Classical Mechanics (Kibble & Berkshire) · Problem 13.7

Hamilton's equations and phase space — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · calculus-of-variations

Problem

Soap film problem Consider the soap film problem for which it is required to minimise

J[y] = \int_{-a}^{a} y\left(1 + \dot{y}^2\right)^{\frac{1}{2}} dx

with

y(-a) = y(a) = b

. Show that the extremals of

J

have the form

y = c \cosh\left(\frac{x}{c} + d\right),

where

c

d

are constants, and that the end conditions are satisfied if (and only if)

d = 0

and

\cosh \lambda = \left(\frac{b}{a}\right)\lambda,

where

\lambda = a/c

. Show that there are two admissible extremals provided that the aspect ratio

b/a

exceeds a certain critical value and none if

b/a

is less than this critical value. Sketch a graph showing how this critical value is determined. The remainder of this question requires computer assistance. Show that the critical value of the aspect ratio

b/a

is about 1.51. Choose a value of

b/a

larger than the critical value (

b/a = 2

is suitable) and find the two values of

\lambda

. Plot the two admissible extremals on the same graph. Which one looks like the actual shape of the soap film? Check your guess by perturbing each extremal by small admissible variations and finding the change in the value of the functional

J[y]

Related theory

calculus-of-variations

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