Tautochrone curve

A tautochrone or isochrone curve (from Greek prefixes tauto- meaning "same" or iso- "equal", and chrono "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 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 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 was a cycloid. 
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 π/2. In modern terms, this means that the time of descent is formula_1, where "r" is the radius of the circle which generates the cycloid, and "g" 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.
If the particle's position is parametrized by the arclength from the lowest point, the kinetic energy is proportional to formula_2 The potential energy is proportional to the height . One way the curve can be an isochrone is if the Lagrangian is that of a simple harmonic oscillator: the height of the curve must be proportional to the arclength squared.
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 formula_8. This integral is the area under a circle, which can be naturally cut into a triangle and a circular wedge:
To see that this is a strangely parametrized cycloid, change variables to disentangle the transcendental and algebraic parts by defining the angle formula_11. This yields
which is the standard parametrization, except for the scale of and .
"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 formula_14, while a particle on a horizontal plane undergoes zero gravitational acceleration. At intermediate angles, the acceleration due to "virtual gravity" by the particle is formula_15. Note that formula_16 is measured between the tangent to the curve and the horizontal, with angles above the horizontal being treated as positive angles. Thus, formula_16 varies from formula_18 to formula_19.
The position of a mass measured along a tautochrone curve, formula_20, must obey the following differential equation:
Abel's solution.
Niels Henrik Abel attacked a generalized version of the tautochrone problem ("Abel's mechanical problem"), namely, given a function "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 "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 formula_22, and since the particle is constrained to move along a curve, its velocity is simply formula_23, where formula_24 is the distance measured along the curve. Likewise, the gravitational potential energy gained in falling from an initial height formula_25 to a height formula_26 is formula_27, thus:
In the last equation, we've anticipated writing the distance remaining along the curve as a function of height (formula_29, recognized that the distance remaining must decrease as time increases (thus the minus sign), and used the chain rule in the form formula_30.
Now we integrate from formula_31 to formula_32 to get the total time required for the particle to fall:
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 formula_34 would be easy to calculate). But Abel's mechanical problem requires the converse — given formula_35, we wish to find formula_36, 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 formula_34 with formula_38 and thus take the Laplace transform of both sides with respect to variable formula_39:
where formula_41
Since formula_42, we now have an expression for the Laplace transform of formula_34 in terms of formula_35's Laplace transform:
This is as far as we can go without specifying formula_35. Once formula_35 is known, we can compute its Laplace transform, calculate the Laplace transform of formula_34 and then take the inverse transform (or try to) to find formula_34.
For the tautochrone problem, formula_50 is constant. Since the Laplace transform of 1 is formula_51, i.e., formula_52, we find the shape function formula_36:
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 formula_39 to obtain the expression of the path shape.
("Simmons", Section 54).