Question

Show that kepler's law of areas is equivalent to the conservation of angular momentum

Answer 1

really simple approach is to look at the area element in polar: \(dA = \frac{1}{2}r^2(\theta) d\theta\)

so \(\frac{dA}{dt} = const \implies \frac{1}{2}r^2(\theta) \frac{d\theta}{dt} = const \implies r^2 \omega = const \)

that's bag of a cigarette backet. but i think it would lead to something nice if you had the time.

Answer 2

and ellipses in polar are a mare. IMHO.

Answer 3

Hey just a naive quesiton
\(r\) isn't a constant with respect to time for ellipse right ?

Answer 4

Nice one @IrishBoy123

Also check this out https://www.princeton.edu/~wbialek/intsci_web/dynamics3.4.pdf

Answer 5

astro, that article really takes it places. good spot.

ganesh, i agree

i mention ellipses only because i recall reading [Bill Bryson's book, not a physics book!] that Newton worked out that the orbit **had to be ** elliptical, and it came to light in some curious episode involving Hooke and Wren and quite some shenanigans .

my suspicion is that if you used r(theta) for an ellipse in the MIT stuff you'd get to see something magical. ellipses are mentioned only 3 times in the article, in the second to lat paragraph of the piece. maybe that is where one could start looking at it.

doing it just for a laugh, which i would if i could, is sadly beyond my abilities.....