# Derivation of Newton’s law of motion from Kepler’s laws of planetary motion

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## Abstract

Newton’s law of motion is derived from Kepler’s laws of planetary motion. This is achieved by applying a simple system identification method using numerical data from the planet’s orbits in conjunction with the inverse square law for the attractive force between celestial bodies and the concepts of the derivative and differential equation. The identification procedure yields the differential equation of motion of a body under the action of an applied force as stated by Newton. Moreover, the employed procedure, besides validating the inverse square law, permits the evaluation of the gravitational mass (standard gravitational parameter), paving thus the way for establishing Newton’s law of universal gravitation. As the employed mathematical tools and the theory were available before 1686, we are allowed to state that the equation of motion for a body with constant mass could have been established from Kepler’s law of planetary motion, before Newton had published his law of motion.

### Keywords

Newton’s law of motion Kepler’s laws Planetary motion Inverse square law System identification### References

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