Oscillation / Phase Space Question

[union68]

Homework Statement

Thornton and Marion, chapter 3, problem 21:

Use a computer to produces a phase space diagram similar to Figure 3-11 for the case of critical damping. Show analytically that the equation of the line that the phase paths approach asymptotically is $$\dot{x}=-\beta x$$. Show the phase paths for at least three initial positions above and below the line.

$$\beta>0$$ is the usual damping parameter.

Homework Equations

Equation of motion for critically damped oscillator:

$$x = A\exp \left(-\beta t\right) + Bt\exp \left(-\beta t\right)$$.

And,

$$\dot{x} =-A\beta\exp\left(-\beta t \right) +B\exp\left(-\beta t \right)-B\beta t \exp\left(-\beta t \right)$$.

The Attempt at a Solution

The phase diagram is done and correct. My problem is in showing the equation of the asymptote. My first inclination was to examine the limits of $$x$$ and $$\dot{x}$$ as $$t \to \infty$$. But they both go to zero, correct?

But I took at peak at the solution and they have

$$\lim_{t \to \infty} x = Bt\exp \left(-\beta t\right)$$

and

$$\lim_{t\to \infty} \dot{x} = -B\beta t \exp\left(-\beta t \right)$$.

Therefore, $$\dot{x} = -\beta x$$ as $$t \to \infty$$.

What?! Aren't those limits zero?! Am I so sleep deprived that I can't even take limits anymore? What am I doing wrong?

Thanks!

 

[ideasrule]

Those limits are indeed zero, but the limit of x-dot divided by the limit of x is -beta. That's exactly what the question wanted you to prove.

 

[union68]

Oh, OK. So you're taking the limit of the ratio $$\dot{x}/x$$. Why would be interested in that limit? I'm having a hard time finding motivation for all these things that the problems want you to do.