3 motion under a central force, kepler's laws

3 motion under a central force, kepler's laws
When the force acting on a particle is always directed towards a fixed point, the motion is called central force motion.
This type of motion is particularly relevant when studying the orbital movement of planets and satellites.

The laws which govern this motion were first postulated by Kepler around 1630 and deduced from observation. Later when Newton enunciated his theory of gravitation and laws of motion, it was possible to deduce the characteristics of central force motion from Newton's Laws.

Before Kepler, another astronomer Copernicus had first stated around 1540 that the universe is not earth-centric but sun-centric and it is the earth that moves around the sun and not vice-versa as was the belief. But this idea had not caught momentum. Only after Kepler's observations and postulates and mathematical deduction that tallied with observation about planetary motion that this was accepted.
The simplest center-force motion is obviously the uniform circular motion of a particle around a fixed center. We learn about such motion under "centripetal force". Let us complicate that study by bringing in two factors. First, imagine two bodies of masses M1 and M2 which are moving with respect to each other. It can be shown that their motion is equivalent to the motion of their center of mass. Thus we can reduce a two-body-motion problem to a one-body-motion problem, thus partly reducing the complication. Further, if we have a very large body at the center and a tiny mass moving round it. Then M1 >> M2, so the system will behave as if first mass is stationary and the center of mass is at same position as that of M1. This model was useful for Kepler as he was trying to study the motion of planets around the sun, whose mass and size is very large wrt that of planets. Later we can go to a slightly complicated situation where masses of the two bodies are comparable to each other.
The second complication can be brought in by considering a path which is not circular but elliptical. Kepler postulated from various astronomical observations that the planetary path around the sun was elliptical. It is interesting to note that the two body problem was studied by Kepler (1571-1630) well before Newton was born. Kepler postulated the following 3 laws:
1.- The orbits of the planets are ellipses with the Sun at one focus
2.- The line joining a planet to the Sun sweeps out equal areas in equal intervals of time
3.- The square of the period of a planet is proportional to the cube of the major axis of its elliptical orbit.
Then comes the issue of force. The planetary motion is result of some force, which will obviously be less for far away planets. Applying Newton's law of Gravitation, the force is inversely proportional to square of distance.
we have F = - k by r square into vector e where vector e is a unit vector along the line between the sun and the planet.. The - sign indicates that the force is attractive but a generalisation can be made to include repulsive force also (k>0).
The corresponding scalar potential (the potential energy of the planet) is:
V(r) = k by r

Equations of Motion
The equation of motion (F = ma), is
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Since the only force in the system is directed towards point O, the angular momentum of m with respect
to the origin will be constant. Therefore, the position and velocity vectors, r and rÿ , will be in a plane
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