(John Bernoulli’s proof of Snel’s law using the mechanical equilibrium of a string under tension.) Two weights and hang from frictionless pulleys at and , meaning that the tension on the portions and is and . The point of contact, , slides without friction along . The potential energy of the system, , is minimized at equilibrium, where the horizontal components of the tensions and cancel, giving Snel’s law: . (Figure reproduced from Ref. 14.)
An alternative version of Bernoulli’s setup: (a) Two weights hanging from frictionless points and are assigned zero potential energy. (b) The two weights are then lifted, and the ends of the strings are joined at point along the line . The work done is equal to the increase in the potential energy: .
Frictionless pulleys that can slide in horizontal lines with a string passing through them a sufficient number of times gives the trajectory of the particle if is identified with at each segment. Because the string can only pass through each pulley an integer number of times, the ratios of the velocities are approximated by the ratio of the times the rope passes through each segment.
(a) Space–time trajectory of an otherwise free one-dimensional particle acted on by an impulsive force at . (b) Equivalent equilibrium configuration of two segments of a stretchable string with spring constant and an external force .
Analogies used in the principle of least action between mechanics, geometric optics, and the equilibrium of a nonstretchable string.
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