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  $P_1=(x_1,y_1)$  and  $P_2=(x_2,y_2)$.

We have an isoperimetric problem

$\text{to minimise}\mathit{\hspace{1em}}{\displaystyle {\int }_{P_1}^{P_2}}y𝑑s$ (1)

under the constraint

${\displaystyle {\int }_{P_1}^{P_2}}𝑑s=l,$ (2)

where both the path integrals are taken along some curve $c$.  Using a Lagrange multiplier $\lambda$, the task changes to a free problem

${\displaystyle {\int }_{P_1}^{P_2}}(y-\lambda )𝑑s={\displaystyle {\int }_{x_1}^{x_2}}(y-\lambda )\sqrt{1+y^{\prime 2}}|dx|=\text{min}!$ (3)

(cf. example of calculus of variations).

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

$$(y-\lambda )\sqrt{1+y^{\prime 2}}-y^{\prime }\cdot (y-\lambda )\cdot \frac{y^{\prime }}{\sqrt{1+y^{\prime 2}}}\equiv \frac{y-\lambda }{\sqrt{1+y^{\prime 2}}}=a,$$

where $a$ is a constant of integration.  After solving this equation for the derivative $y^{\prime }$ and separation of variables, we get

$$\pm \frac{dy}{\sqrt{{(y-\lambda )}^2-a^2}}=\frac{dx}{a}$$

which may become clearer by notating  $y-\lambda :=u$;  then by integrating

$$\pm \frac{du}{\sqrt{u^2-a^2}}=\frac{dx}{a}$$

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

$$\pm {\int }_a^u\frac{du}{\sqrt{u^2-a^2}}={\int }_b^x\frac{dx}{a}$$

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

$$\ln \frac{u+\sqrt{u^2-a^2}}{a}=+\frac{x-b}{a},\ln \frac{u-\sqrt{u^2-a^2}}{a}=-\frac{x-b}{a},$$

i.e.

$$\frac{u+\sqrt{u^2-a^2}}{a}=e^{+\frac{x-b}{a}},\frac{u-\sqrt{u^2-a^2}}{a}=e^{-\frac{x-b}{a}}.$$

Adding these allows to eliminate the square roots and to obtain

$$u=\frac{a}{2}\left(e^{\frac{x-b}{a}}+e^{-\frac{x-b}{a}}\right),$$

or

$y-\lambda =a\cosh {\displaystyle \frac{x-b}{a}}.$ (4)

This is the sought form of the equation of the chain curve.  The constants $\lambda ,a,b$ can then be determined for putting the curve to pass through the given points $P_1$ and $P_2$.

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