Envelopes

5.10 ENVELOPES
We now have the general form for the distribution of particles amongst energy levels, but
in order to determine the two constants µ, C we recall that
R
C exp µEk = N
k=1
and
R
CEk exp µEk = E.
k=1
This is known as the Boltzmann distribution and is a well-known result from statistical
mechanics. 5.10 Envelopes
As noted at the start of this chapter, many of the functions with which physicists,
chemists and engineers have to deal contain, in addition to constants and one
or more variables, quantities that are normally considered as parameters of the
system under study. Such parameters may, for example, represent the capacitance
of a capacitor, the length of a rod, or the mass of a particle – quantities that
are normally taken as fixed for any particular physical set-up. The corresponding
variables may well be time, currents, charges, positions and velocities. However,
the parameters could be varied and in this section we study the effects of doing so;
in particular we study how the form of dependence of one variable on another,
typically y = y(x), is affected when the value of a parameter is changed in a
smooth and continuous way. In effect, we are making the parameter into an
additional variable.
As a particular parameter, which we denote by α, is varied over its permitted
range, the shape of the plot of y against x will change, usually, but not always,
in a smooth and continuous way. For example, if the muzzle speed v of a shell
fired from a gun is increased through a range of values then its height–distance
trajectories will be a series of curves with a common starting point that are
essentially just magnified copies of the original; furthermore the curves do not
cross each other. However, if the muzzle speed is kept constant but θ, the angle
of elevation of the gun, is increased through a series of values, the corresponding
trajectories do not vary in a monotonic way. When θ has been increased beyond
45◦ the trajectories then do cross some of the trajectories corresponding to θ < 45◦ .
The trajectories for θ > 45◦ all lie within a curve that touches each individual
trajectory at one point. Such a curve is called the envelope to the set of trajectory
solutions; it is to the study of such envelopes that this section is devoted.
For our general discussion of envelopes we will consider an equation of the
form f = f(x, y, α) = 0. A function of three Cartesian variables, f = f(x, y, α),
is defined at all points in xyα-space, whereas f = f(x, y, α) = 0 is a surface in
this space. A plane of constant α, which is parallel to the xy-plane, cuts such
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PARTIAL DIFFERENTIATION
P1
y
P
P2
f(x, y, α1 ) = 0
f(x, y, α1 + h) = 0
x
Figure 5.4 Two neighbouring curves in the xy-plane of the family f(x, y, α) =
0 intersecting at P . For fixed α1 , the point P1 is the limiting position of P as
h → 0. As α1 is varied, P1 delineates the envelope of the family (broken line).
a surface in a curve. Thus different values of the parameter α correspond to
different curves, which can be plotted in the xy-plane. We now investigate how
the envelope equation for such a family of curves is obtained.
5.10.1 Envelope equations
Suppose f(x, y, α1 ) = 0 and f(x, y, α1 + h) = 0 are two neighbouring curves of a
family for which the parameter α differs by a small amount h. Let them intersect
at the point P with coordinates x, y, as shown in figure 5.4. Then the envelope,
indicated by the broken line in the figure, touches f(x, y, α1 ) = 0 at the point P1 ,
which is defined as the limiting position of P when α1 is fixed but h → 0. The
full envelope is the curve traced out by P1 as α1 changes to generate successive
members of the family of curves. Of course, for any finite h, f(x, y, α1 + h) = 0 is
one of these curves and the envelope touches it at the point P2 .
We are now going to apply Rolle’s theorem, see subsection 2.1.10, with the
parameter α as the independent variable and x and y fixed as constants. In this
context, the two curves in figure 5.4 can be thought of as the projections onto the
xy-plane of the planar curves in which the surface f = f(x, y, α) = 0 meets the
planes α = α1 and α = α1 + h.
Along the normal to the page that passes through P , as α changes from α1
to α1 + h the value of f = f(x, y, α) will depart from zero, because the normal
meets the surface f = f(x, y, α) = 0 only at α = α1 and at α = α1 + h. However,
at these end points the values of f = f(x, y, α) will both be zero, and therefore
equal. This allows us to apply Rolle’s theorem and so to conclude that for some
θ in the range 0 ≤ θ ≤ 1 the partial derivative ∂f(x, y, α1 + θh)/∂α is zero. When
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5.10 ENVELOPES
h is made arbitrarily small, so that P → P1 , the three defining equations reduce
to two, which define the envelope point P1 :
f(x, y, α1 ) = 0
and
∂f(x, y, α1 )
= 0.
∂α
(5.42)
In (5.42) both the function and the gradient are evaluated at α = α1 . The equation
of the envelope g(x, y) = 0 is found by eliminating α1 between the two equations.
As a simple example we will now solve the problem which when posed mathematically reads ‘calculate the envelope appropriate to the family of straight lines
in the xy-plane whose points of intersection with the coordinate axes are a fixed
distance apart’. In more ordinary language, the problem is about a ladder leaning
against a wall.
A ladder of length L stands on level ground and can be leaned at any angle against a
vertical wall. Find the equation of the curve bounding the vertical area below the ladder.
We take the ground and the wall as the x- and y-axes respectively. If the foot of the ladder
is a from the foot of the wall and the top is b above the ground then the straight-line
equation of the ladder is
x y
+ = 1,
a
b
where a and b are connected by a2 + b2 = L2 . Expressed in standard form with only one
independent parameter, a, the equation becomes
f(x, y, a) =
x
y
− 1 = 0.
+ 2
a
(L − a2 )1/2
(5.43)
Now, differentiating (5.43) with respect to a and setting the derivative ∂f/∂a equal to
zero gives
x
ay
− 2 + 2
= 0;
a
(L − a2 )3/2
from which it follows that
a=
Lx1/3
(x2/3 + y 2/3 )1/2
and (L2 − a2 )1/2 =
Ly 1/3
.
(x2/3 + y 2/3 )1/2
Eliminating a by substituting these values into (5.43) gives, for the equation of the
envelope of all possible positions on the ladder,
x2/3 + y 2/3 = L2/3 .
This is the equation of an astroid (mentioned in exercise 2.19), and, together with the wall
and the ground, marks the boundary of the vertical area below the ladder. Other examples, drawn from both geometry and and the physical sciences, are
considered in the exercises at the end of this chapter. The shell trajectory problem
discussed earlier in this section is solved there, but in the guise of a question
about the water bell of an ornamental fountain.
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