Why are triangular orbits not possible in our universe?
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$\begingroup$ I wouldn't say they're impossible, only that I know of no equation of motion that yields triangular solutions in this context. Could you say why you focus on triangular trajectories? $\endgroup$– MiyaseCommented Jun 10, 2022 at 21:42
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2$\begingroup$ The answer is pretty much the same as physics.stackexchange.com/q/692338 $\endgroup$– shai horowitzCommented Jun 10, 2022 at 21:55
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$\begingroup$ If you get very close to a black hole, orbits can become very strange looking, including some with somewhat triangular features. $\endgroup$– RC_23Commented Jun 11, 2022 at 2:15
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2 Answers
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Triangular orbits would have sharp corners. This implies sudden changes of direction, which in turn implies sudden changes of velocity and therefore accelerations. Acceleration is defined by
$$\text {acceleration}=\frac{\text{change in velocity}}{\text{time taken to change}}$$
The top line of the fraction has a finite magnitude, but the bottom line is immeasurably small if the corner is sharp and the body is moving at a finite speed. So the acceleration at the corners is infinite. An infinite acceleration needs the orbiting body to be acted upon by an infinite force. What could supply such a force?
answered Jun 10, 2022 at 22:00
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A triangle has corners, three of 'em. At those corners, acceleration must be high, to alter the direction of a moving object abruptly. So, there are three high-force (F= mA) events in each orbit. To the best of my knowledge, there are no central forces consistent with an isotropic (same-physics-in-all-places) universe that can exert such force. One could have three reflectors and propel a rubber ball in a triangular 'orbit', but the only way a central force would have that effect, is with an equilateral triangle orbit around a center of constant potential energy except with a large potential up-step at some radius R; this is called a 'spherical square well' in quantum mechanics.
Expected central force fields in this three dimensional universe (Yukawa potentials) are exponential in radius R, times R^(-2). That doesn't create any region with straight-line trajectories in multiple directions, so is inconsistent with triangle orbits.
