catenary

A wire takes a form resembling an arc of a parabola when suspended at its ends.  The arc is not from a parabola but from the graph of the hyperbolic cosine (http://planetmath.org/HyperbolicFunctions) function in a suitable coordinate system.

Let’s derive the equation  $y=y(x)$  of this curve, called the catenary, in its plane with $x$-axis horizontal and $y$-axis vertical.  We denote the of the wire by $\sigma$.

In any point  $(x,y)$  of the wire, the tangent line of the curve forms an angle $\phi$ with the positive direction of $x$-axis.  Then,

$$\tan \phi =\frac{dy}{dx}=y^{\prime }.$$

In the point, a certain tension $T$ of the wire acts in the direction of the value $a$.  Hence we may write

$$T=\frac{a}{\cos \phi },$$

whence the vertical of $T$ is

$$T\sin \phi =a\tan \phi$$

and its differential (http://planetmath.org/Differential)

$$d(T\sin \phi )=ad\tan \phi =ad{y}^{\prime }.$$

But this differential is the amount of the supporting   $\sigma \sqrt{1+{(y^{\prime }(x))}^2}dx$ (see the arc length).  Thus we obtain the differential equation

$\sigma \sqrt{1+y^{\prime 2}}dx=ad{y}^{\prime },$ (1)

which allows the separation of variables:

$$\int 𝑑x=\frac{a}{\sigma }\int \frac{d{y}^{\prime }}{\sqrt{1+y^{\prime 2}}}$$

This may be solved by using the substitution (http://planetmath.org/SubstitutionForIntegration)

$$y^{\prime }:=\sinh t,d{y}^{\prime }=\cosh tdt,\sqrt{1+y^{\prime 2}}=\cosh t$$

giving

$$x=\frac{a}{\sigma }t+x_0,$$

i.e.

$$y^{\prime }=\frac{dy}{dx}=\sinh \frac{\sigma (x-x_0)}{a}.$$

This leads to the final solution

$$y=\frac{a}{\sigma }\cosh \frac{\sigma (x-x_0)}{a}+y_0$$

of the equation (1).  We have denoted the constants of integration by $x_0$ and $y_0$.  They determine the position of the catenary in regard to the coordinate axes.  By a suitable choice of the axes and the equation

$y=a\cosh {\displaystyle \frac{x}{a}}$ (2)

of the catenary.

Some of catenary

• $\tan \phi =\sinh \frac{x}{a},\sin \phi =\tanh \frac{x}{a}$  (cf. the Gudermannian)

• The arc length of the catenary (2) from the apex  $(0,a)$  to the point  $(x,y)$  is   $a\sinh \frac{x}{a}=\sqrt{y^2-a^2}$.

• The radius of curvature of the catenary (2) is  $a{\cosh }^2\frac{x}{a}$, which is the same as length of the normal line of the catenary between the curve and the $x$-axis.

• The catenary is the catacaustic of the exponential curve (http://planetmath.org/ExponentialFunction) reflecting the vertical rays.

• If a parabola rolls on a straight line, the focus draws a catenary.

• The involute (a.k.a. the evolvent) of the catenary is the tractrix.

Title	catenary
Canonical name	Catenary
Date of creation	2014-10-26 21:25:30
Last modified on	2014-10-26 21:25:30
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	29
Author	pahio (2872)
Entry type	Derivation
Classification	msc 53B25
Classification	msc 51N05
Synonym	chain curve
Related topic	EquationOfCatenaryViaCalculusOfVariations
Related topic	LeastSurfaceOfRevolution
Related topic	HyperbolicFunctions
Related topic	Tractrix
Related topic	EqualArcLengthAndArea
Defines	catenary