The University of Georgia

The historical roots of the calculus of variations trace back to
Isaac Newton's *Principia Mathematica* problem concerning the
shape of a solid of revolution that experiences minimal resistance to
rapid motion through a "rare medium" consisting of elastic particles.
Newton's stated solution to this problem barely hinted at the
complexity and sophistication of his underlying analysis, and
remained enigmatic to most readers of the *Principia*, at least
until the late nineteenth century when rigorous foundations for the
variational calculus began to be laid. This article exploits modern
computer algebra to explore the meaning and origin of Newton's
analysis and solution. This is a
** Mathematica notebook version**
of a paper that originally appeared in

The study of motion of bodies *in vacu*o dates back at least
to Galileo [1638]. The effect of air resistance was first analyzed by
Newton in Book 2 of the *Principia* *Mathematica *[Newton
1687]. But his original analysis was so intricate and so sparsely
revealed that much of what Newton said (or hinted at) remained only
partly understood for well over two hundred years. Now the
availability of a computer algebra system using traditional notation
for input and output especially encourages an attempt to retrace his
steps.

**Figure 1** illustrates a solid of revolution consisting of a
symmetrical "nose cone" attached to a cylindrical body, moving
(downward) with velocity *v* through air. Alternatively, we can
consider the solid to be stationary with the air streaming upward.
Newton considered the air to consist of small elastic particles (such
as molecules), each of mass *m*, uniformly distibuted with
*N* particles per unit volume. The density of the air is then
*r* = *Nm* and the flux of air
particles across a unit area normal to the flow is *Nv*
(particles per second per unit area). When one of these particles
strikes the nose cone at angle * f *
to the surface normal, in rebounding elastically it experiences a
change 2*mv *cos(*f*) in its
momentum. The resulting change in the vertical component of momentum
is 2*mv* cos^{2}(*f*).
The rate at which such particles strike an area element *dA* of
the nose cone surface *S* is *Nv *cos(*f*) *dA*, so the vertical force (rate of
change of momentum, by Newton's own second law of motion) of air
resistance experienced by this area element is the product *dF*
= 2 *r* *v*^{2}
cos^{3}(*f*) *dA*. The
total force of air resistance experienced by the solid is therefore
given by the surface integral

evaluated over the surface *S* of the nose cone. We therefore
see that (under Newton's assumptions) the air resistance is
proportional to the density * r* of
the air and to the *square* of the velocity *v*, with
*drag coefficient R* given by

If the nose cone of the solid is a **flat circle** of radius
*r*, not rounded at all, then

the area of the circle. Hence we may regard the drag coefficient
*R* as the body's "effective cross-sectional area for
resistance."

In the first three examples below, we will consider a solid with a
cylindrical body with radius *r* = 1, having a nose cone of
height *h* = 1 that is generated by revolving the curve *y =
y(x)*, 0 < *x* < 1 around the *y*-axis. Using
cylindrical shells and the fact that

where *ds* = (*dx*^{2} +
*dy*^{2})^{1/2} is the curve's arclength
element, it is an elementary calculus exercise to show that the
"reduced" drag coefficient *R*_{p} =* R*/p
is given by the integral

In the computations that follow, we will for convenience delete
the subscript and use *R* to denote this *reduced* drag
coefficient.

**Figure 2** shows a pointed cone defined by

The reduced drag coefficient *R* is given by

**Figure 3** shows a rounded hemispherical nose cone defined by

The drag coefficient is

Thus the pointed conical nose and the rounded hemispherical nose
provide the same air resistance, namely, *half* that of a
cylinder with a flat circular "nose." This observation (for the
hemispherical nose) was Newton's starting point in his *Principia
*discussion of the solid of revolution of least resistance. (See
Proposition 34 in Section VII of Book 2 in [Newton 1687]; this was
Proposition 35 in the first edition of the *Principia*.)

Consider now a parabolic nose cone defined by

Then *R* is

Thus the parabolic nose cone offers less air resistance (about 20% less) than either the actual cone or the hemisphere.

Newton then raised the question of the shape of the optimal nose cone that provides the least possible air resistance. This was the first historical instance of a "calculus of variations" problem (predating the brachistochrone problem of the 1690s). He had the prescience to consider the possibility of a flat-tipped nose cone.

**Figure 4 **shows a "conical frustum" nose cone with a flat
tip of radius *a* and with a denoting
the angle of inclination of its side from the vertical. If the
conical frustum has height *h* and top radius *b*, then

and cos f = sin a. Adding the reduced drag coefficient
*a*^{2} of the flat tip and the integral corresponding
to the curved frustum, we find that the reduced drag coefficient is
given by

At this point Newton asked for the optimal conical frustum of
height *h* and top radius *b*. Differentiating with respect
to a,

we therefore see that the air resistance is minimal when
tan(2a ) = 2*b/h*:

Thus a flat-tipped nose cone can offer less air resistance than either a pointed one or a smoothly rounded one!

Note that if the height *h -> 0* while the base radius
*b* remains fixed, then the relation

Thus an "infinitesimally thin" conical frustum nose cone has exterior base angle a = p/4. This result will play an important role in the following sections.

Ignoring the flat tip, the reduced drag coefficient of the curved surface itself is

But , where is the
*slant height* of the conical frustum, so

This formula for the reduced drag coefficient *R* of the
curved surface of a conical frustum of slant height *s* and base
radii *a* and *b* is crucial to the investigation that
follows.

After calculating the drag coefficient of the conical frustum,
Newton reasoned that the optimal nose cone of least resistance would
have a flat circular tip joined to the cylindrical body by a
curvilinear band. What should be the radius *a* of the flat tip
and what should be the shape *y = y(x)* of the arc generating
this optimal band by revolution around the *y*-axis? Nowadays we
would regard this as a calculus of variations problem (complicated by
a variable endpoint condition) and proceed to set up the appropriate
Euler-Lagrange equation.

Not having this Euler-Lagrange equation at his disposal, Newton
proceeded directly, envisioning the curvilinear band as comprised of
very narrow conical frusta (see his notes and preliminary analyses in
[Newton 1974, 456--465]. Consider an adjacent pair of such conical
frusta, obtained by revolution around the *y*-axis of the
segment from * P*_{0}* = *(*x-h*,
*y*+*z-k*) to *P*_{1}* = *(*x*,
*y*) and the segment from *P*_{1} to
*P*_{2} = (*x*+*h*,
*y*+*z*+*k*) (**Figure 5**). Intuitively, Newton
wanted to impose the condition that -- in the limit as * h, k
->* 0 with (*x*, *y*) fixed -- these two frusta offer
miminal combined air resistance. Applying the conical frustum drag
resistance formula, we find that their separate reduced drag
coefficients are

and

Hence the total resistance coefficent of the two frusta is

We want to minimize *R*_{12} as a function of
*z* with *h*, *k*, *s*, and *x* held
constant, so we calculate the derivative with respect to *z*. To
simplify the expression, we first divide by the factor
4*h*^{3}.

Let's collect coefficients of powers of *z* in the numerator.

Now, reasoning (as Newton does) that *z *is very very small
if both h and *k* are very small, let's "blot out" the higher
powers of *z*,

set the result equal to 0, and solve for z.

We get our differential equation by noting from **Figure 5**
that when* h* is very small,

(thinking of the slope of the segment
*P*_{0}*P*_{2})* *and

(thinking of the second-degree term in the "Newton-Taylor series"
which gives the departure *z *from the linear approximation).

So finally Newton's differential equation finally takes the "normal" form

How did Newton state this differential equation, given that Euler
had not yet introduced the functional notation we are using here? In
the Scholium to Proposition 34 in Book II of the *Principia*
[Newton 1687, 333], we see a drawing equivalent to **Figure 6**,
where *BG* designates the radius *x*_{0} of the
nose cone's flat tip, and the curve *GN* (which we parametrize
by *y = y(x)*) generates its curved surface. Newton asserts
(without proof) that the shape of the optimal curved surface is
determined by the fact that, given the fixed length *BR*, the
point *N* where the tangent line is parallel to the segment
*GR* is determined by the condition

If we write this condition in the form

and then substitute

the result is the first-order differential equation

(where *C* = *BG/*4 is constant). Then differentiation
with respect to *x *yields precisely the second-order
differential equation **deq** derived above.

Thus Newton's geometrically stated condition amounts to a first
integral of **deq**. He asserts further that the base angle
indicated in Figure 6 is , consistent
with our result in the previous section.

Let us digress from Newton's line of investigation to explore a modern numerical solution of his differential equation for the surface of least resistance. (Though Newton did not have a computer algebra system at his disposal, he was no stranger to numerical integration, as one might infer from the nomenclature of "Newton-Cotes" methods.)

Suppose the cylindrical body has radius and height *r* =
*h* = 1, and denote by *x*_{0} the (unknown) radius
of the flat tip of the optimal nose cone. Then our solution *
y(x)* should satisfy the endpoint conditions

Moreover, since the inclination of the curved surface at the tip
is , we want to determine the initial point *
x*_{0} so that the solution on [ *x*_{0},1]
satisfying these endpoint conditions also satisfies the additional
initial condition

y'(x_{0}) = 1.

(This latter condition was the first historical instance of a "transversality condition" in a calculus of variations problem.) Thus our task is to find the solution of the initial value problem

such that *y*(1) = 1.

Let's first try * x*_{0} = 0.5 as an initial guess,
solve our system using **NDSolve**, and then evaluate *y*(1).

From the differential equation with *y'* (*x*) > 0 we
see that *y''* (*x*) > 0, so the curve *y* =
*y*(*x*) is convex upward, with its graph appearing as
pictured in **Figure 6**. Evidently, we need to start "sooner" in
order to end higher at *x* = 1. So let's try
*x*_{0} = 0.3:

Now we've started too soon. The method of bisection would call for
us to split the difference between * x*_{0} = 0.3 and
* x*_{0 }= 0.5, and try * x*_{0} = 0.4
next. Instead, let's first define the pertinent function *f*
such that * f* (*x*_{0}) = *y*(1), so we can
apply **FindRoot** to find the desired value of *
x*_{0}.

First we check the two values calculated previously,

to verify that there is, indeed, a solution of the equation *
f* (*x*_{0}) = 1 between * x*_{0} = 0.3
and *x*_{0 }= 0.5. We can use **FindRoot** to find
this solution.

Thus it appears that the desired value -- the radius of the circular flat spot at the tip of the optimal nose cone -- is approximately

(Recall that we are investigating a nose-cone of fixed radius and height both equal to 1.)

Finally, we solve Newton's initial value problem with the value of
* x*_{0} just found, and then calculate the
corresponding minimal resistance coefficient.

Thus the minimum possible value of the reduced drag coefficient is about 75% that of a rounded hemispherical nose.

To retrace in modern notation the steps we believe Newton may have taken, let us write the differential equation,

Noting that the dependent variable** ***y*** **is
missing from the equation, we begin with the usual reduction-of-order
substitution * u = y'.*

Obviously we can separate the variables.

We'll let *Mathematica* do the integration by partial
fractions.

Recalling our substitution, we get

Note that the dependent variable *y* is missing from this
first-order equation which gives *x* explicitly in terms of *
y'* (*x*) . We can solve in a standard fashion for *x
*and* y* in terms of the parameter ** v = y'**.

The usual chain rule device gives

Finally we want to impose the boundary conditions

The second of these says that * x* = *x*_{0}
when *v* = 1, and then the first says that *y* = 0 when
*v* = 1. To satisfy the left endpoint condition, we need to
choose *v*_{1} so that x(*v*_{1}) =
y(*v*_{1}) = 1. First, we find the values of
*C*_{2} and *C*_{3}.

With these substitutions, our two equations

define the desired solution curve parametrically in terms of the
slope *v = y'* once * x*_{0} has been determined.
We proceed to solve numerically, initially estimating *
x*_{0} = 0.5 (something between 0 and 1) and *v* = 2
(something larger than 1) for the value of the parameter where
*x* = *y* = 1.

So we pick out the initial *x*-value (tip radius) and the
final *v*-value (slope)

and plot the axial profile shape of the optimal nose cone.

Finally, let's plot a 3-dimensional picture of Newton's optimal nose cone of least resistance, flat tip and all.

On what basis can we plausibly hope to have retraced -- using a
computer algebra system -- the steps that Newton himself took? As
Whiteside remarks (see page 466 of [4]), "The immediate reaction of
Newton's contemporaries to this scholium [on the solid of least
resistance] on its publication in the 1687 *Principia* was one
of near-total incomprehension." Sometime in the mid-1690s, Newton
composed an intended addendum to clarify the matter (though it never
appeared in any subsequent edition of the *Principia). ***
Figure 9 **is an elaboration of Figure 6 above and corresponds
closely to Newton's figure as reproduced by Whiteside [Newton 1974,
479] (though with the

Newton provides the following steps for the geometric construction
of the point *N*. We translate Whiteside's Latin transcription
of the hand-written manuscript, and use *Mathematica* to
evaluate the results as we proceed through the construction.

**1**. Noting that

pick the point *S* on the *x*-axis such that

Then *x* = *BS* will be the x-coordinate of the desired
point *N*. Note that this construction agrees with the
expression for * x(v) *given in our evaluation of **eq1**
above. The *y*-coordinate of *N* is determined by the
remaining steps.

**2**. Pick the point F on the *x*-axis such that

so

**3**. Pick the point *A* on the positive *z*-axis
(the negative *y*-axis) such that * BA* = *BG* =
*x*_{0}. Then

**4**. Erect at *A* the perpendicular *AH* of length
*AH* = *BG* + *FS*, that is,

and then complete the rectangle *HAEI* shown in Figure 9.

**5**. Construct the rectangular hyperbola *xz = c* that
is tangent to the segment *AG*. An elementary computation shows
that the hyperbola will be tangent to the hypotenuse *AG* of the
triangle *ABG* provided that

Then denote by *K* and *L* the indicated points of
intersection of this hyperbola with the vertical sides of the
rectangle *HAEI*.

**6**. Finally, erect the segment *NS* perpendicular to
the *x*-axis with length *y* = *NS* given by

The curvilinear area in the numerator on the right-hand side is given by

so the *y*-coordinate of *N* is

When we compare this with our previous result

se see that Newton's geometric construction (which he sets forth without any indication of its origins) duplicates the parametrization that we have (and surely he) derived by solution of his differential equation.

In this article we have used formidable computational machinery to
explicate just a few pages of Newton's published work and preliminary
notes on the solid of least resistance. Paraphrasing Whiteside, we
can hardly fail to note how little justice the published *Principia
Mathematica* does to the complexity and sophistication of the
preliminary analyses on which that founding document of modern exact
science was based, especially when we consider the geometric
machinery he himself employed [Newton 1974, 457]. As William Whewell
wrote in 1837 (Whewell 1837, 167],

"The ponderous instrument of synthesis, so effective in his [Newton's] hands, has never since been grasped by [any]one who could use it for such purposes; and we gaze at it with admiring curiosity, as on some gigantic implement of war, which stands idle among the memorials of ancient days, and makes us wonder what manner of man he was who could wield as a weapon what we can hardly lift as a burden."

Galileo Galilei. 1638. *Dialogues Concerning the Two New
Sciences*. Translated by Henry Crew and Alfonso de Salvio, Great
Books of the Western World vol. 28. Encyclopedia Britannica, Inc.,
1952.

Newton, Isaac. 1687. *Philosophiae Naturalis Principia
Mathematica* (Mathematical Principles of Natural Philosophy).
Translated by Andrew Motte, revised by Florian Cajori. University of
California Press, 1966.

Newton, Isaac. 1974. *The Mathematical Papers of Isaac Newton*
Volume VI. Edited by D. T. Whiteside. Cambridge University Press.

Whewell, William. 1837. *History of the Inductive Sciences, from
the Earliest to the Present Time. * London.