The Chemistry Maths Book, Second Edition

9.4 Stationary points 257

where a

1

, 1 a

2

,1=, 1 a

m

are constants. The procedure is to construct the auxiliary

function

(9.13)

and to solve then 1 + 1 mequations

(9.14)

for the nvariables x

i

and the mmultipliers λ

k

.

The following examples demonstrate one application of the method of Lagrangian

multipliers in geometry (or packaging) and one important application in chemistry.

Another application, the derivation of the Boltzmann distribution, is discussed in

Chapter 21.

EXAMPLE 9.8Find the dimensions of the rectangular box of largest volume for

given surface area.

The volume of a box of sides x,y, and zis V 1 = 1 xyzand its surface area is

A 1 = 1 2(xy 1 + 1 yz 1 + 1 zx). The problem is therefore to find the maximum value of Vsubject

to the constraintA 1 = 1 constant. By the method of Lagrangian multipliers we form the

auxiliary functionφ 1 = 1 V 1 − 1 λA, and solve the set of equations

Multiplication of the first equation by x, the second by y, and the third by zgives

xyz 1 − 12 λ(xy 1 + 1 xz) 1 = 1 xyz 1 − 12 λ(xy 1 + 1 yz) 1 = 1 xyz 1 − 12 λ(xz 1 + 1 yz) 1 = 10

It follows thatx 1 = 1 y 1 = 1 z, and the box is a cube of side

0 Exercise 29

EXAMPLE 9.9Secular equations

Variation principles in the physical sciences often lead to the problem of finding the

stationary values of a ‘quadratic form’ in the nvariables,x

1

, x

2

,1=, x

n

,

fx x ...x Cxx

n

i

n

j

n

ij i j

()

12

11

,,, =

==

∑∑

A6.

∂

∂

=− +=

φ

λ

z

xy 20()x y

∂

∂

=− +=,

∂

∂

=− +=,

φ

λ

φ

λ

x

yz y z

y

20 20()xz ()x z

ga k ...m

kk

==,1,,, 23 ,

∂

∂

=

∂

∂

−

∂

∂

==,,,,

=

∑

φ

λ

x

f

x

g

x

i...n

ii

k

m

k

k

i

1

01 , 23

φλλλ=− − − − −λ=−λ

=

∑

fg g g gf g

mm

k

m

11 2 2 33 kk

1