I know this will be basic but I'm trying to visualise this pendulum equation and it's left me a little stumped.
If we have a mass on the end of a string and the mass is allowed to swing like a pendulum. Here we have $ t $ = time and $ x $ = inclination from the vertical. So we have the following equation of motion:
$$ \ddot x + \omega^2\sin x = 0 $$
It is possible to re-write $ \ddot x $ in terms of $ \dot x $ and $ x $ by using:
$$ \begin{align} \ddot x & = \frac{d \dot x}{dt} \\ & = \frac{d \dot x}{dx} \frac{dx}{dt} \\ & = \frac{d}{dx}\left(\frac{1}{2}\dot x^2\right) \end{align}$$
Ok, I get the $$ \ddot x = \frac{d \dot x}{dt} $$
part. Whats getting me is $$ \frac{d \dot x}{dx} \frac{dx}{dt} $$
Ok, so we have the inclination changing with respect to time which is the $$ \frac{dx}{dt} $$
For the $$ \frac{d \dot x}{dx} $$
term, are we saying that it's the velocity changing with respect to the inclination? I'm struggling to somehow visualise that in my head. Any pointers?
Thankyou in advance.
By writing $$\frac{d\dot x}{dt}=\frac{d\dot x}{dx}\frac{dx}{dt},$$ you have assumed that $\dot x$ can be written as a function of $x$, so that you can apply chain rule. However, in the case of the pendulum, we cannot write $\dot x$ as a function of $x$.
It is easy to see that the pendulum can be either rising or dropping if we are only given the position of the pendulum, so $\dot x$ is not a function of $x$, and writing $$\frac{d\dot x}{dx}$$ makes no sense.