Does small angle mean small angular velocity?

[Happiness]

Why is the term $(\sin\theta\,\dot{\theta})^2$ fourth order in the small $\theta$, as claimed by the sentence below (5.108)?

By small-angle approximation, $(\sin\theta\,\dot{\theta})^2\approx\theta^2\,\dot{\theta}^2$.

For this to be fourth order, it seems like we must have $\theta=\dot{\theta}$. Why is this true? What are the conditions for this to be true?

 

[wrobel]

For this to be fourth order, it seems like we must have θ=˙θ\theta=\dot{\theta}. Why is this true?

why quantities of different dimension must be equal? They must not. David Morin wrote a good book but definitely not for high school

 

[Happiness]

why quantities of different dimension must be equal? They must not. David Morin wrote a good book but definitely not for high school

So is he right or wrong?

 

[wrobel]

So is he right or wrong?

My advice is that you should first think about your own understanding whether it is right or wrong.

 

[Happiness]

My advice is that you should first think about your own understanding whether it is right or wrong.

My understanding is that $(\sin\theta\,\dot{\theta})^2$ is second order in $\theta$. But that would mean that it cannot be ignored, since (5.109) contains second-order terms.

 

[wrobel]

The degree of the term $\theta^2\dot \theta^2$ is equal to 2+2=4. It is a degree of polynomial $P(\theta,\dot \theta)= \theta^2\dot \theta^2$. Both quantities $\theta,\dot \theta$ are assumed to be small. The approximation of the Lagrangian up to the second order terms corresponds to linearization of the Lagrange equations in the vicinity of equilibrium

 

[mfb]

Here it is important that θ is small, but the timescale is not. $\dot \theta$ is "something with θ divided by time", which is small of θ is small.