Derivation of newton's gravitational law from kepler's law of motion
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shall consider Kepler’s Second Law (that the planet sweeps out equal areas in equal times) first, because it has a simple physical interpretation.
Looking at the above picture, in the time Dt during which the planet moves from A to B, the area swept out is the approximately triangular area ABS, where S is the center of the Sun. For the distance AB sufficiently small, this area tends to that of the long thin triangle BSC, which has a base of length rDq and a height r. Using area of a triangle = ½ base´height, it follows immediately that

Now, the angular momentum L of the planet in its orbit is given by

Looking at the above picture, in the time Dt during which the planet moves from A to B, the area swept out is the approximately triangular area ABS, where S is the center of the Sun. For the distance AB sufficiently small, this area tends to that of the long thin triangle BSC, which has a base of length rDq and a height r. Using area of a triangle = ½ base´height, it follows immediately that

Now, the angular momentum L of the planet in its orbit is given by

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