Integral with dot products

nos

Hi everyone,

I'm trying to understand the integral on http://www.phys.lsu.edu/~jarrell/COU...p14/chap14.pdf (page 14)

I get all the steps except the how to get from eq. 48 to eq 49. I'm not really sure how to compute all the dot products. He lets the angle between n and β be θ
and angle between β and [itex]\dot{β}[/itex] be θ(0)

Attempt at solution:

[itex]n\cdotβ=βcos(θ)[/itex]
[itex]β\cdot\dot{β}=β\dot{β}cos(θ(0))[/itex]
[itex]n\cdot\dot{β}=\dot{β}cos(θ-θ(0))[/itex]?

If this is correct, do I proceed by applying the difference identity of cosine in the last dot product and then square the whole thing? There are going to be a lot of terms, so before wasting more time on expanding, let's first see if what i'm doing is in fact the right way to do this integral.

Many thanks!

haruspex

You could only do something like that if you know the three vectors are coplanar. Are they?

nos

Yes that is whats been troubling me. I am not sure. But how else do you go from eq 48 to eq 49?

haruspex

Sorry, I can't follow it either. I strongly suspect a typo, like a β that should be a β-dot or vv., but I haven't been able to find a simple explanation.

nos

Oh well, thanks very much anyway for replying:)