equation of catenary via calculus of variations

Using the mechanical principle that the centre of mass itself as low as possible, determine the equation of the curve formed by a l when supported at its ends in the points  P1=(x1,y1)  and  P2=(x2,y2).

We have an isoperimetric problemMathworldPlanetmath

to minimiseP1P2y𝑑s (1)

under the constraint

P1P2𝑑s=l, (2)

where both the path integrals are taken along some curve c.  Using a Lagrange multiplierMathworldPlanetmath λ, the task changes to a free problem

P1P2(y-λ)𝑑s=x1x2(y-λ)1+y2|dx|=min! (3)

(cf. example of calculus of variationsMathworldPlanetmath).

The Euler–Lagrange differential equationMathworldPlanetmath (http://planetmath.org/EulerLagrangeDifferentialEquation), the necessary condition for (3) to give an extremal c, reduces to the Beltrami identityMathworldPlanetmath


where a is a constant of integration.  After solving this equation for the derivativePlanetmathPlanetmath y and separation of variablesMathworldPlanetmath, we get


which may become clearer by notating  y-λ:=u;  then by integrating


we choose the new constant of integration b such that  x=b  when  u=a:


We can write two equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Equivalent3) results




Adding these allows to eliminate the square roots and to obtain



y-λ=acoshx-ba. (4)

This is the sought form of the equation of the chain curve.  The constants λ,a,b can then be determined for putting the curve to pass through the given points P1 and P2.


  • 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset IV. Johdatus variatiolaskuun.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1946).
Title equation of catenary via calculus of variations
Canonical name EquationOfCatenaryViaCalculusOfVariations
Date of creation 2013-03-22 19:12:07
Last modified on 2013-03-22 19:12:07
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Derivation
Classification msc 49K05
Classification msc 49K22
Classification msc 47A60
Related topic Catenary
Related topic CalculusOfVariations
Related topic LeastSurfaceOfRevolution