Andrew?

>> No, it's not. >> No, is not,

that's the key thing, right?

And the example we had was, think of a spring mass system,

a spring master amp system.

Just floating in space, it would oscillate and settle down as zero equilibrium.

So now, you're studying this equilibrium subject to a disturbance and

you're treating gravity as a disturbance, it's going to settle to this deflection.

But it settles to the same deflection no matter how big you bumped it.

It's always going to come to the same one.

So it doesn't matter on initial conditions, right?

It's just a quick visual way to think of this.

What's the next stronger level of stability?

We have Lagrange, Downtanis, then we have the Lyapunov, good.

So Matt, talk me through the Lyapunov now.

All right, we have some initial delta, [SOUND].

>> So for the Lyapunov, for any initial delta, there's some final epsilon that if

you start in the b sub delta, you will end up and in the b selector.

>> And stay within it, right?

We talked about the separatrix motion.

The astronaut spinning that key.

It kind of stable for

short period but then it flipped around then it will stable here.

And then it flips again and it continues this.

So that would never fit the stability requirements because it might be there for

a short period of time.

But at some point, it leaves again and then it comes back.

And that's a whole different kind of a thing, okay, CK.

>> I had a question on that because when it's flipping around,

it's sort of flipping within these two like defined equilibria.

So could you bound around that and call that-

>> You could call it bounded.

You could come up with boundedness arguments for that one, and say that for

the system, the rates aren't spinning up like crazy.

Something could be unstable like if you look at Europe effects,

people keep studying on asteroids and debris.

There, the spin's going to get bigger, and bigger, and bigger, and

bigger, and bigger, and the cements are going to break apart.

This spinning thing is not going to do that.

It is bounded in its response.

And you could argue some types of Lagrange stability around it,

that I know it's not going to 6 billion RPMs all of a sudden, right?

Exactly, but now, what's the analog we use here for the Lyapunov stability?

because we say, we can pick any epsilon, any epsilon, really, really small.

Now, the corresponding delta might be really, really small too, but

you can find it, right?

What kind of a mechanical system can you pick?

I want to be oscillating within one degree, 0.1 degree,

1 arc second and all this find an initial condition that puts you there.

>> [INAUDIBLE] >> No temper, you're close.

It's just a spring mass system.

If you look at a classic spring mass system, you can deflect it, right?

And the stability, we are always talking about protobation.

You can't go while otherwise, you don't deflect it.

You can do anything but nothing.

[LAUGH] Right?

You can't do nothing, you have to do something.

And so you bump it infinitesimally and whatever infinitesimal bump you gave it,

it's just going to wiggle and there's no damping, right?

It's just going to wiggle.

So that's why we can never set epsilon here to zero.