A doubt from the book Galactic Dynamics from Binney and Tremaine

[krishna mohan]

Hi...

I was reading through the second edition of the book Galactic Dynamics from Binney and Tremaine.

Chapter 3 :eqns 3.8 and 3.9.

3.8 says

$$r^2 \frac{d \phi}{dt}=L=constant$$.

Then 3.9 writes

$$\frac{d}{dt}=\frac{L}{r^2}\frac{d}{d\phi}$$

But is this correct? For example, if I act this on r, the radial coordinate, I get

$$\frac{dr}{dt}=0$$

For a general orbit in a central force, r is not a constant.
So is there a mistake in the book, or am I missing something?

 

[George Jones]

Yes, it's correct, but the notation is a little sloppy.

Let

$$r \left ( t \right) = \tilde{r} \left( \phi \left ( t \right) \right).$$

These are really different functions (of different vraiables) , but this type of notational abuse, denoting different, but related functions by the same letter, is very common in physics.

Using the chain rule, what is $dr/dt$? Using $d/dt$ from the book, what is $dr/dt$?

 

[krishna mohan]

Yes..I understand the mistake I am making...

I was assuming $$r$$ and $$\phi$$ are independent variables...

But once an orbit is determined by the initial conditions, they are no longer independent...

Is that correct?