Catenary

A flexible cord of uniform density ρ and fixed length l is suspended from two points of equal height (see Figure 1.1). The gravitational acceleration is taken to be a constant g in the negative z direction.

a) Write the expressions for the potential energy U and the length for a given curve z = z(x).

b) Formulate the Euler-Lagrange equations for the curve with minimal potential energy, subject to the condition of fixed length.

c) Show that the solution of the previous equation is given by z = A cosh (x/A) + B where A and B are constants. Calculate U and l for this solution.

 Figure 1.1

  Formulae:

 SOLUTION

  a) Write the expressions for the length and potential energy U (see Figure 1.2) using

Figure 1.2

(1)

(2)

b) Here, we are not reproducing the usual Euler–Lagrange equations where we have minimized the action Instead, we look for the minimum of U found in (a), subject to the constraint of constant length l. Utilizing the method of undetermined Lagrange multipliers, λ we may write

(3)

The coefficient preceding simplifies the calculation. From (3)

where

(4)

Before proceeding to (c), note that in this problem, we may immediately extract a first integral of the motion since  does not depend explicitly on x.

(5)

c) We may now substitute z = A cosh (x/A) +B into (5), yielding

(6)

h is constant for λ = -B. Calculate from (1):

(7)

Using (2) and (7), find U:

(8)

Using z(a) = z(-a) = 0, we see that

(8) becomes

(9)

From (7), we have

so