Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.5 Physical variational principles

wherekis a constant; this reduces to

y′^2 =

(

ρgy+λ

k

) 2

− 1.

Making the substitutionρgy+λ=kcoshz, this can be integrated easily to give

k

ρg

cosh−^1

(

ρgy+λ

k

)

=x+c,

wherecis the constant of integration.
We now have three unknowns,λ,kandc, that must be evaluated using the two end
conditionsy(±a) = 0 and the constraintJ=2L. The end conditions give

cosh

ρg(a+c)

k

=

λ

k

=cosh

ρg(−a+c)

k

,

and sincea= 0, these implyc=0andλ/k=cosh(ρga/k). Puttingc=0intothe
constraint, in whichy′= sinh(ρgx/k), we obtain

2 L=

∫a

−a

[

1+sinh^2

(ρgx

k

)] 1 / 2

dx

=

2 k

ρg

sinh

(ρga

k

)

.

Collecting together the values for the constants, the form adopted by the rope is therefore

y(x)=

k

ρg

[

cosh

(ρgx

k

)

−cosh

(ρga

k

)]

,

wherekis the solution of sinh(ρga/k)=ρgL/k. This curve is known as a catenary.

22.5 Physical variational principles

Many results in both classical and quantum physics can be expressed as varia-

tional principles, and it is often when expressed in this form that their physical

meaning is most clearly understood. Moreover, once a physical phenomenon has

been written as a variational principle, we can use all the results derived in this

chapter to investigate its behaviour. It is usually possible to identify conserved

quantities, or symmetries of the system of interest, that otherwise might be found

only with considerable effort. From the wide range of physical variational princi-

ples we will select two examples from familiar areas of classical physics, namely

geometric optics and mechanics.

22.5.1 Fermat’s principle in optics

Fermat’s principle in geometrical optics states that a ray of light travelling in a

region of variable refractive index follows a path such that the total optical path

length (physical length×refractive index) is stationary.