Microworld: Calculus
in Action: Chapter 4, Section 2: The Grand Deduction
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above to visit the Microworld.
Author:James
E. White
In
this last section of the final Chapter, we will continue the work of the previous
section where we derived John Kepler's Second Law of planetary motion, and
follow Newton in the deduction of Kepler's First and Third laws from Isaac
Newton's single hypothesis of Universal Gravitation.
Recall
that Kepler's First law asserts:
" The shape of each planet's orbit is an ellipse with the sun at one focus."
In
one beautiful calculation, using his Calculus, Newton showed that all of these
would follow if the planets, in their motion around the Sun, obeyed his Universal
Law of Gravitation. And, of course, he developed a theory of Lunar motion
on the same principles. Thus, these laws about the motion of the planets,
derived from empirical observation, could now be deduced mathematically, with
the aid of the Calculus.
Modeling
planetary motion with his gravitational law gave a geometrically compelling
but complicated picture of the evolution as a two-step process in the plane
of the ecliptic. The model is called a second-order ordinary differential
equation.
We
will show how, when we apply the two grand conservation laws: conservation
of angular momentum, and conservation of energy, Newton's theory of second
order differential equations in the line gives a dramatic reduction of the
picture. This latter is intimately related to the notion of "integration"
which is, as we saw a central theme in all of the Calculus. We will introduce
the small piece of this theory that we will need in this section, in order
to find a polar equation 
for a planetary orbit when the planet moves under Newtonian gravity. We will
discover that whether the orbit is an ellipse or a hyperbola depends only
on the (constant) energy.
Kepler's
third law of planetary motion states: "The ratio of the cube of the semi-major
axis of the ellipse (i.e., the average distance of the planet from the sun)
to the square of the planet's period (the time it needs to complete one revolution
around the sun) is the same for all the planets."
It
was one of the most mysterious quantitative laws in all of science in its
time, and it is still mysterious today, even though, with Newton's law of
Universal Gravitational Attraction, we will see in this section why it is
true.
The
reason Kepler's Third Law is true rests in two things: Kepler's Second
Law that states: "Each planet, as it moves around the Sun, sweeps out
equal areas in equal times." We proved this in the first section: Geometry
of Planetary Orbits. The second thing that it rests on is the Geometry
of Conic Sections.
The
following is the Table of Contents for this 6 page Microworld
- A Noteworthy Equation
- Second Order Differential Equations in the Line
- They Must Be Conics: Kepler's First Law
- Kepler's Third Law and the Music of the Spheres
- Testing Kepler's Third Law
- Symbolic Calculator
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| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
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