Timeline for Can an object accelerate to infinite speed in a finite time-interval in non-relativistic Newtonian mechanics?
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Apr 22, 2020 at 18:28 | comment | added | Ilmari Karonen | … Also, since pulling the binary closer together reduces its orbital period, it's possible to arrange the system so that a suitable encounter is always possible even though the smaller body keeps bouncing more and more frequently between the two binaries. In the limit, the semi-major axis of each binary tends to zero (which, for pointlike bodies under Newton's laws, makes their potential energy tend to minus infinity) while each binary is pushed away from the other at a velocity that tends to infinity as a result of ever more frequent encounters with the small body shuttling between them. | |
Apr 22, 2020 at 18:22 | comment | added | Ilmari Karonen | @AlanRominger: I know this is a very old discussion, but since nobody's responded to it: the trick in the paper is that the orbital periods are tuned so that each time the small body encounters one of the binary systems of large bodies, it steals a little bit of energy from the binary, thus pulling the bodies in the binary closer together (and so reducing their mutual potential energy) while getting flung back with more velocity (and thus more kinetic energy) than it had before. Of course, since momentum is conserved, each encounter also pushes the two binaries further away from each other. | |
Aug 17, 2011 at 17:02 | comment | added | Alan Rominger | @Blue I take the image here to show a mass moving through the CM of two binary pairs, this means that it does not have the possibility of coming infinitely close to one of the large $M$ masses. The 1/t sin(1/t) still sees an even greater magnitude derivative and 2nd derivative than 1/t. If it does not come infinitely close to any other mass, there is no plausible mechanism to have the acceleration increase infinitely. I've only glanced at the paper a little bit, but on the surface these claims are still incoherent. | |
Aug 17, 2011 at 16:52 | comment | added | BlueRaja - Danny Pflughoeft |
@Zassounotsukushi: Imagine the left-half of the position-vs-time graph 1/t , with t < 0 . No matter what speed you give me, I can find a time t < 0 where the object is moving at an even greater speed; thus at the limit, the object's speed is infinite. Xia found a case where, under Newton's laws, this happens (though the graph behaves a bit more like (1/t) sin(1/t) ). At t=0 , there's what we call a singularity; I'm not really sure how to answer the question of where the object is, or what it looks like, at or after t=0 , partly because singularities don't make physical sense.
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Aug 7, 2011 at 3:04 | vote | accept | BlueRaja - Danny Pflughoeft | ||
Aug 4, 2011 at 22:51 | comment | added | mmc | @Zassounotsukushi Check page 8 of the paper. | |
Aug 4, 2011 at 20:32 | comment | added | Alan Rominger | I'm having trouble understanding your diagram. I'm fine with the fact that $m$ is a comparatively small mass, but where does it go when it oscillates to infinite speeds? Does it get out of the gravity well eventually? And how do the 2 binary systems maintain the distance from each other? Is it supposed to be implied that they're rotating, or that they're oscillating as well? | |
Aug 4, 2011 at 19:57 | history | edited | BebopButUnsteady | CC BY-SA 3.0 |
added a picture
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Aug 4, 2011 at 19:07 | comment | added | kuzand | We can tell the same for atracting point charges. As the distance r decreases the force between them increases. Taking limit $r\to 0$ you get $F\to \infty$, therefore $u\to \infty$. But this means that the speed aproaches the infinity, not that the velocity is infinite. Take a look at definition of the limit. | |
Aug 4, 2011 at 18:26 | comment | added | BlueRaja - Danny Pflughoeft | Amazing! Just the answer I was looking for. Here is a link to the paper mentioned, and here is an article trying to explain it in laymen's terms. Could you perhaps add some more information about the "five bodies interacting?" | |
Aug 4, 2011 at 18:04 | history | answered | BebopButUnsteady | CC BY-SA 3.0 |