View source for Abel problem

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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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.H. Abel,   "Solutions de quelques problèmes à l'aide d'intégrales défines" , ''Oeuvres complètes, nouvelle éd.'' , '''1''' , Grondahl &amp; Son , Christiania  (1881)  pp. 11–27  (Edition de Holmboe)</TD></TR></table>



====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|cycloid]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.J. Jerri,   "Introduction to integral equations with applications" , M. Dekker  (1985)  pp. Sect. 2.3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Hochstadt,   "Integral equations" , Wiley  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.L. Moiseiwitsch,   "Integral equations" , Longman  (1977)</TD></TR></table>