Derivation of Kepler's first law

An adaptation/explanation of Stewart's derivation of Kepler's First Law from Newton's Law of Gravitation

[jdtaramona]

thanks once again David. Despite being too tiny this extra quantity the derivation is a very interesting matter

[md.rezaulislam]

First i see this proof it seems quite difficult now its clear, thanks.

[Gonza-lhvo]

Great explanation! I just don't get what the c vector is. Is there anyway to calculate it in terms of the initial conditions?

[DiffQ_Bro]

In the previous problem, a x h = -GMu x (u x u') was the proof that was to be demonstrated. If (v x h)' = a x h, then how can (v x h)' = GMu' (?)

[samuelhe]

I don't understand at 3:55 why this term equals 0, can you explain me why we get that result ?

[glencheckisthename]

Hi thanks for the clear explanation from S.K. !!

i have one question.
doesnt' c^ needs to be parallel to r^?
v x h // u ( because v⊥u and h⊥u)
if you plus u^ with c^ (whatever that isnt parellel to u^)
u^+c^ is not parellel to v^ x h^ anymore In your video graph,

[fenghualiu]

The first five minutes is essentially conservation of angular momentum right?

[johnwilliams]

how do we know that c lies in the same plane as r and v?

[jdtaramona]

Thanks a lot once again David, maybe one of the most beautiful solutions in the history of physics/mathematics. Maybe I did not understand very well the physical meaning of vector C yielded during integration of motion equation (V x h)' = GM.U'. I suspect that C is the eccentricity vector. On the other hand, some books include an additional angle substracted from angle theta (true anomaly) as an argument of cosine function, and I do not know which angle is that.. r=p/1+ecos(theta-phi), where p is semi-lactus rectum and phi and C as well are determined by the initial conditions.
Finally, Have you thought to derive this Kepler law for a more general case as a trajectory of a massive particle in the schwarzschild geometry in the same way as yo have done here by means of vectors?

[khushburaza]

Hi! Quick question: I am a junior in high school currently taking Math Pre-Calc HL, so I havent learned calculus yet. I wanted to write an exploration/research paper on deriving and applying Kepler's 3 laws but bc i didnt learn calc, im having trouble understanding online proofs of it. do you think i could still learn these proofs, including yours, easily if i learn very simple calculus? or should I just try to find another topic on my level?

[Itsjustjenah]

that was difficult but you made it understandable