The Curve of Least Energy

SUMMARY We have found the simple equation ± w = Vcos (ip — y), and solved it to find theequation of the curve of least energy in Whewell form s == ± V2 c F{ cos" 1 Vcos(^ — <p), — | .\ ^^, / We also developed a differential equation for the curve in Cartesian coordinatesaligned with the axis of symmetry, dy _ ^/l-(y/2c) 4 dx (y/2c} 2 ' The solution of this equation, for given initial conditions, led to x=^/2c\2E(cos- l ^,- l -\-F(cos- l ^,- l -}~\.[ \ 2c'^) \ 2c'V2/J We considered the curve of least energy connecting the point (—1, 0) to thepoint (+1, 0) with vertical initial and final orientations. This curve has minimumradius of curvature _ r(j) 2 c (2^) 3/2 ' rises to a heighthas arc lengthand energy X= Jf=2c, cr^^ird) 4 2^2 (27r) 2 '._ 1 _ w ' c 2 ro) 4 ' or about 91.39 percent of that of the simple semicircle approximation. We havealso given a method for finding approximations, consisting of circular arcs, to thecurve of least energy.Note that the curve found here is