We’ll swiftly continue from the previous section to derive Kepler’s laws of planetary motion from Newton’s law of gravitation concerning a small particle of mass m (usually taken to be the planet Earth), under the attraction of a fixed large particle of mass M (otherwise known as the sun).
To order to derive Kepler’s laws, a systematic way of modeling the planets interaction is required. Hence, we’ll do some preliminaries before going to the derivation proper.
KEPLER'S LAWS PRELIMINARIES.1
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KEPLER'S LAWS PRELIMINARIES.2
IN SEARCH FOR THE ORBIT
KEPLER'S SECOND LAW
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VIDEO LECTURE part 1
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LESSON
We place a fixed particle M at the origin of a polar coordinate system and express the position vector of the moving particle m in the form
where
and also the unit vector
Using these definitions, we have the diagram below. As you can see, we have defined an ‘unconventional’ coordinate system for our calculations. Here is the rationale. One, we do not currently know the orbit of the particle m and so in order to do any sort of calculations, we have to use an axis system which changes as m moves around M. Two, anticipating that our calculations will involve Here comes the calculations. A word of caution: The algebra from here on gets highly complicated so I advise careful reading of the mathematics. Differentiating the unit vectors we have
by differentiating the trigonometry functions and substituting accordingly. The derivative
Now the geometrical meaning of these derivatives is not as important as using them for substitution into other equations which we will do now. Let’s find the velocity and acceleration vector of particle m, with careful use of product rule.
and
through the use of the derivatives we previously had. We now group the
using the equation Here comes the power in our coordinate axis we defined earlier. From the diagram we can resolve any force we acts on the particle m as
which allows use to nicely equate the unit vectors through Newton’s second law
or more appropriately called equations of motion. These equations govern the motion of particle m regardless of the nature of the force All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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is the unit vector in the direction of 
. It is obvious that
, perpendicular to 
and in the direction of increasing 
is given by
, the force the particle m experiences, we can resolve 
to vector components that are parallel and perpendicular our axis 
, simplifying the algebra.
and 
can be interpreted as the rotating the 
in the clockwise direction and vice versa. It is more important to find the derivatives with respect to t. Employing chain rule,
and 
and 
vectors together and write the acceleration vector in the form
as substitution.
giving us
applied to the particle, or in this case, planet Earth. We are now ready to derive Kepler’s laws.
