Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.5 PHYSICAL VARIATIONAL PRINCIPLES

y

O

dx l x

Figure 22.9 Transverse displacement on a taut string that is fixed at two

points a distancelapart.

states that in moving from one configuration at timet 0 to another at timet 1 the

motion of such a system is such as to make

L=

∫t 1

t 0

L(q 1 ,q 2 ...,qn, ̇q 1 , ̇q 2 ,...,q ̇n,t)dt (22.21)

stationary. TheLagrangianLis defined, in terms of the kinetic energyTand

the potential energyV(with respect to some reference situation), byL=T−V.

HereVis a function of theqionly, not of the ̇qi. Applying the EL equation toL

we obtainLagrange’s equations,

∂L

∂qi

=

d

dt

(

∂L

∂ ̇qi

)

,i=1, 2 ,...,n.

Using Hamilton’s principle derive the wave equation for small transverse oscillations of a

taut string.

In this example we are in fact considering a generalisation of (22.21) to a case involving
one isolated independent coordinatet,togetherwithacontinuumin which theqibecome
the continuous variablex. The expressions forTandVtherefore become integrals overx
rather than sums over the labeli.
Ifρandτare the local density and tension of the string, both of which may depend on
x, then, referring to figure 22.9, the kinetic and potential energies of the string are given
by

T=

∫l

0

ρ

2

(

∂y

∂t

) 2

dx, V=

∫l

0

τ

2

(

∂y

∂x

) 2

dx

and (22.21) becomes

L=

1

2

∫t 1

t 0

dt

∫l

0

[

ρ

(

∂y

∂t

) 2

−τ

(

∂y

∂x

) 2 ]

dx.