In
the Polar Coordinates microworld, we used Newton's moving frames to show
that if a satellite (of mass 1 kg) is subject to a central force, that
is, it is accelerated at each instant toward (or away from the Earth) and the
acceleration depends only on its position at that instant, then a quantity called
angular momentum is constant, or conserved throughout its motion.
Recall that on the page Polar
Curves and their Moving Frames, you proved that when the acceleration of
the satellite is represented in the 
coordinates of the frame then
Angular and Radial Acceleration
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where the first coordinate
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and the second coordinate
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The argument went this way. We saw that in the 
coordinates of our moving frame:
Now when we described the
acceleration also in terms of our moving frame, we saw that the angular acceleration
must be 0, since we assumed that the acceleration was central. So it followed
that
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Next, we considered the quantity
associated with the satellite at each point in time:
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and saw that
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We thus applied the Mean
Value Theorem to conclude that the quantity 
is constant throughout the motion. It is a conserved quantity for the
orbit in the plane.
Notice that the only property
of gravity that we needed to prove that angular momentum was conserved is the
fact that the acceleration is radial and depends only on position (not velocity).
The precise magnitude of gravity was not needed. We say that gravity is a central
force (always attracting to the center), and we conclude that for any
central force, the angular momentum must be conserved.
We closed that microworld
with a mystery. For satellite orbits whose time evolution is governed by Newtonian
gravity, we saw in the pictures of the velocity components of those satellites
made with reference to their accompanying moving frames, an interesting fact.
Unlike the general curves
traced by velocity vectors in the moving frames for arbitrary motion around
an ellipse:
the pictures of the framed velocity vectors for motion under Newtonian gravity always seem to be circles!
In fact, those circles always seem to have their centers on the 
axis of the moving frame. And for the closed (elliptical) orbits we examined,
the circle did not contain the origin of the moving frame. It is now time to
examine what this condition means and what it implies about the form
of Newton's gravitational law.
Let us assume at the outset
that the acceleration is central, and that during the time that 
is different from 0, the quantity
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is a positive constant. As long as 
is different from 0, we see that 
must remain either positive or negative, and we choose a positive sense of motion.
It will follow from other considerations (from the precise form of Newton's
law) that if
is once different from 0, it will always be different from 0.
Now in the 
coordinates of our moving frame:
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We may therefore write, for constant 
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We may formulate the condition
that the 
lie in a circle as indicated in the pictures thus:
Circularity conditions (Conservation of Energy)
Circle Condition 1: There is a positive constant K and a non-negative radius R such that
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In this case the center of the circle is at the point 
in the moving frame. Expanding this, we see that
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We may thus restate the circle
condition in the following way:
Circle Condition 2: There is a positive constant K and a constant E such that
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We also require that
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In the second form, we see that the circle condition first stipulates that for some constant K,
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and then imposes a condition on that constant.
Now it may seem a strong condition
to impose on a central force that the aforementioned quantity should be constant.
But let us look at the force law that Newton suggested. We approximate the mass
of the Earth to be
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and the the radius of the Earth by
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Then, according to Newton's
Gravitational Law, there is a universal constant G
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with 
such that if 
represents the distance of a satellite from the center of the Earth, then the
acceleration due to gravity is:
measured in 
That is the acceleration is
radial and, by your earlier calculation in Section 1: Polar
Coordinates,
so
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Now let us see what this
specific form of the gravitational law implies. Suppose we calculate the derivative
of the expression from Circle Condition 2 above with respect to time:
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and this is equal to
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but since
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it is easy to see from
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that
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So we see that with the form
of the Gravitational Law that Newton proposed, the first part of Circle Condition
2 is satisfied:
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Call that constant 
so that
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then for the circle condition, we observe that since
, there is at least one time 
on its orbit where
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This is because
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and so
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would be unbounded above (hence could not be constant) if 
could become arbitrarily small. Therefore there is some time 
where 
attains its minimum value, and at that time, 
.
At that time,
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Then
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So
the second criterion for Circle Condition 2 is automatically satisfied
by Newton's law
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This conserved quantity E
is called the total energy of the satellite. Let us write it in the form:
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Recall that in the 
coordinates of our moving frame:
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Question 1: Show that if the velocity 
has Cartesian components 
then since
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it follows that
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The part of the total energy
E: 
is called the kinetic energy of the satellite at time t.
It is 
.
Recall that the mass of the satellite is 1 kg.
The other part of the total
energy: 
is called the gravitational potential energy of the satellite.
As you can see, it is associated with the satellite position rather than its
speed.
Principle of Reciprocity (Conservation of Energy)
Our principle of reciprocity
states that as the satellite moves through its orbit:
kinetic energy + gravitational potential energy is constant
Notice that if we assume the
satellite is moving along a line radially away from the center of the Earth
then 
and the above argument does not work. However in that 1-dimensional case, we
showed in Section 1: Polar Coordinates by direct differentiation that
if r is the distance from the center, and 
, the quantity
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is conserved under Newtonian gravitation also.
The calculation we made in
Motion in a Gravitational Field in the Polar Coordinates Chapter
showed that in that case (satellite moving radially along a line) it was sufficient
that the Energy 
for the satellite to "escape".
In the present case, we can
ask when the orbit will be bounded. We will see later that this happens when
.
We will see why in Harmony of the Spheres when we characterize the possible
orbits as conic sections. Here, we mention that the idea is visible in the circles
traced by the velocities in the moving frames:
A similar idea was apparently discovered by Physicist William Rowan Hamilton, and rediscovered by Richard Feynman in his delightful "lost lecture" to High School Students. The circle of velocities may
1)
have radius less than 
(the 
coordinate of the center) as pictured above. In this case the orbit is an ellipse.
Here the Energy < 0
2)
have radius equal to 
, so it is tangent to the 
axis. In this case, the orbit is a parabola. Here the Energy = 0
3)
have radius greater than 
. The circle crosses the 
axis. In this case, the orbit is a branch of a hyperbola. Here Energy > 0.
On
this page, you may experiment with satellite orbits. We use Cartesian instead
of polar coordinates to represent the plane motion. And we use the reciprocity
principle to construct the simulation as follows.
Given
a physical system that can be represented at each time by a pair of vectors
in the plane
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so that 
Suppose
we call the kinetic energy 
and suppose there is a function of position only: 
with
the property that
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Then
we call 
the potential energy of the system
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and the principle of reciprocity says that kinetic energy + potential energy is constant.
Now
recall the discussion of gravitational potential surfaces in the discussion
of Galileo's inclined planes in the Introduction. That discussion is
based on the following idea, first understood by Newton, but brought into modern
form by Euler, Lagrange, and Hamilton:
Whenever
a system in the x-y plane, such as our satellite, is governed by a reciprocity
principle, we may construct the family of curves in the x-y plane:
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These
are called equipotential curves, and for our satellite system they resemble:
You
may draw these curves by going to the Symbolic Calculator and executing in the
yellow command line:
u(x,y) := -100/sqrt(x^2+y^2);
draw the levels of u(x,y) using width 2 refinement 19 levels 20 range [-99,-1];
At any point 
(black dot) the velocity of the system 
(black arrow) changes instantaneously in such a way that the acceleration 
(red arrow) cuts the equipotential curves as quickly as possible. For the picture
below, the direction of acceleration is toward the origin (Earth). The magnitude
of the acceleration is determined by the number of curves the red arrow crosses
in a unit time. The details of that are a modern topic requiring the notion
of differential forms, and we will not develop them here, but for our
model system with potential energy function:
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we observe that the magnitude of this acceleration is given by
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This allows us to represent
Newtonian gravitational force on the satellite in our model system.
In
our model system, we can sum up the account of reciprocity with a trick. We
draw the graph of the gravitational potential energy function
above the x-y plane. As we said, for simplicity, we let K = 100 and graph
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Now imagine that there is
a constant vertical gravitational acceleration, just as Galileo did with
his inclined planes, that points down to the x-y plane. A particle is allowed
to slide without friction along the graph surface. At any instant, it experiences
an acceleration in the gradient direction, the direction in which the surface
falls most steeply away.
Next,
imagine that a light is situated high above the x-y plane and it projects the
shadow of the moving particle to that plane. The shadow moves in the x-y plane
in exactly the way a satellite moving under Newtonian gravity would move.
The closer to the center,
the steeper will be the acceleration. We give the "shadow" an initial
position and velocity in the x-y plane, and then trace this shadow
in order to understand what the satellite would do. This trick was developed
in the 19th century by William Rowan Hamilton, based on earlier work
of Euler and Lagrange. It is called the "Hamiltonian" model.
Exploration: Gravitational potential wells
The potential energy function
that we use in the model is already entered:
When
you come to this page, you have a birds eye view, looking straight down on the
x-y plane so that you see the projection (or "shadow") of the satellite,
with the blue Earth at the center.
You may change the energy
function later, if you like, to any differentiable function of x and y. If you
do, then press Enter on the field to draw the new graph. At any time you may
make the surface transparent by turning off the check in the Draw filled
? checkbox. One amusing choice for potential is: 3*cos((x^2+y^2)/10)
Now before you do the experiment, you may want to set the initial values for the satellite in the control panel.
The
screen extends from -20 to 20 units in both directions. The top row gives
the initial position of the satellite, and the second row gives the initial
velocity. Each has 2 coordinates.
The refinements determine
how finely the graph is drawn, and the step size determines how carefully the
system solves Newton's differential equation of motion. The smaller the step
size, the more accurate the solution, and the longer it takes to draw.
Now one important point.
When you want to stop drawing, press the "q" key (for quit).
You may also press the Esc key. This is the only way to stop the simulation.
When you do that, you will see a picture of the orbit of the satellite in the
white Graph2D on the right.
Press the Go button 
to start, then press the q key to quit.
Now what are some experiments
to perform? Recall the condition for closed orbits. We observed above that
1) If the Total Energy < 0, then the orbit is an ellipse.
2) If the Total Energy = 0, then the orbit is a parabola.
3) If the Total Energy > 0, then the orbit is on a branch of a hyperbola.
You can check these things.
For example with the preset values: 
you see that the kinetic energy is 
and the potential energy is -20, so the total energy is indeed negative. Try
some other values. While we have not proved it yet, but will show it in Harmony
of the Spheres, these orbits are conic sections, except in the case that
the satellite moves radially away on a straight line.
Question 2: Calculate the "escape velocity" for a satellite
launched vertically away from the Earth at 
if it is subject to our model gravitational potential 
.
Be careful. You may have to adjust the step size in the model, and in any case,
the system returns if the satellite is more than 40 units in any direction from
the Earth.
,
the radial acceleration 
,
the angular acceleration.
