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Originally Posted by sjastro
What you will find is that the partial derivatives in Euler LaGrange equation vanish as there is no horizontal component. This reduces the equation to the trivial case 0=0. There is no stationary value.
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However, the function [image that won't paste] is particularly nice since x does not appear explicitly. Therefore, [another image] and we can immediately use the Beltrami identity ...
http://mathworld.wolfram.com/Brachis...neProblem.html
Are you suggesting there is no limit as delta x approaches zero, and the the vertical line is a discontinuity? Physics and discontinuities don't get on well together.
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From a physics perspective since the wire is frictionless, the wire does not provide a constraint when the endpoints are vertical. The bead falls purely under the effect of gravity which is in a straight line.
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Doesn't that assume a non-rotating frame of reference? A falling object in a rotating frame follows a parabola. (Ignoring any effects of air resistance.)