WOLFRAM

Wolfram Language & System Documentation Center

Minimal Arc Length Solution

Plot the arc length of the solution against a parameter value in a parametric equation:
Wolfram Language code: psol = ParametricNDSolve[{y''[t] + y[t] == 3a Sin[y[t]], y[0] == y'[0] == 1, α'[t] == Sqrt[1 + y[t] ^ 2], α[0] == 0}, {y, α}, {t, 0, 10}, {a}];
Wolfram Language code: GraphicsRow[{Plot[Evaluate@Table[y[a][t] /. psol, {a, 0, 1, .1}], {t, 0, 10}, ImageSize -> Small], Plot[Evaluate[α[a][10] /. psol], {a, 0, 1}, MaxRecursion -> 0, PlotPoints -> 100, AxesLabel -> {a, α}, ImageSize -> Small]}]
Find the solution corresponding to the locally minimal arc length near a=.725:
Wolfram Language code: lmin = FindMinimum[α[a][10] /. psol, {a, 0.72}, AccuracyGoal -> 7]
Wolfram Language code: Show[{Plot[Evaluate@Table[y[a][t] /. psol, {a, .65, .8, .005}], {t, 0, 10}, PlotStyle -> Lighter[Gray, 0.5]], Plot[Evaluate[y[a][t] /. psol /. lmin[[2]]], {t, 0, 10}, PlotStyle -> Thick]}, ImageSize -> Medium]