Even the stability and the stabilization properties of a simple pendulum system can be less obvious

than we think, right?

And this is the computation I mentioned yesterday that Lord Maxwell did.

He took a harmonic oscillator, a pendulum, linearized one.

He observed, right, we did it yesterday, just multiplying this equation by the velocity

x prime, something that you also do in the context of partial differential equations

when dealing with wave-like problems.

He observed that the energy was conserved, what is completely compatible with the fact

that solutions are simply combinations of complex exponentials, right?

Dealing or living to a motion which is completely periodic and with constant amplitude.

Then he said, okay, how do we stabilize the oscillation properties of the system?

We can do it by adding this feedback term.

So you see, this is our feedback, right?

So this is a feedback or a damping term, right?

We are damping, we are dissipating energy on one hand.

On the other, we are doing it using a feedback mechanism.

Why I do refer to the feedback?

Well, because if you just call the right-hand side of this equation, you just call it, say,

f of t, right?

In the context of mechanics, f of t is the force applied to the pendulum, to the harmonic

oscillator, right?

So you have a pendulum which is oscillating, but you can also push it up and make the oscillations

being more accelerated, larger amplitude.

Or you can, when it's coming to me, you can push it backwards so that these oscillations

are damp, are dissipated, right?

So here you see that this is a feedback mechanism.

This is an important concept, as I mentioned yesterday in control theory, because what

you do is actually that at every time instant, you measure the velocity x prime of t of the

pendulum in oscillation, you measure the velocity, and then you apply a force which is proportional

to the velocity x prime of t with a minus sign.

So it's going against the velocity.

When the velocity is positive, I push with a negative force.

When the velocity is negative, I'm pushing with a positive force.

So I am always doing, you know, I am pushing against the tendency of the motion to evolve,

right?

And I am doing it with, say, multiplicative constant k, right?

So in principle, our intuition, our first intuition says, oh, if I do this, so I'm always

pushing with a force which is proportional to the velocity, but with the opposite sign,

the solutions will decay, right?

The solutions will decay faster and faster whenever I increase the multiplicative constant k.

Because larger constant k, it means that larger is the force I apply, right?

This was, you know, this is the first intuition.

In fact, I think I didn't say yesterday, the fact that this term here,

this term here is a damping term, you can see very nicely by doing the same thing, right?

So if here what we did in order to get the energy conservation was simply multiply the equation by

x prime of t, right?

And observe that this equation leads to the conservation of energy d,

e t dt equals zero.

Here, if you do the same, then you get a dissipation law saying that the derivative of the energy

is minus k times x prime t squared.