A tautochrone curve or isochrone curve 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 π times the square root of the radius (of the circle which generates the cycloid) over the acceleration of gravity. The tautochrone curve is related to the brachistochrone curve, which is also a cycloid.
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.
The cycloid is given by a point on a circle of radius
r
x
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
\pi/2
r
g
This solution was later used to solve the problem of the brachistochrone curve. Johann Bernoulli solved the problem in a paper (Acta Eruditorum, 1697).
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 escapements could greatly reduce this source of inaccuracy.
Later, the mathematicians Joseph Louis Lagrange and Leonhard Euler provided an analytical solution to the problem.
If the particle's position is parametrized by the arclength from the lowest point, the kinetic energy is proportional to
s |
2.
where the constant of proportionality has been set to 1 by changing units of length.
The differential form of this relation is
which eliminates, and leaves a differential equation for and . To find the solution, integrate for in terms of :
where
u=\sqrt{y}
To see that this is a strangely parametrized cycloid, change variables to disentangle the transcendental and algebraic parts by defining the angle
\theta=\arcsin2u
which is the standard parametrization, except for the scale of and .
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
g
g\sin\theta
\theta
\theta
-\pi/2
\pi/2
The position of a mass measured along a tautochrone curve,
s(t)
which, along with the initial conditions
s(0)=s0
s'(0)=0
It can be easily verified both that this solution solves the differential equation and that a particle will reach
s=0
\pi/2\omega
s0
The explicit appearance of the distance,
s
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
\theta
dx
dy
ds
Replacing
ds
dx
x
\theta
Likewise, we can also express
ds
dy
y
\theta
Substituting
\phi=2\theta
x
y
r
(Cx+r\phi,Cy)
Note that
\phi
-\pi\le\phi\le\pi
Cx=0
Cy=r
Solving for
\omega
T=
\pi | |
2\omega |
r
(Based loosely on Proctor, pp. 135–139)
Niels Henrik Abel attacked a generalized version of the tautochrone problem (Abel's mechanical problem), namely, given a function
T(y)
T(y)
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 , and since the particle is constrained to move along a curve, its velocity is simply
{d\ell}/{dt}
\ell
y0
y
mg(y0-y)
In the last equation, we have anticipated writing the distance remaining along the curve as a function of height (
\ell(y))
Now we integrate from
y=y0
y=0
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
{d\ell}/{dy}
T(y0)
f(y)={d\ell}/{dy}
{d\ell}/{dy}
{1}/{\sqrt{y}}
y
where
F(s)=l{L}{\left[{d\ell}/{dy}\right]}
{d\ell}/{dy}
T(y0)
This is as far as we can go without specifying
T(y0)
T(y0)
{d\ell}/{dy}
{d\ell}/{dy}
For the tautochrone problem,
T(y0)=T0
{1}/{s}
Making use again of the Laplace transform above, we invert the transform and conclude:
It can be shown that the cycloid obeys this equation. It needs one step further to do the integral with respect to
y
(Simmons, Section 54).