Velocity Vector in Polar Coordinates (Kleppner p.30)

[Von Neumann]

In polar coordinates we have \vec{r} = r \hat{r} \Rightarrow \vec{v} = \frac{d}{dt}({r \hat{r}}) = \dot{r}\hat{r} + r \frac{d \hat{r}}{dt}.

In the book Introduction to Mechanics, K & K says the right term is the component of velocity directed radially outward. (Surely a typo, as the left term is the velocity component associated with the direction \hat{r}.) Then he goes on to say it's a good guess that the other term is the component in the tangential \left( \hat{\theta} \right) direction. He proves this is so in 3 ways; namely by proving \frac{d\hat{r}}{dt} is in the \hat{\theta} direction). The first two ways I understand - It's the third one I'm stuck on.

He starts by drawing two position vectors \vec{r} and \vec{r} + \Delta \vec{r} at the respective times t and t + \Delta t, along with their respective unit vectors \hat{r}_{1} , \hat{r}_{2} , \hat{\theta}_{1} , and \hat{\theta}_{2}. From the geometry, we see that \Delta \hat{r} = \hat{\theta}_{1}sin\Delta\theta - \hat{r}_{1}(1-cos\Delta\theta), where \Delta\theta is the angle between the two position vectors).

From this we see that \frac{\Delta\hat{r}}{\Delta t} = \hat{\theta}_{1}\frac{sin\Delta\theta}{\Delta t} - \hat{r}_{1}\frac{(1-cos\Delta\theta)}{\Delta t} = \hat{\theta}_{1} \left( \frac{\Delta\theta - \frac{1}{6}(\Delta\theta)^3+\cdots}{\Delta t} \right) - \hat{r}_{1} \left( \frac{\frac{1}{2}(\Delta\theta)^2 - \frac{1}{24}(\Delta\theta)^4+\cdots}{\Delta t} \right). Almost there, just need to take the limit of this quantity as \Delta t tends to 0. So we need to evaluate \frac{d \hat{r}}{dt} = \displaystyle \lim_{\Delta t \to 0} \frac{\Delta\hat{r}}{\Delta t}.

He make the following argument which concludes the proof:

"In the limit \Delta t \to 0, \Delta\theta approaches zero, but \Delta\theta/\Delta t approaches the limit d\theta/dt. Therefore, \displaystyle \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t}(\Delta\theta)^n for n>0. The term in \hat{r} entirely vanishes in the limit and we are left with \frac{d \hat{r}}{dt}= \dot{\theta} \hat{\theta}."

I understand that \Delta\theta/\Delta t approaches d\theta/dt as \Delta t \to 0, but I'm lost after that. How does one come to the conclusion that \displaystyle \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t}(\Delta\theta)^n for n>0? Then, how does this lead us to the conclusion that \hat{r} entirely vanishes in the limit?

I've been trying my hardest to work through this text, but I tend to get snagged for quite some time on explanations like the above. Usually I can fill in the missing steps myself. I feel as though I cannot thoroughly penetrate this textbook as I have others.

Thanks in advance for any guidance.

 

[TSny]

How does one come to the conclusion that \displaystyle \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t}(\Delta\theta)^n = 0 for n>0?

Maybe you can make use of the product law for limits if you think of $\frac{\Delta \theta}{\Delta t}$ and $(\Delta \theta)^n$ as functions of $\Delta t$.

 

[Von Neumann]

Well, since \displaystyle \lim_{x \to a} f(x)g(x)= \lim_{x \to a}f(x) \cdot \lim_{x \to a}g(x) then \Rightarrow \displaystyle \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t}(\Delta\theta)^n = \lim_{\Delta t \to 0}\frac{\Delta\theta}{\Delta t} \cdot \lim_{\Delta t \to 0} (\Delta\theta)^n = \lim_{\Delta t \to 0}\frac{\Delta\theta}{\Delta t} \cdot 0 = 0.

How does this help though?

 

[TSny]

Well, since \displaystyle \lim_{x \to a} f(x)g(x)= \lim_{x \to a}f(x) \cdot \lim_{x \to a}g(x) then \Rightarrow \displaystyle \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t}(\Delta\theta)^n = \lim_{\Delta t \to 0}\frac{\Delta\theta}{\Delta t} \cdot \lim_{\Delta t \to 0} (\Delta\theta)^n = \lim_{\Delta t \to 0}\frac{\Delta\theta}{\Delta t} \cdot 0 = 0.

How does this help though?

I hope I'm not misunderstanding your original question. I am assuming you want to show that $\displaystyle \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t}(\Delta\theta)^n$ equals zero. If so, isn't that what you have essentially shown?

 

[Von Neumann]

The goal is to prove that \frac{d \hat{r}}{dt} = \dot{\theta}\hat{\theta}.

I do now understand why the previous limit holds. However, why is it relevant to proving that \frac{d \hat{r}}{dt} = \dot{\theta}\hat{\theta}?

Since a limit of products is identical to the product of the separated limits, any limit containing a factor of (\Delta\theta)^n will result in 0. Is this why \hat{r} vanishes? We can factor out a (\theta)^2 from the second term of \frac{\Delta\hat{r}}{\Delta t}. And, since n=2,>0, then the limit is 0. Right?

But this reasoning suggests that we can factor out a (\theta) from the first term of \frac{\Delta\hat{r}}{\Delta t}. And since n=1,>0 then the limit is 0 for this one as well. Which is certainly not correct, since \hat{\theta} is not supposed to vanish.

EDIT: Hold on! I figured it out, I just need to type out all the details.

 

[TSny]

OK. Good.

 

[Von Neumann]

I think I got it:

\begin{align*}<br /> <br /> \frac{d\hat{r}}{dt} &amp;= \displaystyle \lim_{\Delta t \to 0} \left( \frac{\Delta\hat{r}}{\Delta t} \right) \\<br /> <br /> &amp;= \lim_{\Delta t \to 0} \left [ \hat{\theta}_{1} \left( \frac{\Delta\theta - \frac{1}{6}(\Delta\theta)^3+\cdots}{\Delta t} \right) - \hat{r}_{1} \left( \frac{\frac{1}{2}(\Delta\theta)^2 - \frac{1}{24}(\Delta\theta)^4+\cdots}{\Delta t} \right) \right] \\<br /> <br /> &amp; = \lim_{\Delta t \to 0} \hat{\theta}_{1} \cdot \left( \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t} - \frac{1}{6} \lim_{\Delta t \to 0} \frac{(\Delta\theta)^3}{\Delta t} + \cdots \right) - \lim_{\Delta t \to 0} \hat{r}_{1} \cdot \left( \frac{1}{2} \lim_{\Delta t \to 0} \frac{(\Delta\theta)^2}{\Delta t} - \frac{1}{24} \lim_{\Delta t \to 0} \frac{(\Delta\theta)^4}{\Delta t} + \cdots \right) \\<br /> <br /> &amp; = \lim_{\Delta t \to 0} \hat{\theta}_{1} \cdot \left[ \dot{\theta} - \frac{1}{6} \lim_{\Delta t \to 0} \left( \frac{\Delta\theta}{\Delta t} \cdot (\Delta\theta)^2 \right) + \cdots \right] - \lim_{\Delta t \to 0} \hat{r}_{1} \cdot \left[ \frac{1}{2} \lim_{\Delta t \to 0} \left ( \frac{\Delta\theta}{\Delta t} \cdot \Delta\theta \right) - \frac{1}{24} \lim_{\Delta t \to 0} \left( \frac{\Delta\theta}{\Delta t} \cdot (\Delta\theta)^3 \right) + \cdots \right] \\<br /> <br /> &amp; = \dot{\theta} \cdot \lim_{\Delta t \to 0} \hat{\theta}_{1} - \lim_{\Delta t \to 0} \hat{r}_{1} \cdot 0 \ \left( \text{since} \displaystyle \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t}(\Delta\theta)^n \ \text{for} \ n&gt;0 \right) \\<br /> <br /> &amp;= \dot{\theta} \hat{\theta}<br /> <br /> \end{align*}

Correct?

 

[TSny]

I think I got it:

\begin{align*}<br /> <br /> \frac{d\hat{r}}{dt} &amp;= \displaystyle \lim_{\Delta t \to 0} \left( \frac{\Delta\hat{r}}{\Delta t} \right) \\<br /> <br /> &amp;= \lim_{\Delta t \to 0} \left [ \hat{\theta}_{1} \left( \frac{\Delta\theta - \frac{1}{6}(\Delta\theta)^3+\cdots}{\Delta t} \right) - \hat{r}_{1} \left( \frac{\frac{1}{2}(\Delta\theta)^2 - \frac{1}{24}(\Delta\theta)^4+\cdots}{\Delta t} \right) \right] \\<br /> <br /> &amp; = \lim_{\Delta t \to 0} \hat{\theta}_{1} \cdot \left( \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t} - \frac{1}{6} \lim_{\Delta t \to 0} \frac{(\Delta\theta)^3}{\Delta t} + \cdots \right) - \lim_{\Delta t \to 0} \hat{r}_{1} \cdot \left( \frac{1}{2} \lim_{\Delta t \to 0} \frac{(\Delta\theta)^2}{\Delta t} - \frac{1}{24} \lim_{\Delta t \to 0} \frac{(\Delta\theta)^4}{\Delta t} + \cdots \right) \\<br /> <br /> &amp; = \lim_{\Delta t \to 0} \hat{\theta}_{1} \cdot \left[ \dot{\theta} - \frac{1}{6} \lim_{\Delta t \to 0} \left( \frac{\Delta\theta}{\Delta t} \cdot (\Delta\theta)^2 \right) + \cdots \right] - \lim_{\Delta t \to 0} \hat{r}_{1} \cdot \left[ \frac{1}{2} \lim_{\Delta t \to 0} \left ( \frac{\Delta\theta}{\Delta t} \cdot \Delta\theta \right) - \frac{1}{24} \lim_{\Delta t \to 0} \left( \frac{\Delta\theta}{\Delta t} \cdot (\Delta\theta)^3 \right) + \cdots \right] \\<br /> <br /> &amp; = \dot{\theta} \cdot \lim_{\Delta t \to 0} \hat{\theta}_{1} - \lim_{\Delta t \to 0} \hat{r}_{1} \cdot 0 \ \left( \text{since} \displaystyle \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t}(\Delta\theta)^n \ \text{for} \ n&gt;0 \right) \\<br /> <br /> &amp;= \dot{\theta} \hat{\theta}<br /> <br /> \end{align*}

Correct?

Yes, that looks OK. In going from the second to third line you don't really need to take the limits of $\hat{\theta}_1$ and $\hat{r}_1$ since they are fixed vectors that don't depend on $\Delta t$. But, what you did is fine.

 

[Von Neumann]

Yes, that looks OK. In going from the second to third line you don't really need to take the limits of $\hat{\theta}_1$ and $\hat{r}_1$ since they are fixed vectors that don't depend on $\Delta t$. But, what you did is fine.

I meant it implicitly in going from the 5th to the 6th line. Thanks TSny!