Erratum :
https://en.wikipedia.org/wiki/Kepler_problem instead of https://en.wikipedia.org/wiki/Kepler's_equation in my #21 post .
In Maxima :
assume(ε>0,ε<1);
integrate(1/(1+ε*cos(θ))^2,θ);
assume(ε>0,ε<1,θ>-π,θ<π);
solve(t=integrate(1/(1+ε*cos(θ))^2,θ),θ);
https://en.wikipedia.org/wiki/Kepler's_equation just gives at the end the Kepler's 1st law of my #17 post in general ( ##-\frac{k}{m}##instead of ##GM## ) .
The problem is solved , we've r(θ) & t(θ) , the summary with " the conversation ultimately ends without a solution " needs to be edited ...
I'm back , sorry for long time .
In short :
##L=r^2ω=constant## ( Kepler's 2nd law )
##r=\frac{L^2}{GM(1+εcos(θ))}## ( Kepler's 1st law )
⇒##\frac{1}{ω}=\frac{dt}{dθ}=\frac{L^3}{(GM(1+εcos(θ)))^2}##
⇔##t(θ)=\frac{L^3}{(GM)^2}∫\frac{1}{(1+εcos(θ))^2}dθ##
##∫\frac{1}{(1+εcos(θ))^2}dθ## ...
I can't edit my messages #6 and #9 but r=c²/(G M(1+ε cos(θ))) , c²/r(t)³-G M/r(t)²-r''(t)=0 can be simplified :
c²/r(t)-r(t)²r''(t)=G M
c²G M(1+ε cos(θ))/c²-c⁴/(G M(1+ε cos(θ)))²ε cos(θ)(G M)³(1+ε cos(θ))²/c⁴=G M .
Sorry pasmith but I don't understand from -L du/dθ in #7 ...
Well I found how...
I found t(θ) :
1/θ'(t)=dt/dθ=t'(θ)=(p/(1+ε cos(θ)))²/c
t(θ)=see https://www.wolframalpha.com/input?i2d=true&i=Divide[a²*antiderivative+\(40)Divide[1,1+b*cos\(40)x\(41)]\(41)²,c] .
Now we need to from t(θ) to θ(t) , how to do this ? With WolframAlpha ?
I can't edit my previous messages to...
Well sorry I thought equations will be rendered automatically like WolframAlpha , it seems to need LaTeX but I never used this and it seems tiresome ...
In fact I want to verify if r(t) is solution to this differential equation to prove the Kepler's 1st law :
c , G , M constants :
c²/r(t)^3-G...
Hi
I posted this differential equation to WolframAlpha https://www.wolframalpha.com/input?i2d=true&i=Power[\(40)Divide[a,1+b*cos\(40)y\(40)x\(41)\(41)]\(41),2]*y'\(40)x\(41)=c but no solution , " Standard computation time exceeded... Try again with Pro computation time "
Should I ( buy and )...