A doubt from the book Galactic Dynamics from Binney and Tremaine

In summary, the conversation was about equations 3.8 and 3.9 in Chapter 3 of the book Galactic Dynamics from Binney and Tremaine. The equations discussed the relationship between the radial coordinate r and the angular coordinate phi in a central force orbit. The notation in the book was slightly confusing, but after using the chain rule, it was determined that the equations were correct. The conversation also touched on the fact that r and phi are not truly independent variables once an orbit is determined by initial conditions.
  • #1
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

[tex]r^2 \frac{d \phi}{dt}=L=constant[/tex].

Then 3.9 writes

[tex]\frac{d}{dt}=\frac{L}{r^2}\frac{d}{d\phi}[/tex]

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

[tex]\frac{dr}{dt}=0[/tex]

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?
 
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  • #2
Yes, it's correct, but the notation is a little sloppy.

Let

[tex]r \left ( t \right) = \tilde{r} \left( \phi \left ( t \right) \right). [/tex]

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 [itex]dr/dt[/itex]? Using [itex]d/dt[/itex] from the book, what is [itex]dr/dt[/itex]?
 
  • #3
Yes..I understand the mistake I am making...

I was assuming [tex]r[/tex] and [tex]\phi[/tex] are independent variables...

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

Is that correct?
 

1. What is the main focus of the book Galactic Dynamics from Binney and Tremaine?

The main focus of the book is to provide a comprehensive understanding of the dynamics of galaxies, including their formation, evolution, and interactions with other galaxies.

2. Who are the authors of the book Galactic Dynamics from Binney and Tremaine?

The authors of the book are James Binney and Scott Tremaine, both renowned astrophysicists with extensive experience and contributions in the field of galactic dynamics.

3. What are some of the key topics covered in the book Galactic Dynamics from Binney and Tremaine?

Some of the key topics covered in the book include galactic potential theory, orbits and resonances, stellar dynamics, spiral structure, and galaxy interactions and mergers.

4. Is the book suitable for beginners in the field of galactic dynamics?

While the book is considered a classic in the field and may be challenging for beginners, it is still a valuable resource for those with a strong foundation in astrophysics and mathematics.

5. Are there any updated editions of the book Galactic Dynamics from Binney and Tremaine?

Yes, the second edition of the book was published in 2008, which includes updates and new material on recent advances in the field of galactic dynamics.

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