Quote:
Originally Posted by mithrandir
I think you will find that the vertical wire is the limit case of the brachistochrone when the points become vertically aligned. I don't feel up to remembering enough of my Calculus of Variations course - more than 30 years ago - to prove it.
Andrew
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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.
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.
Regards
Steven