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I'm working with the equation of motion for a pendulum as follows: $$x''+ \frac{g}{l} \sin (x)=0$$ Where $x$ is the angle between the pendulum and the vertical rest position.

I am required to use the complex variable $w=e^{ix}$ to rewrite the equation of motion in the form $(w')^2= Q (w)$, where $Q$ is a cubic polynomial. So in the form $(u')^2=u^3 + au + b$, with $a$, $b$ constants.

I'm not sure where to start with the question, can anybody help me get going? Homework help

Moira
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1 Answers1

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Multiply the equation through by $x'$ and integrate once to get

$$x'^2-\frac{2 g}{\ell} \cos{x} = C$$

where $C$ is a constant of integration. Now, if $w=e^{i x}$, then $\cos{x}=(w+w^{-1})/2$ and

$$w' = i x' e^{i x} \implies x'=-i w'/w$$

Then the equation is equivalent to

$$-\frac{w'^2}{w^2} - \frac{g}{\ell} \left (w+\frac{1}{w}\right)=C$$

Then, multiplying through by $-w^2$, we get

$$w'^2+\frac{g}{\ell} w^3 + C w^2+\frac{g}{\ell} w=0$$

which is not quite the form specified, but is an equation of the form $w'^2+Q(w)=0$, where $Q$ is a cubic in $w$.

Ron Gordon
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  • Furthermore, substitute $w = -\frac{l}{g},v$ to get a monic cubic polynomial, and $v = \frac{C}{3} + u$ to zero the coefficient of $u^2$. – ccorn Jul 18 '13 at 11:27
  • This looks like an OK solution, but as you said, it isn't quite of the correct form. I'm confused in your first step, you say you multipied through by x' but it looks like you just integrated. Also, can you explain why you integrate in the first step? – Moira Jul 18 '13 at 11:29
  • @Moira: 1) as ccorn states, just transform to get the cubic in the form you need, 2) yes, as I stated, I integrated. Why? It is a standard manipulation with an equation like this; it becomes a statement of conservation of energy. In any case, it was a helpful step when the goal is to get an equation such as the one you specified. If you go at it directly, then you get a term of $w''$ that is difficult to eliminate directly. – Ron Gordon Jul 18 '13 at 11:41
  • Thank you for your help :) – Moira Jul 18 '13 at 11:42