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

We have an isoperimetric problem

to minimize ∫ P1P2 y ds

(1)

under the constraint

∫ P1P2 ds = l,

(2)

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

∫ P1P2 (y − λ) ds	= ∫ x1x2 (y − λ) |dx|	(3)
	= minimum.	(4)

The Euler–Lagrange differential equation, the necessary condition for the preceding functional to give an extremal c, reduces to the Beltrami identity

where a is a constant of integration. After solving this equation for the derivative y^′ and separating variables, we get

This may become clearer by writing u := y − λ; then

Choose the new constant of integration b such that x = b when u = a. Then

We can write two equivalent results:

Thus

Adding these equations eliminates the square roots and gives

or

y − λ = a cosh .

(5)

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

References