Wikipedia / Tautochrone curve
Wikipedia / Tautochrone curve
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, divided by the square root of the acceleration of gravity. The tautochrone curve is related to the brachistochrone curve, which is also a cycloid.
The tautochrone problem
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.
is the gravity of Earth, or more accurately, the earth's gravitational acceleration.
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.
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 , and the potential energy is proportional to the height . 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:
. Compared to the simple harmonic oscillator's Lagrangian 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
which eliminates , and leaves a differential equation for and . This is the differential equation for a cycloid when the vertical coordinate is counted from its vertex (the point with a horizontal tangent) instead of the cusp.
To find the solution, integrate for in terms of :
and yield:
This is the standard parameterization of a cycloid.
"Virtual gravity" solution
.
, must obey the following differential equation:
, has solution:
. 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:
, is troublesome, but we can differentiate to obtain a more manageable form:
This equation relates the change in the curve's angle to the change in the distance along the curve. We now use trigonometry:
:
:
, we see that these parametric equations rolling along a horizontal line (a cycloid:
so that the lowest point on the curve coincides with the origin. Therefore:
:
(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 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, thus:
, recognized that the distance remaining must decrease as time increases (thus the minus sign), and used the chain rule.
to get the total time required for the particle to fall:
This is called Abel's integral equation, 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 and thus take the Laplace transform:
:
.
:
Making use again of the Laplace transform above, we invert the transform and conclude:
(Simmons, Section 54).
, we can rewrite the above as:
Let:
So that:
In order to integrate, substitute:
as:
Integrating both sides arrives at:
as:
:
axis.
(Based on O'neil, pp. 263-264)
See also
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