I am attempting the attached problem from Emmy Noether's Wonderfull theorem by Dwight Neuenschwander.

I am happy with parts a. and b.

For part c I get:

$\displaystyle H=x_k' p_k - L = \frac {n x_k' x_k'} {r'} - n r' = \frac {n r'^2} {r'} - n r' = 0$

It seems quite reasonable to me that H and hence E could be zero. But, what are they saying when they ask "Do you need to make a distinction between H as a function and the numerical value of H?".

Then how can part d be solved? Surely if H = 0 then K = -U? I suspect that their equation U=-0.5n^2 is wrong and should read U = -0.5 r ^2? But even assuming this to be true implies that $\displaystyle L = \frac 12 r'^2 + \frac 12 r^2$ and not nr'.