Of Logarithmic Spirals and Planetary Orbits
Chambered
Nautilus, gnomons and fractals
Nature presents us
with many excellent examples of mathematical concepts. We should say
that mathematics is the art that strives (in its finest moments) to reflect
upon and illuminate our characteristic and human perception of Nature. The planets
are (essentially) spheres, and they move in curves that have an exceedingly
simple description, as we will see. Flowers, plants and animals grow according
to principles that can sometimes disclose simple mathematical relationships
also. As an example, consider the contour of Oliver Wendell Holmes' "Chambered
Nautilus" shell. This is an example of a construction of Jacob Bernoulli
that he called the "equiangular spiral."
We can imagine, following
an idea of D'Arcy Wentworth Thompson in his wonderful book, On Growth and
Form, that it acquired its shape in the following way. As the shell grew
from the center, the angle that the growing point made with the radius to the
center, O, is always the same angle y, as in the picture below. This just means that the principle
of growth is the same, and does not differ from time to time. The figure is
then "self-similar." It has the same shape at any point. Such figures
were sometimes called "gnomons" and are nowadays called "fractals."
We will see below
that we can approximate such growth with similar triangles:
Differential
equation strategy
The picture above
is at the heart of a method for representing curves by means of differential
approximation. That method is called a "differential equation" model
and it does more than represent the curves. It often offers explanatory principles
that can help us understand how complex shapes can evolve from simple local
laws.
In fact, this picture,
which is built from a growing sequence of similar triangles, contains the germ
of the Euler sequence that we studied in the previous part and will, by means
of that sequence provide a very simple description of the shape of a Nautilus
shell in terms of exponential growth. In that model we will also find a standard
method for solving differential equations in general. That method is called,
unsurprisingly, Euler's method.
Now
the real reason for studying the equiangular spiral is as heuristic preparation
for Newton's deduction of Kepler's laws from universal gravitation. As we mentioned,
Newton set up and solved the basic differential equation that provides the key
to those laws. But his equation took into account the fact that gravity acts
on velocities, and not directly on positions, as Galileo observed. Momentum,
therefore, plays a key role in his equations, and makes them for that reason
somewhat subtler than the spiral equations. In the picture below, we see that
the red acceleration vector (which is determined by the position of the satellite)
directly modifies the green velocity vector in each instant of time. The satellite
moves along the modified velocity vector. Such a differential equation is called
a "second-order ordinary differential equation." We will examine that
process also in the exploration in order to understand Newton's deduction when
we come to it in Harmony of the Spheres.
Self
similarity of the equiangular spiral
If we call the origin
of polar coordinates point O in the spiral picture, then each point on
the curve has a description in terms of the polar coordinates. Recall that the
first coordinate of a point on the curve is an angle, 
,
measured in the counterclockwise sense (for positive angles) from the positive
x-axis, and the second coordinate is
a nonnegative number, 
, which is the distance of the point from the origin, O. Thus, these
polar coordinates specify the point 
in the picture below.
If we think of 
as the independent variable, such a curve is a "polar graph" - the
graph of the function
or simply, 
.
Now a point P on this polar graph is entirely determined by q, and it has "polar coordinates"
Sometimes, it is
useful to write down the Cartesian coordinates of such a point. We saw that
when we examined conic sections. That is easy to do. The point
may be represented in Cartesian coordinates as
where
We may also go the
other way. Assuming the point (represented in Cartesian coordinates) 
is not the origin, what are its polar coordinates in terms of x and y? Suppose
that 
. Then 
is a well-defined angle between 
. The polar coordinates are 
where
and 
is the angle just found. If 
, 
is a well-defined angle between 
.
Let 
. The polar coordinates are 
where 
again. If 
,
then 
and r = y. If 
, then 
and r = -y.
If both x and y are 0, then q is not defined, and so the coordinates are
not defined in this case.
If we think of both 
as functions of time, then we may describe the curve (in polar coordinates)
as a polar curve
Here, we distinguish
between graphs and curves.
Having described
the principle of growth for the curve above by saying that at each point
P on the curve, the radius OP makes an angle 
with the tangent to the curve, measured, as indicated above, in the clockwise
sense from the radius to the tangent, we may ask: What is the mathematical
description of this curve in polar coordinates?
We will first describe
in the exploration below an approximation to this curve with a discrete
exponential function, and then we will return to give the exact description
as the solution of a differential equation. For that purpose, we will choose
a constant small angle 
and we will assume that 
has a fixed positive value in order to get started.
Let us use the idea
of "self-similarity" to see if we can make some headway. Suppose we
watch the curve grow from angle 
to angle 
where 
is now some fixed small angle. We would have a picture:
We are interested in the relation between the lengths OA and
OB
In this picture,
the triangle 
is helpful. Its three angles are: 
.
The constant angle 
is now equal to our constant angle 
. That is: 
.
The crucial observation is that for this 
the shape (but not the size) of the triangle 
is entirely determined, since all of the angles are determined.
This is what self-similarity
means. The growth law is the same from whatever angle it is measured. It is
a local or "differential" law, and knows nothing of what came before
(the actual distance from O). We see that this is true because it is
formulated only in terms of the angle between the radius and the tangent at
each point. It is plausible that a Nautilus shell would grow approximately in
this way - but by no means necessary. History, after all, does matter.
In any case, we will
assume that it is true of this discrete process (constant
)
that the shape of 
is the same for each new triangle. To move from one triangle to the next, we
simply make the side OB the new OA for the next triangle. What
do we observe ?
If we accept this
self-similarity, we see that the "law of sines" has something useful
to say. Consider the figure above. The law of sines says that:
This is because the altitude from O to AB is equal to 
and it is also equal to 
. So
If
we construct in this fashion a sequence of triangles, all with the same shape,
where the side OA of the next triangle was the side OB of the
previous, then we can conclude that if for each non-negative integer n,
we denote the numbers
Now
by a familiar identity, and so we may write:
Next, notice that
depends only on 
(and not on n). This leads us directly to the conclusion that 
is a discrete exponential function of n, and k is
simply 
.
Exploration:
A discrete geometric model for the equiangular spiral
You may simulate
this discrete model for fixed angles 
. On the control panel of the Spiral graph2D
enter an angle 
in degrees using the slider. We chose 100 degrees.
After that, choose
a starting distance from the origin on the x-axis. This is the Radius.
We chose 1 for simplicity.
Next, choose a Step size. We chose 0.5 in the picture above.
This determines with the other 2 choices the other angles
These choices determine the size and shape of
the first triangle, hence the size and shapes of all of the others. Of course,
the smaller the step size, the smoother the curve will appear, but we will take
that matter up later, when we describe the exact curve with a differential equation.
To start the simulation,
press Begin Spiral once. And then press Continue repeatedly for
each new step.
When you press Begin Spiral, you will see the first triangle:
As each new one is added, the previous is filled in yellow.
Now to compare your
discrete simulation with the actual spiral, press the animate button
after you have drawn a few steps.
This time, we will simulate with step size 0.05 after pressing Reset
for better accuracy:
The blue sequence
of points is the way the spiral would grow if it grew according to the law:
"At each point P on the curve, the radius OP makes an angle
with the tangent to the curve, measured in the clockwise sense from the
radius to the tangent".
Now our questions
are these:
(1) What is the
relation between this discrete approximation and the "actual" curve
?
(2) What is
the equation of the "actual" curve ?
At this point, we have no equation, only a sequence of discrete points that
only approximate the actual curve. So we take up these questions now.
Thus suppose that
we have a polar curve 
given by a differentiable function 
such that for each angle 
,
is the distance from the origin. We may as well assume that 
has a fixed value. Suppose that it satisfies:
At each point P on the curve, the radius OP makes a constant angle 
with the tangent to the curve, measured in the clockwise sense from the radius
to the tangent.
Then what will 
be in general? It is not obvious how to determine the slope of a tangent line
to a polar curve. So let us do that first. The Cartesian coordinates for the
general point on the curve are
And we know how to calculate the derivative of such a curve. That derivative
is a vector and the ratio of its y-coordinate to x-coordinate is the slope of
the tangent line. The derivative is:
Therefore the slope
of the tangent line (when it is defined) is
Now the slope of the ray through the origin and 
is simply
If two lines: 
have slope A and slope B respectively then the tangent of the
angle from 
in the counterclockwise sense is
Question 2: Prove this from the formula for the tangent of a difference
of angles.
End of Question
We
therefore see that
Why the minus sign? It is because we measure 
in the clockwise sense from the radius to the tangent as indicated in our picture:
Now
the expression above simplifies quite a bit
or as we prefer to write it
Now in this form,
we recognize that we have an equation that expresses the derivative of 
with respect to 
as a function of 
.
That equation is
In this case, the derivative is expressed in terms of 
alone. If it were expressed in terms of 
alone, e.g.
then we could use ordinary integration (anti-differentiation, really) to solve
for 
:
But this problem is not like that. Still it is easy to make a change of variable
to make the new problem like that.
Before we do that,
we give a working definition of first-order differential equations.
First
order ordinary differential equations
Definition 1: Suppose that x and t are variables and that
is a continuous real-valued function defined on the x-t plane. A first-order
ordinary differential equation is an equation of the form
What it means to
solve such an equation is to find a differentiable function 
that has the property that
for all 
End of Definition
Usually,
one stipulates that for some value of t, say t=0, the value of
x is specified to be some number, for example,
In this form, the differential equation is called an "initial-value problem"
and then, under fairly light conditions on F, a unique solution can be
found locally, at least, on some open interval containing 0. We will not enter
into the technicalities here.
Recalling our observations
in the previous part, and stipulating of course that 
never vanishes, because then we could not use polar coordinates at all, and
our tangent condition would not make sense in any case, we see that for the
natural logarithm function log that we just defined, this means that
Now we can integrate
to find that
for an arbitrary constant k. Now, exponentiating both sides we see that
or
and writing 
as a non-zero constant C, we may conclude that
where the value of C is easily seen to be
We thus have an
answer to Question 2 above: " What is the equation of the "actual"
curve ?"
Now we take up Question 1: "What is the relation between our discrete
approximation and the "actual" curve ?"
Euler's
Method
Most differential
equations do not have solutions that are as neatly packaged as this one. When
computers give approximate solutions to these equations, they often use the
method devised by Euler. If we have an initial value problem
and
we may ask for example, what is 
for the unique solution (assuming it exists) ? If we choose a positive integer
N and let
be a constant increment, then we can apply the following recursive procedure:
and for 
we approximate
We finish after N
steps with an approximation for 
.
This is only an approximation because at each intermediate step
since
Now this is what
we did with our discrete approximation to the spiral, although perhaps we did
not know that we were applying Euler's method to approximate the solution to
a differential equation.
To see this, let
us follow the steps of the construction above. First, let 
be the supplement of the angle we chose with the slider. And suppose for simplicity
that 
. Then we chose a step size that we will call h. The first step of our
construction was to build a triangle with vertices 
.
Each new triangle was geometrically similar to this one.
Now choose a natural
number N and suppose that h was selected to satisfy
so in particular, 
was not equal to 0. In that case, our construction led in the first step to
the triangle with vertices
Now if we apply Euler's method to the initial value problem
and
and we let 
, then Euler's method gives the sequence of numbers:
So it gives 
We recognize the
Euler sequence here. To see this, suppose we choose a positive integer N
and let 
.
Again assume that 
. Then we have
This is what our exponential solution to the differential equation tells us
is the exact value of 
.
Now when the step
size h is chosen so that 
in our geometric construction, then we see that after N steps, we will
have
which is a good approximation to 
when N is large.
You
may wonder what all of this has to do with planetary orbits. Satellite orbits
will also be modeled by a differential equation. But in this case, the state
of a satellite is not determined by its position alone. At each instant of time,
the state of a satellite is described by two vectors in the plane: its position
and its velocity. We may summarize this by saying that at each time t,
the state is described by the pair:
The differential
equation that we will have to solve will give an ordinary curve in the plane
subject to two conditions:
(1) 
(2) 
Here 
is the gravitational acceleration. It is a vector in the plane and it depends
only on the position (not the velocity) of the satellite, and it directly affects
the way velocity changes with time. But it is important to understand that there
are now 4 independent variables: 
. All vary with time and the conditions (1) and (2) above guarantee that if
we specify all 4 of the initial values of those variables, say at time 0, 
then there is a unique curve in this 4-dimensional space that will satisfy the
conditions (1) and (2). We know that curve by its position portrait 
as the orbit. This is because once we know 
then 
is determined by the equations.
We often write those
conditions as the single condition
This is the second-order differential equation in the plane that Newton studied.
We will not say
more now about how the equation is to be solved. Instead, in the next exploration,
we would like simply to say how to visualize it.
Exploration:
A second-order differential equation in the plane
The basic idea
is to approximate the orbit in small steps as we did above. The state of the
satellite is now given by four numbers: 
at each time.
We
show the position 
on the screen and in each step, we draw first the velocity
at the beginning of the step (green arrow). Next, we show the acceleration 
(red arrow) first along the orbit with tail on the satellite. Here, you will
see that it always points toward the origin. Next, we translate that acceleration
vector to the tip of the velocity vector, as you see it in the picture above.
It is then "added" to the initial velocity vector to give the final
velocity vector at the end of the interval. The result is represented by the
black line segment. We call that the new velocity.
The
satellite actually moves along the black line segment in the next step. Its
velocity is thus modified in each step by gravity. This two-step process is
what the exploration simulates.
The
control panel for the satellite orbit appears on the right side of the screen:
Its
operation is very much like that of the spiral control panel. In this case you
may do a step-by-step discrete simulation of a satellite orbiting the Sun, which
is at the center of the coordinate system. The initial condition for the satellite
is already determined. You may control the step size for the discrete simulation
however, by entering the value in the field:
As usual, the smaller the step size, the more accurate
(and slower) the discrete simulation. Now to start the simulation, press the
Begin Orbit button. 
The initial step will
be drawn as described above:
The
satellite is the small red dot at position (1,0) where distance is now measured
in "astronomical units" and this satellite is a solar probe launched
from the vicinity of the Earth (which is 1 A.U. from the Sun). As we mentioned,
first the initial velocity is drawn as a green arrow. This is rather large since
it contains also the velocity of the Earth itself in addition to the velocity
of the probe (the speed is about 3/8 of the Earth's speed around the Sun and
the heading is about 137 degrees). The red arrow is the Sunward acceleration
at the beginning of the step, and the black line is the sum of the acceleration
with the velocity the represents the velocity at the end of the step.
When
you press the Continue button repeatedly, you will see the satellite
move a short distance along the new velocity vector, then the velocity vector
will shift to the new position indicated by the black line at the end of the
previous step. Then the acceleration vector will be drawn first from the satellite,
pointing towards the Sun, and then from the tip of the velocity vector to compute
the new velocity.
In
this intricate dance, you will see the orbit of the satellite traced step-by-step
in a discrete simulation. If you would like to see the differential equation
solved continuously, press the Animation button. To stop this animation, you
must press the p key, or the Esc button. The animation is done
on the computer using Euler's method, as described above, in 4 dimensions.
You are now ready to enter the next Chapter, Harmony
of the Spheres and follow Newton's calculation.
