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To find, in a vertical plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a0101601.png" />, a curve such that a material point moving along it under gravity from rest, starting from a point with ordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a0101602.png" />, will meet the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a0101603.png" />-axis after a time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a0101604.png" />, where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a0101605.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a0101606.png" /> is the angle formed by the tangent of the curve being sought with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a0101607.png" />-axis, then
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a0101608.png" /></td> </tr></table>
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$$\frac{ds}{d\tau}=-\sqrt{2g(x-s)}\sin\omega.$$
  
Integrating this equation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a0101609.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a01016010.png" /> and putting
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Integrating this equation between $0$ and $x$ and putting
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a01016011.png" /></td> </tr></table>
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$$\frac1{\sin\omega}=\phi(s),\quad-\sqrt{2g}\Phi(x)=f(x),$$
  
 
one obtains the integral equation
 
one obtains the integral equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a01016012.png" /></td> </tr></table>
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$$\int\limits_0^x\frac{\phi(s)\,ds}{\sqrt{x-s}}=f(x)$$
  
for the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a01016013.png" />, 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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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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a01016014.png" /></td> </tr></table>
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$$\phi(x)=\frac1\pi\left[\frac{f(0)}{\sqrt x}+\int\limits_0^x\frac{f'(\tau)\,d\tau}{\sqrt{x-\tau}}\right].$$
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
In the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010160/a01016015.png" />, this is the famous tautochrone problem first solved by Chr. Huyghens, who showed that this curve is then a [[Cycloid|cycloid]].
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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====
 
====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>
 
<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>

Latest revision as of 14:27, 14 February 2020

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)
How to Cite This Entry:
Abel problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel_problem&oldid=12327
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article