Newtonian Gravity in Higher Dimensions - Ray Goerke
This project has been completed with great success. Visit the webpage for the results.
Introduction: What if there were four or five or more spatial dimentions? Then newtonian gravity would fall off as the inverse cube or the inverse fourth of the distance. What kind of implications could this have for a two or three body system? I could try to create a computer simulation of some system like this and attempt to derive the equations themselves if possible. This one I like because I've often thought about it. I've heard that no stable orbit exists in the four dimensional case, I could try to confirm this.
Good luck. Actually, if I may distort the content of String Theory somewhat, there supposedly are 11 or so dimensions, but all but the 4 we know and love are "compactified" (curled up) so we can't notice them. (Don't ask me.) The problem with doing computation on more than 3D (which is hard enough to simulate on a 2D screen!) is, how do you display your conclusion? "Yep, it works!" is somehow unsatisfying to me, but it may make the theorists happy if it's really valid. -- Jess 15:44, 20 September 2008 (PDT)
You can certainly rederive the standard Newtonian theory and pretend that the world has a inverse cubic relationship, or better still an inverse nth order relationship. Not a bad idea from a theoretical point of view, since in making those changes you will have a much better understanding of the world as it is. -- Jason
In regards to displaying my conclusion (please correct me if I'm wrong), even if the bodies are sitting in four spacial dimensions, all the action would still only take place on a single plane (assuming I set up the initial conditions properly) because there's only two force vectors, so it's possible to present my results in two dimensions. Likewise with three bodies confined to a single volume. I'd get in trouble of course if I tried to include four bodies, but I doubt I'd get that far anyways. Thanks for all the comments -- Ray 10:53, 21 September 2008 (PDT)
I believe that's correct, independent of the number of dimensions; the 3-body case would be restricted (all the velocities must also lie in the mutual plane -- call it the "ecliptic") but not nearly as artificially restricted as the 4-body case would have to be. BTW, you can construct a nice argument for the r-(N-1) force law in N dimensions by invoking a generalization of Gauss' Law. You might also be interested in a paper Lorne Whitehead wrote when he was an undergraduate here, on spinning tops in N dimensions (or something like that). -- Jess 11:04, 14 October 2008 (PDT)
That's actually exactly how I derived it myslef, with generalized Gauss' Law. I'll check out that paper. --Ray 10:15, 23 October 2008 (PDT)
I have discovered a differential equation which I can solve for a given force law which whose solutions are the possible orbits. I will use numerical integration methods to solve for orbits of higher dimensions and try to see if I can figure out anything about what happens when n gets very large.--Ray 00:22, 6 November 2008 (PST)
