Wolfram Language

Symbolic & Numeric Calculus

Solve the Tautochrone Problem

The tautochrone problem requires finding the curve down which a bead placed anywhere will fall to the bottom in the same amount of time. Expressing the total fall time in terms of the arc length of the curve and the speed yields the Abel integral equation . Defining the unknown function by the relationship and using the conservation of energy equation yields the following explicit equation.

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abeleqn = T == 1/Sqrt[2 g] \!\( \*SubsuperscriptBox[\(\[Integral]\), \(0\), \(y\)]\( \*FractionBox[\(h[z]\), SqrtBox[\(y - z\)]] \[DifferentialD]z\)\);

Use DSolveValue to solve the integral equation.

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dsdy = DSolveValue[abeleqn, h[y], y]

Using the relationship , solve for .

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dxdy = Sqrt[dsdy^2 - 1]

Starting the curve from the origin and integrating yields as a function of . Notice that the assumptions ensure the integrand is real valued.

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x[y_] = Integrate[dxdy, {y, 0, y}, Assumptions -> (2 g (T^2) )/(\[Pi]^2 y) > 1 && y > 0]

Using a time of descent of two seconds and substituting in the value of the gravitational acceleration, plot the maximal curve for the tautochrone. (The branch comes from the solution for the derivative of .)

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Show[ParametricPlot[{{x[y], y}, {-x[y], y}} /. {g -> 9.8, T -> 2}, {y, 0, (2 (9.8) 2^2)/\[Pi]^2}], ImageSize -> Medium]

Making the change of variables gives a simple, nonsingular parametrization of the curve with .

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c[\[Theta]_] = ( g T^2)/\[Pi]^2 {Sin[\[Theta]] + \[Theta], 1 - Cos[\[Theta]]} ;

Combining the conservation of energy equation and the chain rule produces the following differential equation for as a function of , which can be solved numerically.

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\[Theta]' == \[PlusMinus]FullSimplify[ Sqrt[ 2 g (Last[c[\[Theta]Max]] - Last[c[\[Theta]]])] /Sqrt[ c'[\[Theta]].c'[\[Theta]]] , g > 0 && T > 0]

Visualize the motion along the tautochrone.

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