Classical Mechanics (Goldstein) · Problem 7

Variational Principles and Lagrange's Equations — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · minimum-surface-of-revolution-catenary

Problem

In Example 2 of Section 2.1 we considered the problem of the minimum surface of revolution. Examine the symmetric case

x_1 = x_2

y_2 = -y_1 > 0

, and express the condition for the parameter

a

as a transcendental equation in terms of the dimensionless quantities

k = x_2/a

, and

\alpha = y_2/x_2

. Show that for

\alpha

greater than a certain value

\alpha_0

two values of

k

are possible, for

\alpha = \alpha_0

only one value of

k

is possible, while if

\alpha < \alpha_0

no real value of

k

(or

a

) can be found, so that no catenary solution exists in this region. Find the value of

\alpha_0

, numerically if necessary.