(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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 [tex] \dot{x}=-\beta x[/tex]. Show the phase paths for at least three initial positions above and below the line.

[tex]\beta>0[/tex] is the usual damping parameter.

2. Relevant equations

Equation of motion for critically damped oscillator:

[tex] x = A\exp \left(-\beta t\right) + Bt\exp \left(-\beta t\right)[/tex].

And,

[tex] \dot{x} =-A\beta\exp\left(-\beta t \right) +B\exp\left(-\beta t \right)-B\beta t \exp\left(-\beta t \right)[/tex].

3. 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 [tex]x[/tex] and [tex]\dot{x}[/tex] as [tex]t \to \infty[/tex]. But they both go to zero, correct?

But I took at peak at the solution and they have

[tex] \lim_{t \to \infty} x = Bt\exp \left(-\beta t\right) [/tex]

and

[tex] \lim_{t\to \infty} \dot{x} = -B\beta t \exp\left(-\beta t \right)[/tex].

Therefore, [tex] \dot{x} = -\beta x [/tex] as [tex]t \to \infty[/tex].

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!

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# Homework Help: Oscillation / Phase Space Question

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