(assuming a uniform gravitational field)
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A ball sliding along a 'Lemniscate' curve takes the same time to arrive at a given point as a ball that takes the direct pathpic.twitter.com/a8FtMonFiE
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How are these curves found
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calculus of variations!
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Hey, is this a brachistochrone as the ball falls?
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Nope, different problem
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This soothed my soul, thank you.
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Very cool. The shape/evolution of the green ball ensemble seems interesting too. Distance s ball travels in time t, s= (g/2) sinθ t^2, so I'm thinking it maintains the same shape, just scales rapidly with t? Guessing r = sinθ. (Maybe why when I google it comes up polar curves.)
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My guess/interpretation : If you try to maximize the horizontal (in gif’s frame of reference) acceleration then the solution would be at theta = - pi/4. The further the angle is from it, the lesser the horizontal acc, resulting in that semicircle.
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I've never done/seen the calculus of variations on this sucker...always the brachistochrone. So standard! Maybe I'll do the math on this one tomorrow
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impossible not to anthropomorphize that green ball
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