Tautochrone curve

Lagrangian solution

For a simple harmonic oscillator released from rest, regardless of its initial displacement, the time it takes to reach the lowest potential energy point is always a quarter of its period, which is independent of its amplitude. Therefore, the Lagrangian of a simple harmonic oscillator is isochronous.

In the tautochrone problem, if the particle's position is parametrized by the arclength s(t) from the lowest point, the kinetic energy is then proportional to ${\displaystyle {\dot{s}}^2}$, and the potential energy is proportional to the height h(s). One way the curve in the tautochrone problem can be an isochrone is if the Lagrangian is mathematically equivalent to a simple harmonic oscillator; that is, the height of the curve must be proportional to the arclength squared:

${\displaystyle h(s)=s^2/(8r),}$

where the constant of proportionality is ${\displaystyle 1/(8r)}$. Compared to the simple harmonic oscillator's Lagrangian, the equivalent spring constant is ${\displaystyle k=mg/(4r)}$, and the time of descent is ${\displaystyle T/4=\frac{\pi }{2}\sqrt{\frac{m}{k}}=\pi \sqrt{\frac{r}{g}}.}$ However, the physical meaning of the constant ${\displaystyle r}$ is not clear until we determine the exact analytical equation of the curve.

To solve for the analytical equation of the curve, note that the differential form of the above relation is

${\displaystyle \begin{array}{cc}dh& =sds/(4r),\\ d{h}^2& =s^{2}d{s}^2/(16r^2)=h\left(d{x}^2+d{h}^2\right)/(2r),\\ {\left(\frac{dx}{dh}\right)}^2& =\frac{2r}{h}-1\end{array}}$

which eliminates s, and leaves a differential equation for dx and dh. This is the differential equation for a cycloid when the vertical coordinate h is counted from its vertex (the point with a horizontal tangent) instead of the cusp.

To find the solution, integrate for x in terms of h:

${\displaystyle \begin{array}{cc}\frac{dx}{dh}& =-\frac{\sqrt{2r-h}}{\sqrt{h}},\\ x& =-4r\int \sqrt{1-u^2}du,\end{array}}$

where ${\displaystyle u=\sqrt{h/(2r)}}$, and the height decreases as the particle moves forward ${\displaystyle dx/dh<0}$. This integral is the area under a circle, which can be done with another substitution ${\displaystyle u=\cos (t/2)}$ and yield:

${\displaystyle \begin{array}{cc}x& =r(t-\sin t),\\ h& =r(1+\cos t).\end{array}}$

This is the standard parameterization of a cycloid with ${\displaystyle h=2r-y}$. It's interesting to note that the arc length squared is equal to the height difference multiplied by the full arch length ${\displaystyle 8r}$.

"Virtual gravity" solution

The simplest solution to the tautochrone problem is to note a direct relation between the angle of an incline and the gravity felt by a particle on the incline. A particle on a 90° vertical incline undergoes full gravitational acceleration ${\displaystyle g}$, while a particle on a horizontal plane undergoes zero gravitational acceleration. At intermediate angles, the acceleration due to "virtual gravity" by the particle is ${\displaystyle g\sin \theta }$. Note that ${\displaystyle \theta }$ is measured between the tangent to the curve and the horizontal, with angles above the horizontal being treated as positive angles. Thus, ${\displaystyle \theta }$ varies from ${\displaystyle -\pi /2}$ to ${\displaystyle \pi /2}$.

The position of a mass measured along a tautochrone curve, ${\displaystyle s(t)}$, must obey the following differential equation:

${\displaystyle \frac{d^{2}s}{{dt}^2}=-{\omega }^{2}s}$

which, along with the initial conditions ${\displaystyle s(0)=s_0}$ and ${\displaystyle s^{\prime }(0)=0}$, has solution:

${\displaystyle s(t)=s_0\cos \omega t}$

It can be easily verified both that this solution solves the differential equation and that a particle will reach ${\displaystyle s=0}$ at time ${\displaystyle \pi /2\omega }$ from any starting position ${\displaystyle s_0}$. The problem is now to construct a curve that will cause the mass to obey the above motion. Newton's second law shows that the force of gravity and the acceleration of the mass are related by:

${\displaystyle \begin{array}{cc}-g\sin \theta & =\frac{d^{2}s}{{dt}^2}\\ & =-{\omega }^{2}s\end{array}}$

The explicit appearance of the distance, ${\displaystyle s}$, is troublesome, but we can differentiate to obtain a more manageable form:

${\displaystyle \begin{array}{cc}g\cos \theta d\theta & ={\omega }^{2}ds\\ ⟹ds& =\frac{g}{{\omega }^2}\cos \theta d\theta \end{array}}$

This equation relates the change in the curve's angle to the change in the distance along the curve. We now use trigonometry to relate the angle ${\displaystyle \theta }$ to the differential lengths ${\displaystyle dx}$, ${\displaystyle dy}$ and ${\displaystyle ds}$:

${\displaystyle \begin{array}{c}ds=\frac{dx}{\cos \theta }\\ ds=\frac{dy}{\sin \theta }\end{array}}$

Replacing ${\displaystyle ds}$ with ${\displaystyle dx}$ in the above equation lets us solve for ${\displaystyle x}$ in terms of ${\displaystyle \theta }$:

${\displaystyle \begin{array}{cc}ds& =\frac{g}{{\omega }^2}\cos \theta d\theta \\ \frac{dx}{\cos \theta }& =\frac{g}{{\omega }^2}\cos \theta d\theta \\ dx& =\frac{g}{{\omega }^2}{\cos }^2\theta d\theta \\ & =\frac{g}{2{\omega }^2}\left(\cos 2\theta +1\right)d\theta \\ x& =\frac{g}{4{\omega }^2}\left(\sin 2\theta +2\theta \right)+C_x\end{array}}$

Likewise, we can also express ${\displaystyle ds}$ in terms of ${\displaystyle dy}$ and solve for ${\displaystyle y}$ in terms of ${\displaystyle \theta }$:

${\displaystyle \begin{array}{cc}ds& =\frac{g}{{\omega }^2}\cos \theta d\theta \\ \frac{dy}{\sin \theta }& =\frac{g}{{\omega }^2}\cos \theta d\theta \\ dy& =\frac{g}{{\omega }^2}\sin \theta \cos \theta d\theta \\ & =\frac{g}{2{\omega }^2}\sin 2\theta d\theta \\ y& =-\frac{g}{4{\omega }^2}\cos 2\theta +C_y\end{array}}$

Substituting ${\displaystyle \varphi =2\theta }$ and $r=\frac{g}{4{\omega }^2}$, we see that these parametric equations for ${\displaystyle x}$ and ${\displaystyle y}$ are those of a point on a circle of radius ${\displaystyle r}$ rolling along a horizontal line (a cycloid), with the circle center at the coordinates ${\displaystyle (C_x+r\varphi ,C_y)}$:

${\displaystyle \begin{array}{cc}x& =r\left(\sin \varphi +\varphi \right)+C_x\\ y& =-r\cos \varphi +C_y\end{array}}$

Note that ${\displaystyle \varphi }$ ranges from ${\displaystyle -\pi \le \varphi \le \pi }$. It is typical to set ${\displaystyle C_x=0}$ and ${\displaystyle C_y=r}$ so that the lowest point on the curve coincides with the origin. Therefore:

${\displaystyle \begin{array}{cc}x& =r\left(\varphi +\sin \varphi \right)\\ y& =r\left(1-\cos \varphi \right)\end{array}}$

Solving for ${\displaystyle \omega }$ and remembering that ${\displaystyle T=\frac{\pi }{2\omega }}$ is the time required for descent, being a quarter of a whole cycle, we find the descent time in terms of the radius ${\displaystyle r}$:

${\displaystyle \begin{array}{cc}r& =\frac{g}{4{\omega }^2}\\ \omega & =\frac{1}{2}\sqrt{\frac{g}{r}}\\ T& =\pi \sqrt{\frac{r}{g}}\end{array}}$

(Based loosely on Proctor, pp. 135–139)

Abel's solution

Niels Henrik Abel attacked a generalized version of the tautochrone problem (Abel's mechanical problem), namely, given a function ${\displaystyle T(y)}$ that specifies the total time of descent for a given starting height, find an equation of the curve that yields this result. The tautochrone problem is a special case of Abel's mechanical problem when ${\displaystyle T(y)}$ is a constant.

Abel's solution begins with the principle of conservation of energy – since the particle is frictionless, and thus loses no energy to heat, its kinetic energy at any point is exactly equal to the difference in gravitational potential energy from its starting point. The kinetic energy is $\frac{1}{2}m{v}^2$, and since the particle is constrained to move along a curve, its velocity is simply ${\displaystyle d\ell /dt}$, where ${\displaystyle \ell }$ is the distance measured along the curve. Likewise, the gravitational potential energy gained in falling from an initial height ${\displaystyle y_0}$ to a height ${\displaystyle y}$ is ${\displaystyle mg(y_0-y)}$, thus:

${\displaystyle \begin{array}{cc}\frac{1}{2}m{\left(\frac{d\ell }{dt}\right)}^2& =mg(y_0-y)\\ \frac{d\ell }{dt}& =\pm \sqrt{2g(y_0-y)}\\ dt& =\pm \frac{d\ell }{\sqrt{2g(y_0-y)}}\\ dt& =-\frac{1}{\sqrt{2g(y_0-y)}}\frac{d\ell }{dy}dy\end{array}}$

In the last equation, we have anticipated writing the distance remaining along the curve as a function of height (${\displaystyle \ell (y))}$, recognized that the distance remaining must decrease as time increases (thus the minus sign), and used the chain rule in the form $d\ell =\frac{d\ell }{dy}dy$.

Now we integrate from ${\displaystyle y=y_0}$ to ${\displaystyle y=0}$ to get the total time required for the particle to fall:

${\displaystyle T(y_0)={\int }_{y=y_0}^{y=0}dt=\frac{1}{\sqrt{2g}}{\int }_0^{y_0}\frac{1}{\sqrt{y_0-y}}\frac{d\ell }{dy}dy}$

This is called Abel's integral equation and allows us to compute the total time required for a particle to fall along a given curve (for which ${\displaystyle d\ell /dy}$ would be easy to calculate). But Abel's mechanical problem requires the converse – given ${\displaystyle T(y_0)}$, we wish to find ${\displaystyle f(y)=d\ell /dy}$, from which an equation for the curve would follow in a straightforward manner. To proceed, we note that the integral on the right is the convolution of ${\displaystyle d\ell /dy}$ with ${\displaystyle 1/\sqrt{y}}$ and thus take the Laplace transform of both sides with respect to variable ${\displaystyle y}$:

${\displaystyle \mathcal{L}[T(y_0)]=\frac{1}{\sqrt{2g}}\mathcal{L}\left[\frac{1}{\sqrt{y}}\right]F(s)}$

where ${\displaystyle F(s)=\mathcal{L}\left[d\ell /dy\right]}$. Since $\mathcal{L}\left[1/\sqrt{y}\right]=\sqrt{\pi /s}$, we now have an expression for the Laplace transform of ${\displaystyle d\ell /dy}$ in terms of the Laplace transform of ${\displaystyle T(y_0)}$:

${\displaystyle \mathcal{L}\left[\frac{d\ell }{dy}\right]=\sqrt{\frac{2g}{\pi }}{s}^{\frac{1}{2}}\mathcal{L}[T(y_0)]}$

This is as far as we can go without specifying ${\displaystyle T(y_0)}$. Once ${\displaystyle T(y_0)}$ is known, we can compute its Laplace transform, calculate the Laplace transform of ${\displaystyle d\ell /dy}$ and then take the inverse transform (or try to) to find ${\displaystyle d\ell /dy}$.

For the tautochrone problem, ${\displaystyle T(y_0)=T_0}$ is constant. Since the Laplace transform of 1 is ${\displaystyle 1/s}$, i.e., $\mathcal{L}[T(y_0)]=T_0/s$, we find the shape function $f(y)=d\ell /dy$:

${\displaystyle \begin{array}{cc}F(s)=\mathcal{L}\left[\frac{d\ell }{dy}\right]& =\sqrt{\frac{2g}{\pi }}{s}^{\frac{1}{2}}\mathcal{L}[T_0]\\ & =\sqrt{\frac{2g}{\pi }}{T}_0{s}^{-\frac{1}{2}}\end{array}}$

Making use again of the Laplace transform above, we invert the transform and conclude:

${\displaystyle \frac{d\ell }{dy}=T_0\frac{\sqrt{2g}}{\pi }\frac{1}{\sqrt{y}}}$

(Simmons, Section 54).

Using the fact that ${\displaystyle {\left(\frac{d\ell }{dy}\right)}^2={\left(\frac{dx}{dy}\right)}^2+1}$, we can rewrite the above as:

${\displaystyle {\left(\frac{dx}{dy}\right)}^2+1=T_0^2\frac{2g}{{\pi }^2}\frac{1}{y}}$

Let:

${\displaystyle r=\frac{T_0^{2}g}{{\pi }^2}}$

So that:

${\displaystyle \begin{array}{cc}{\left(\frac{dx}{dy}\right)}^2+1& =\frac{2r}{y}\\ {\left(\frac{dx}{dy}\right)}^2& =\frac{2r-y}{y}\\ dx& =\sqrt{\frac{2r-y}{y}}dy\end{array}}$

In order to integrate, substitute:

${\displaystyle \begin{array}{cc}y& =2r{\sin }^2\frac{\theta }{2}\\ dy& =2r\sin \frac{\theta }{2}\cos \frac{\theta }{2}d\theta \end{array}}$

We can now rewrite the equation for ${\displaystyle dx}$ as:

${\displaystyle \begin{array}{cc}dx& =\sqrt{\frac{2r-2r{\sin }^2\frac{\theta }{2}}{2r{\sin }^2\frac{\theta }{2}}}2r\sin \frac{\theta }{2}\cos \frac{\theta }{2}d\theta \\ & =2r\sqrt{\frac{1-{\sin }^2\frac{\theta }{2}}{{\sin }^2\frac{\theta }{2}}}\sin \frac{\theta }{2}\cos \frac{\theta }{2}d\theta \\ & =2r\sqrt{\frac{{\cos }^2\frac{\theta }{2}}{{\sin }^2\frac{\theta }{2}}}\sin \frac{\theta }{2}\cos \frac{\theta }{2}d\theta \\ & =2r{\cos }^2\frac{\theta }{2}d\theta \\ & =r(\cos \theta +1)d\theta \end{array}}$

Integrating both sides arrives at:

${\displaystyle x=r(\theta +\sin \theta )}$

And as we parameterized ${\displaystyle y}$ as:

${\displaystyle \begin{array}{cc}y& =2r{\sin }^2\frac{\theta }{2}\\ & =r(1-\cos \theta )\end{array}}$

we now have equations for both ${\displaystyle x}$ and ${\displaystyle y}$ parameterized by ${\displaystyle \theta }$:

${\displaystyle \begin{array}{cc}x& =r(\theta +\sin \theta )\\ y& =r(1-\cos \theta )\end{array}}$

which trace a tautochrone sitting on top of the ${\displaystyle x}$ axis.

(Based on O'neil, pp. 263-264)