Editing Tautochrone curve

{{short description|Curve for which the time to roll to the end is equal for all starting points}}
[[File:Tautochrone curve.gif|300px|right|thumb|Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time. The blue arrows show the points' acceleration along the curve. On the top is the time-position diagram.]]
[[File:Objects representing tautochrone curve 03.gif|thumb|300px|Objects representing tautochrone curve]]

A '''tautochrone curve''' or '''isochrone curve''' ({{ety|grc|''ταὐτό'' ([[wikt:tauto-|tauto-]])|same||''ἴσος'' ([[wikt:iso-|isos-]])|equal||''χρόνος'' ([[wikt:chrono-#English|chronos]])|time}}) is the [[curve]] for which the time taken by an object sliding without [[friction]] in uniform [[gravity]] to its lowest point is independent of its starting point on the curve.  The curve is a [[cycloid]], and the time is equal to [[Pi|π]] times the [[square root]] of the radius of the circle which generates the cycloid, over the [[Gravitational acceleration|acceleration of gravity]].  The tautochrone curve is related to the [[brachistochrone curve]], which is also a cycloid.

== The tautochrone problem ==
[[File:Huygens - Horologium oscillatorium, sive De motu pendulorum ad horologia aptato demonstrationes geometricae, 1673 - 869780.jpeg|thumb|upright|[[Christiaan Huygens]], ''[[Horologium Oscillatorium|Horologium oscillatorium sive de motu pendulorum]]'', 1673]]
{{Quote box|width=30%|
quote=It was in the left hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time.
|source=''[[Moby Dick]]'' by [[Herman Melville]], 1851}}
The tautochrone problem, the attempt to identify this curve, was solved by [[Christiaan Huygens]] in 1659. He proved geometrically in his ''[[Horologium Oscillatorium]]'', originally published in 1673, that the curve is a [[cycloid]].

{{Blockquote|On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after having departed from any point on the cycloid, are equal to each other&nbsp;...<ref>{{cite book |last=Blackwell |first=Richard J. |title=Christiaan Huygens' The Pendulum Clock |publisher=Iowa State University Press |date=1986 |location=Ames, Iowa |isbn=0-8138-0933-9 |at= Part II, Proposition XXV, p.&nbsp;69}}</ref>}}

The cycloid is given by a point on a circle of radius <math>r</math> tracing a curve as the circle rolls along the <math>x</math> axis, as:
<math display="block">\begin{align}
  x &= r(\theta - \sin \theta) \\
  y &= r(1 - \cos \theta),
 \end{align}</math>

Huygens also proved that the time of descent is equal to the time a body takes to fall vertically the same distance as diameter of the circle that generates the cycloid, multiplied by <math>\pi / 2</math>. In modern terms, this means that the time of descent is <math display="inline">\pi \sqrt{r/g}</math>, where <math>r</math> is the radius of the circle which generates the cycloid, and <math>g</math> is the [[gravity of Earth]], or more accurately, the earth's gravitational acceleration.

[[File:Isochronous cycloidal pendula.gif|thumb|Five isochronous cycloidal pendulums with different amplitudes]]
This solution was later used to solve the problem of the [[brachistochrone curve]].  [[Johann Bernoulli]] solved the problem in a paper (''[[Acta Eruditorum]]'', 1697).

[[File:CyloidPendulum.png|right|thumb|Schematic of a [[cycloidal pendulum]]]]
The tautochrone problem was studied by Huygens more closely when it was realized that a pendulum, which follows a circular path, was not [[isochronous]] and thus his [[pendulum clock]] would keep different time depending on how far the pendulum swung. After determining the correct path, Christiaan Huygens attempted to create pendulum clocks that used a string to suspend the bob and curb cheeks near the top of the string to change the path to the tautochrone curve. These attempts proved unhelpful for a number of reasons. First, the bending of the string causes friction, changing the timing. Second, there were much more significant sources of timing errors that overwhelmed any theoretical improvements that traveling on the tautochrone curve helps. Finally, the "circular error" of a pendulum decreases as length of the swing decreases, so better clock [[escapement]]s could greatly reduce this source of inaccuracy.

Later, the mathematicians [[Joseph Louis Lagrange]] and [[Leonhard Euler]] provided an analytical solution to the problem.

== Lagrangian solution ==
For a [[Simple harmonic motion#Dynamics|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 timing|isochronous]].

In the tautochrone problem, if the particle's position is parametrized by the [[arclength]] {{math|''s''(''t'')}} from the lowest point, the kinetic energy is then proportional to <math>\dot{s}^2</math>, and the potential energy is proportional to the height {{math|''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:

{{block indent|1=<math> h(s) = s^2/(8r), </math>}}

where the constant of proportionality is <math>1/(8r)</math>. Compared to the simple harmonic oscillator's [[Lagrangian mechanics#Lagrangian|Lagrangian]], the equivalent spring constant is <math>k=mg/(4r)</math>, and the time of descent is <math>T/4=\frac{\pi}{2} \sqrt{\frac{m}{k}}=\pi \sqrt{\frac{r}{g}}.</math> However, the physical meaning of the constant <math>r</math> 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

{{block indent|1=<math>\begin{align}
dh &= s \,ds/(4r), \\
dh^2 &= s^2 \,ds^2/(16r^2) = h \left(dx^2 + dh^2\right)/(2r),\\
\left(\frac{dx}{dh}\right)^2&=\frac{2r}{h}-1
\end{align}</math>}}

which eliminates {{math|''s''}}, and leaves a differential equation for {{math|''dx''}} and {{math|''dh''}}. This is the differential equation for a [[Cycloid#Equations|cycloid]] when the vertical coordinate {{math|''h''}} is counted from its vertex (the point with a horizontal tangent) instead of the [[Cusp (singularity)|cusp]].

To find the solution, integrate for {{math|''x''}} in terms of {{math|''h''}}:

{{block indent|1=<math>\begin{align}
\frac{dx}{dh} &= -\frac{\sqrt{2r-h}}{\sqrt{h}}, \\
x &= -4r\int \sqrt{1-u^2} \, du,
\end{align}</math>}}

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

{{block indent|1=<math>\begin{align}
x &=r(t - \sin t), \\
h &=r(1 + \cos t).
\end{align}</math>}}

This is the standard parameterization of a [[cycloid]] with <math>h=2r-y</math>. It's interesting to note that the arc length squared is equal to the height difference multiplied by the full arch length <math>8r</math>.

== "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 <math>g</math>, while a particle on a horizontal plane undergoes zero gravitational acceleration.  At intermediate angles, the acceleration due to "virtual gravity" by the particle is <math>g\sin\theta</math>. Note that <math>\theta</math> is measured between the tangent to the curve and the horizontal, with angles above the horizontal being treated as positive angles. Thus, <math>\theta</math> varies from <math>-\pi/2</math> to <math>\pi/2</math>.

The position of a mass measured along a tautochrone curve, <math>s(t)</math>, must obey the following differential equation:

{{block indent|1=<math>\frac{d^2s}{{dt}^2} = - \omega^2s</math>}}

which, along with the initial conditions <math>s(0)=s_0</math> and <math>s'(0)=0</math>, has solution:

{{block indent|1=<math>s(t) = s_0 \cos \omega t </math>}}

It can be easily verified both that this solution solves the differential equation and that a particle will reach <math>s=0</math> at time <math>\pi/2\omega</math> from any starting position <math>s_0</math>.  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:

{{block indent|1=<math>
\begin{align}
-g \sin \theta & = \frac{d^2s}{{dt}^2} \\
& = - \omega^2 s \,
\end{align}
</math>}}

The explicit appearance of the distance, <math>s</math>, is troublesome, but we can [[derivative|differentiate]] to obtain a more manageable form:

{{block indent|1=<math>\begin{align}
g \cos \theta \,d\theta &= \omega^2 \,ds \\
\Longrightarrow ds &= \frac{g}{\omega^2} \cos \theta \,d\theta
\end{align}</math>}}

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 <math>\theta</math> to the differential lengths <math>dx</math>, <math>dy</math> and <math>ds</math>:

{{block indent|1=<math>
\begin{align}
ds = \frac{dx}{\cos \theta} \\
ds = \frac{dy}{\sin \theta}
\end{align}
</math>}}

Replacing <math>ds</math> with <math>dx</math> in the above equation lets us solve for <math>x</math> in terms of <math>\theta</math>:

{{block indent|1=<math>
\begin{align}
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{align}
</math>}}

Likewise, we can also express <math>ds</math> in terms of <math>dy</math> and solve for <math>y</math> in terms of <math>\theta</math>:

{{block indent|1=<math>
\begin{align}
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{align}
</math>}}

Substituting <math>\phi = 2\theta</math> and <math display="inline">r = \frac{g}{4\omega^2}\,</math>, we see that these [[parametric equations]] for <math>x</math> and <math>y</math> are those of a point on a circle of radius <math>r</math> rolling along a horizontal line (a [[cycloid]]), with the circle center at the coordinates <math>(C_x + r\phi, C_y)</math>:

{{block indent|1=<math>
\begin{align}
x & = r \left( \sin \phi + \phi \right) + C_x \\
y & = -r \cos \phi + C_y
\end{align}
</math>}}

Note that <math>\phi</math> ranges from <math>-\pi \le \phi \le \pi</math>. It is typical to set <math>C_x = 0</math> and <math>C_y = r</math> so that the lowest point on the curve coincides with the origin. Therefore:

{{block indent|1=<math>
\begin{align}
x & = r \left( \phi + \sin \phi\right)\\
y & = r \left( 1 - \cos \phi\right)\\
\end{align}
</math>}}

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

{{block indent|1=<math>
\begin{align}
r & = \frac{g}{4\omega^2} \\
\omega & = \frac{1}{2} \sqrt{\frac{g}{r}} \\
T & = \pi \sqrt{\frac{r}{g}}\\
\end{align}
</math>}}

(Based loosely on ''Proctor'', pp.&nbsp;135–139)

== {{Anchor|Abel problem}}Abel's solution ==
[[Niels Henrik Abel]] attacked a generalized version of the tautochrone problem (''Abel's mechanical problem''), namely, given a function <math>T(y)</math> 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 <math>T(y)</math> 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 energy|gravitational potential energy]] from its starting point.  The kinetic energy is <math display="inline">\frac{1}{2} mv^2</math>, and since the particle is constrained to move along a curve, its velocity is simply <math>{d\ell}/{dt}</math>, where <math>\ell</math> is the distance measured along the curve. Likewise, the gravitational potential energy gained in falling from an initial height <math>y_0</math> to a height <math>y</math> is <math>mg(y_0 - y)</math>, thus:

{{block indent|1=<math>
\begin{align}
\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{align}
</math>}}

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

Now we integrate from <math>y = y_0</math> to <math>y = 0</math> to get the total time required for the particle to fall:

{{block indent|1=<math>
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
</math>}}

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 <math>{d\ell}/{dy}</math> would be easy to calculate). But Abel's mechanical problem requires the converse – given <math>T(y_0)\,</math>, we wish to find <math>f(y) = {d\ell}/{dy}</math>, 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 <math>{d\ell}/{dy}</math> with <math>{1}/{\sqrt{y}}</math> and thus take the [[Laplace transform]] of both sides with respect to variable <math>y</math>:

{{block indent|1=<math>
\mathcal{L}[T(y_0)] = \frac{1}{\sqrt{2g}} \mathcal{L} \left [ \frac{1}{\sqrt{y}} \right ]F(s)
</math>}}

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

{{block indent|1=<math>
\mathcal{L}\left [ \frac{d\ell}{dy} \right ] = \sqrt{\frac{2g}{\pi}} s^{\frac{1}{2}} \mathcal{L}[T(y_0)]
</math>}}

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

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

{{block indent|1=<math>
\begin{align}
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{align}
</math>}}

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

{{block indent|1=<math>\frac{d\ell}{dy} = T_0 \frac{\sqrt{2g}}{\pi}\frac{1}{\sqrt{y}}</math>}}

It can be shown that the cycloid obeys this equation. It needs one step further to do the integral with respect to <math>y</math> to obtain the expression of the path shape.

(''Simmons'', Section 54).

== See also ==
* [[Beltrami identity]]
* [[Brachistochrone curve]]
* [[Calculus of variations]]
* [[Catenary]]
* [[Cycloid]]
* [[Equations of motion#Equations of uniformly accelerated linear motion|Uniformly accelerated motion]]

== References ==
{{Reflist}}

== Bibliography ==
* {{cite book |last=Simmons |first=George |title=Differential Equations with Applications and Historical Notes |publisher=McGraw–Hill |year=1972 |isbn=0-07-057540-1}}
* {{cite book |url=https://books.google.com/books?id=6UIZBvs0diIC |title=A Treatise on the Cycloid and All Forms of Cycloidal Curves, and on the Use of Such Curves in Dealing with the Motions of Planets, Comets, etc., and of Matter Projected from the Sun |last1=Proctor |first1=Richard Anthony |year=1878}}

== External links ==
* [https://mathworld.wolfram.com/TautochroneProblem.html Mathworld]

[[Category:Plane curves]]
[[Category:Mechanics]]

[[de:Zykloide#Die Tautochronie der Zykloide]]