# Abel problem

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:

#### 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 , 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) |

**How to Cite This Entry:**

Abel problem.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Abel_problem&oldid=12327