# What is the connection between Newton's laws of motion and his law of gravitation?

What is the connection between Newton's laws of motion and his law of gravitation? Is there a connection where one suggests the other, or are they just two separate ideas that are assumed to be true?

• They are completely separate, but neither was "assumed", not even at Newton's time. Some of the most important concepts of mechanics were already developed by Galileo and the motion of the planets suggested a law like the one that is named after Newton. There is a little bit of a controversy whether Newton was the first to notice that a 1/r potential could be used to explain planetary notion. Feb 16 '16 at 7:31
• I meant assumed as in only proven experimentally and taken as laws of the universe without some mathematical derivation from another source. Like is the law of gravitation true just because we observe it without fail or can it be derived from Newton's motion laws. Or are the motion laws somehow related to gravity. Feb 16 '16 at 7:37
• They are different but there was also uncertainty as to whether inertial mass was different from gravitational mass. en.wikipedia.org/wiki/Mass#Inertial_vs._gravitational_mass Feb 16 '16 at 7:41
• Laws in physics (or, more correctly, symmetries and equations of motion) are always derived from observation. They can not and are not being derived mathematically. Feb 16 '16 at 7:42
• Well, you can apply newtons laws of motion while considering the gravitational forces. But , They are not actually derived from one another, if that was what you meant. Neither of them influences others existence! Feb 16 '16 at 7:48

The $F= \frac{GMm}{r^2}$ law (Or equivalently $U=-\frac{G M m}{r}$ potential law) can't be deduced from Newton's laws of motion. It's a little secret that parts of physics like this are, from a purely mathematical perspective, totally arbitrary.
For example, it's perfectly reasonable to play build-a-reality and plug in whatever interaction potentials you want. $U=\frac{A}{r^{12}}-\frac{B}{r^6}$ is a completely valid interaction in classical mechanics. It doesn't contradict anything.