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
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Integrating this equation between and
and putting
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one obtains the integral equation
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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:
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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) |
Abel problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel_problem&oldid=12327