Physical variational principles

22.5 PHYSICAL VARIATIONAL PRINCIPLES
where k is a constant; this reduces to
y =
2
ρgy + λ
k
2
− 1.
Making the substitution ρgy + λ = k cosh z, this can be integrated easily to give
ρgy + λ
k
= x + c,
cosh−1
ρg
k
where c is the constant of integration.
We now have three unknowns, λ, k and c, that must be evaluated using the two end
conditions y(±a) = 0 and the constraint J = 2L. The end conditions give
cosh
ρg(a + c)
λ
ρg(−a + c)
= = cosh
,
k
k
k
and since a = 0, these imply c = 0 and λ/k = cosh(ρga/k). Putting c = 0 into the
constraint, in which y = sinh(ρgx/k), we obtain
a
ρgx 1/2
1 + sinh2
dx
2L =
k
−a
2k
ρga
.
=
sinh
ρg
k
Collecting together the values for the constants, the form adopted by the rope is therefore
ρgx ρga k cosh
− cosh
,
y(x) =
ρg
k
k
where k is 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 variational 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 principles 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.
787
CALCULUS OF VARIATIONS
y
B
θ2
n2
x
n1
θ1
A
Figure 22.8 Path of a light ray at the plane interface between media with
refractive indices n1 and n2 , where n2 < n1 .
From Fermat’s principle deduce Snell’s law of refraction at an interface.
Let the interface be at y = constant (see figure 22.8) and let it separate two regions with
refractive indices n1 and n2 respectively. On a ray the element of physical path length is
ds = (1 + y 2 )1/2 dx, and so for a ray that passes through the points A and B, the total
optical path length is
B
2
n(y)(1 + y )1/2 dx.
P =
A
Since the integrand does not contain the independent variable x explicitly, we use (22.8)
to obtain a first integral, which, after some rearrangement, reads
−1/2
2
n(y) 1 + y = k,
where k is a constant. Recalling that y is the tangent of the angle φ between the
instantaneous direction of the ray and the x-axis, this general result, which is not dependent
on the configuration presently under consideration, can be put in the form
n cos φ = constant
along a ray, even though n and φ vary individually.
For our particular configuration n is constant in each medium and therefore so is
y . Thus the rays travel in straight lines in each medium (as anticipated in figure 22.8,
but not assumed in our analysis), and since k is constant along the whole path we have
n1 cos φ1 = n2 cos φ2 , or in terms of the conventional angles in the figure
n1 sin θ1 = n2 sin θ2 . 22.5.2 Hamilton’s principle in mechanics
Consider a mechanical system whose configuration can be uniquely defined by a
number of coordinates qi (usually distances and angles) together with time t and
which experiences only forces derivable from a potential. Hamilton’s principle
788
22.5 PHYSICAL VARIATIONAL PRINCIPLES
y
O
l
dx
x
Figure 22.9 Transverse displacement on a taut string that is fixed at two
points a distance l apart.
states that in moving from one configuration at time t0 to another at time t1 the
motion of such a system is such as to make
L=
t1
L(q1 , q2 . . . , qn , q̇1 , q̇2 , . . . , q̇n , t) dt
(22.21)
t0
stationary. The Lagrangian L is defined, in terms of the kinetic energy T and
the potential energy V (with respect to some reference situation), by L = T − V .
Here V is a function of the qi only, not of the q̇i . Applying the EL equation to L
we obtain Lagrange’s equations,
d
∂L
=
∂qi
dt
∂L
∂q̇i
,
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 coordinate t, together with a continuum in which the qi become
the continuous variable x. The expressions for T and V therefore become integrals over x
rather than sums over the label i.
If ρ and τ are 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
l 2
l 2
ρ ∂y
τ ∂y
dx,
V =
dx
T =
∂t
∂x
0 2
0 2
and (22.21) becomes
L=
1
2
t1
dt
t0
2 l 2
∂y
∂y
−τ
ρ
dx.
∂t
∂x
0
789