A straight line may be the shortest 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 between two points, but a "Brachistochrone" curve is the path of least timepic.twitter.com/aWPoz9X6i9
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There are several assumptions going on there to make that statement correct though.
several? all it assumes is that it takes place on the surface of earth and the particles are point-like .. very common assumptions
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and of course that the beads are constrained to move along the rods ..
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Those are pretty big assumptions right there… Also, it could occur on the surface of the earth and still have completely different results… e.g., instead of moving under the gravitational influence, you could have the motion happening in the floor for instance.
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"pretty big"? .. i mean you could go down this rabbit hole with literally anything .. there's not a single thing that doesn't make "huge" assumptions under your definition
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Even less than that. All the brachistochrone constraints require is the path be a geodesic for a given metric, no?
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I think the issue in reading it your first statement seems to assume all cases of measuring distance so people who read it assume the second one does as well without clarification.
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