant

and I'm pumping up the energy.

So what happens - this one shift slightly.

It basically means you look at

this coordinate here it was fixed.

If I'm looking at this curve inside,

that curve will be recessed inward

from that equilibrium point,

and there's a slight wobble to it

because it's a potato shaped like curve.

That's why you see a little bit of

a wobble in that coordinate, but the other two axes,

the Omega 2 which it spins about this axis.

These coordinates kind of sweep back and forth

that's what you're seeing here.

And the Omega 3 coordinates also go up and down

up and down.

Right. So it goes through the series.

And this is what these answers would look like.

So you can predict these motions given those

second order differential equations we derived

or you can just get them

numerically by solving Euler's original equations.

And that's we just take an energy

and slightly bumped it up.

If we bump it up more

now I'm actually very close to the separatrix condition.

Not quite there but I'm pretty close.

So it's like one of these curves.

And I get there and hang out

here along and then I come back again.

And if you plotted this in a phase space you get

this teardrop looking thing

that's at t=0.

And this is where you hang out a

long time and then you go back around,

because we're close to the separatrix condition.

The Omega 2 part

that one just kind of oscillates back and forth

doesn't look too crazy.

The Omega 3 speed versus velocity starts to pinch,

and that's what you see happening here,

cause that has to do with the slowdown

as [a student] was saying earlier as we get close

to these corner points and the separatrix stuff,

that's where things really slow down all of a sudden.

And so that position versus rate stuff

- not intuitive here - but if

you solved it, that's what you would have.

So good, that's almost separatrix.

If you make it a separatrix you get this.

And the way I get a closed curve

is I had to really cheat and look

at the conditions on both sides and integrate.

Because we know starting from here,

this is basically on one of these separatrix lines.

We know it would take an infinity of time to reach here.

That's only the asymptote that we would reach.

I can get the other part by starting on the other curve.

And that kind of completes the picture.

All right.

So it looks like a bow tie but you never go from

one end of the bow tie to the other end of the bow tie or

propeller or whatever you want to call this shape.

And this is this limiting case

as predicted early from the phase-space plot,

that's what you have with the separatrix.

Good.

If we add more energy now we're getting to some of these

curves up here it starts to look like before.

This one's a little pinched dogbone shape still got an eyelid.

This other one I lose one of these.

Now I'm either on left or the right hand side.

Which one depends also on initial conditions.

This is where you go through this infinite sensitivity stuff.

It's hard to predict when you're doing just tumbles.

Are you going to spin about positive or negative.

That's the chaotic part of this system.

But here the numerics took me to one side.

That's what I'm left with, then as I pump up energy now

it starts to tighten.

Now we get to these inner curves

and at the very end I pumped up energy as much as I can.

I intercept here so you can see the phase space plots Omega ones

and twos are just zero steady points.

And I'm back in an equilibrium.

That's doing a pure spin about axes of least inertia.

So I hope we get it.

This is just an illustration of how

we can use phase-space plots to actually

analyze whole ranges of motion.

Pole holds was one of them.

This is another approach that you see quite conveniently.

And we can also validate because you can

predict these motions with those

that make second order differential equations

or you can solve the coupled non-linear equations that we had earlier.

They should all give you the same answer.

This is often a nice way to validate

that all this math we did, with momentum and energy constraints, we did it correctly.

Because otherwise these things wouldn't agree.

Any questions on this?

This is the general inertia case.

So we don't quite get the nice sines and cosines

analytic answer, but we do get three de-coupled duffing equations

which is kind of elegant, as a sub-result.