Abel problem

To find, in a vertical plane $(s,\tau)$, a curve such that a material point moving along it under gravity from rest, starting from a point with ordinate $x$, will meet the $\tau$-axis after a time $T=f(x)$, where the function $f(x)$ is given in advance. The problem was posed by N.H. Abel in 1823, and its solution involves one of the first integral equations — the Abel integral equation — which was also solved. In fact, if $\omega$ is the angle formed by the tangent of the curve being sought with the $\tau$-axis, then

$$\frac{ds}{d\tau}=-\sqrt{2g(x-s)}\sin\omega.$$

Integrating this equation between $0$ and $x$ and putting

$$\frac1{\sin\omega}=\phi(s),\quad-\sqrt{2g}\Phi(x)=f(x),$$

one obtains the integral equation

$$\int\limits_0^x\frac{\phi(s)\,ds}{\sqrt{x-s}}=f(x)$$

for the unknown function $\phi(s)$, the determination of which makes it possible to find the equation of the curve being sought. The solution of the equation introduced above is:

$$\phi(x)=\frac1\pi\left[\frac{f(0)}{\sqrt x}+\int\limits_0^x\frac{f'(\tau)\,d\tau}{\sqrt{x-\tau}}\right].$$

References

[1]	N.H. Abel, "Solutions de quelques problèmes à l'aide d'intégrales défines" , Oeuvres complètes, nouvelle éd. , 1 , Grondahl & Son , Christiania (1881) pp. 11–27 (Edition de Holmboe)

Comments

In the case that $f(x)=\mathrm{const}$, this is the famous tautochrone problem first solved by Chr. Huyghens, who showed that this curve is then a cycloid.

References

[a1]	A.J. Jerri, "Introduction to integral equations with applications" , M. Dekker (1985) pp. Sect. 2.3
[a2]	H. Hochstadt, "Integral equations" , Wiley (1973)
[a3]	B.L. Moiseiwitsch, "Integral equations" , Longman (1977)