Abel problem

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To find, in a vertical plane , a curve such that a material point moving along it under gravity from rest, starting from a point with ordinate , will meet the -axis after a time , where the function 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 is the angle formed by the tangent of the curve being sought with the -axis, then

Integrating this equation between and and putting

one obtains the integral equation

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


[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)


In the case that , this is the famous tautochrone problem first solved by Chr. Huyghens, who showed that this curve is then a cycloid.


[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)
How to Cite This Entry:
Abel problem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article