jinydu wrote:I checked your calculations of the acceleration at the surface of the 4D planet using a different method, and they do seem correct.
jinydu wrote:However, I'm not sure about the value of G being higher by a factor of 3pi/8. This would also increase the mass of the planet by 3pi/8, and hence end up increasing the acceleration at the surface of the planet by (3pi/8)^2, so not everything to do with gravity would increase by 3pi/8.
jinydu wrote:Bug Detected: For some reason, "[eight])" keeps getting replaced by a smilie.
pat wrote:I've started working more formally on the calculations that I attempted above. Rather than cram them into BBCode and HTML, I have been using LaTeX. You can check out my start here: http://www.csh.rit.edu/~pat/math/quickies/orbit4/orbit4.pdf
I'm to a point where I cannot find any way to simplify the equation. It doesn't immediately appear to me that the only solution is a circle. But, it is immediately apparent that there are circles which are solutions.
PWrong wrote:I'm just wondering though: Is the ending you came to the same as the 3D equivalent except the r(t) doesn't cancel out?
PWrong wrote:It seems like it should be, but your method is a bit different from the one on the Max Francis webpage.
PWrong wrote:Also, how come you don't mention the words like perihilion or use e for eccentricity anywhere? Would it help if you did?
mghtymoop wrote:Integrate the whole thing, it is really that simple, don't remove variables because you believe they are over time because with time being the 4th dimension this makes no sence, only problem is you are left with unsloved constants but this gets you a lot closer to your goal
pat wrote:The integrand on the left hand side is: r(t)<sup>d<sup>2</sup>r</sup>/<sub>dt<sup>2</sup></sub>. But, the bold r is a vector, the italic r its magnitude. I don't know how to integrate that. It's basically like trying to integrate f(t)g''(t). It's not as simple as F(t)g'(t). Am I missing some nice way to write things?
PWrong wrote:Oh, now I can help!
You can do it with integration by parts
pat wrote:
Following the derivation in my college calculus book, I came up with:
<sup>d<sup>2</sup>p</sup>/<sub>dθ</sub> = - α<sup>2</sup> p
pat wrote:But, that doesn't make for any stable orbits since the solution to that differential equation is:
p(θ) = a * sin( α θ + θ<sub>0</sub> )
PWrong wrote:Doesn't it violate all the laws that you've already proven?
For instance, if the planet spirals inward, how can it sweep out equal areas in equal times?
Also, what's the difference between spinning around a few times then flying off, and simply spiralling outwards? :?
1. Could you explain how you came to equation 6.3? You suddenly introduce e into the equation. It seems like a huge jump to make. Is that the maths constant e, or a variable?
2. Why is it that the equations for Beta=0, Beta>0 and Beta<0 are all completely different?
3. The graph of equation 6.4 is an ugly star-shaped orbit. Why is that?
PWrong wrote:I'm fairly good with complex numbers, but I can't see why you would use them here.
PWrong wrote:Well, I guess I'll have to accept that orbits don't work properly in 4D.
jinydu wrote:If you had to explain, in one sentence, why orbits are unstable in 4D but not in 3D, what would the sentence be, please?
Return to Where Should I Post This?
Users browsing this forum: No registered users and 2 guests