Editing Brachistochrone curve (section)

==Jakob Bernoulli's solution==

Johann's brother [[Jacob Bernoulli|Jakob]] showed how 2nd differentials can be used to obtain the condition for least time. A modernized version of the proof is as follows. If we make a negligible deviation from the path of least time, then, for the differential triangle formed by the displacement along the path and the horizontal and vertical displacements,

:<math>ds^2=dx^2+dy^2</math>.

On differentiation with ''dy'' fixed we get,

:<math>2ds\ d^2s=2dx\ d^2x</math>.

And finally rearranging terms gives,

:<math>\frac{dx}{ds}d^2x=d^2s=v\ d^2t</math>

where the last part is the displacement for given change in time for 2nd differentials. Now consider the changes along the two neighboring paths in the figure below for which the horizontal separation between paths along the central line is ''d<sup>2</sup>x'' (the same for both the upper and lower differential triangles). Along the old and new paths, the parts that differ are,
[[Image:Path function 2.PNG|right]]

:<math>d^2t_1=\frac{1}{v_1}\frac{dx_1}{ds_1}d^2x</math>
:<math>d^2t_2=\frac{1}{v_2}\frac{dx_2}{ds_2}d^2x</math>

For the path of least times these times are equal so for their difference we get,
:<math>d^2t_2-d^2t_1=0=\bigg(\frac{1}{v_2}\frac{dx_2}{ds_2}-\frac{1}{v_1}\frac{dx_1}{ds_1}\bigg)d^2x</math>

And the condition for least time is,
:<math>\frac{1}{v_2}\frac{dx_2}{ds_2}=\frac{1}{v_1}\frac{dx_1}{ds_1}</math>
which agrees with Johann's assumption based on the [[Snell's law|law of refraction]].