How do you derive equation 2.20 in the dynamics notes?

[Lavace]

Homework Statement

http://damtp.cam.ac.uk/user/dt281/dynamics/two.pdf"

Looking at page 5, equations (2.19) and (2.20)

The Attempt at a Solution

I cannot understand how they derived the (2.20), at first from comparing the solutions I had assumed r(dot)' had disappeared as we were differentiating with respect to dr'.

I then went about the derivative of the cross product:

Omega X r(dot)' + ... But in the solution we find Omega x (Omega X r').


Could anyone please help clear this up for me.

 

[D H]

What that paper means by $\partial L/\partial \mathbf r'$ is

$$\frac{\partial L}{\partial{\mathbf r}'} \equiv<br /> \frac{\partial L}{\partial x'}\hat x' +<br /> \frac{\partial L}{\partial y'}\hat y' +<br /> \frac{\partial L}{\partial z'}\hat z'$$

It might be easier for you to see how (2.20) follows from (2.19) by using (2.19) in its first form,

$$L = \frac 1 2 m \Bigl((\dot x - \omega y')^2 + (\dot y + \omega x')^2 + \dot z^2\Bigl)$$

The result in (2.20) is a much more general result. It applies to any rotation vector $\boldsymbol \omega$, not just the pure z rotation used in that example.