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.
![Click for copyable input](assets.en/solve-the-tautochrone-problem/In_107.png)
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.
![Click for copyable input](assets.en/solve-the-tautochrone-problem/In_108.png)
dsdy = DSolveValue[abeleqn, h[y], y]
![](assets.en/solve-the-tautochrone-problem/O_56.png)
Using the relationship , solve for
.
![Click for copyable input](assets.en/solve-the-tautochrone-problem/In_109.png)
dxdy = Sqrt[dsdy^2 - 1]
![](assets.en/solve-the-tautochrone-problem/O_57.png)
Starting the curve from the origin and integrating yields as a function of
. Notice that the assumptions ensure the integrand is real valued.
![Click for copyable input](assets.en/solve-the-tautochrone-problem/In_110.png)
x[y_] = Integrate[dxdy, {y, 0, y},
Assumptions -> (2 g (T^2) )/(\[Pi]^2 y) > 1 && y > 0]
![](assets.en/solve-the-tautochrone-problem/O_58.png)
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
.)
![Click for copyable input](assets.en/solve-the-tautochrone-problem/In_111.png)
Show[ParametricPlot[{{x[y], y}, {-x[y], y}} /. {g -> 9.8, T -> 2}, {y,
0, (2 (9.8) 2^2)/\[Pi]^2}], ImageSize -> Medium]
![](assets.en/solve-the-tautochrone-problem/O_59.png)
Making the change of variables gives a simple, nonsingular parametrization of the curve with
.
![Click for copyable input](assets.en/solve-the-tautochrone-problem/In_112.png)
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.
![Click for copyable input](assets.en/solve-the-tautochrone-problem/In_113.png)
\[Theta]' == \[PlusMinus]FullSimplify[ Sqrt[
2 g (Last[c[\[Theta]Max]] - Last[c[\[Theta]]])] /Sqrt[
c'[\[Theta]].c'[\[Theta]]] , g > 0 && T > 0]
![](assets.en/solve-the-tautochrone-problem/O_60.png)
Visualize the motion along the tautochrone.
![](assets.en/solve-the-tautochrone-problem/swf_2.png)