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## Deriving Kepler's first law from Newton's law of universal gravitation

The movement of an object with mass orbiting another body with mass is given by Newton's law of gravitation. If it is possible to consider the position of the larger object constant and use it as the origin of our coordinate system. Then the following equation applies for movement of the smaller object:

In order to solve this differential equation we first consider the angular momentum of or object around its orbit.

In the abscense of external toques, because the only force acting on the object is parallel to its position, the angular momentum is conserved.

We now multiply both sides of our equation from the right by the angular momentum and develop the right side of the equation using vector identities.

We observe that each side of our equation is a derivative of a quantity. We know integrate both sides and take the integrations constant into account.

Our objective is now to solve the equation for , so we multiply both sides by :

By applying a cyclic permutation of the resulting triple product and using the known property of the scalar product we now express the equation only in terms of the magnitudes of the vectors.

The last steps are to solve for

Finally we reach our result. Objects orbiting according to Newton's Law of Gravitation follow paths that correspond to the conic sections. Here is Kepler's first Law a special case, where our object has a stable orbit around the larger body.