In this series, I’m discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury’s orbit and thus confirm Einstein’s theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago.
In this post, following from the previous two posts, we will show that if the motion of a planet around the Sun is expressed in polar coordinates , with the Sun at the origin, then under Newtonian mechanics (i.e., without general relativity) the motion of the planet follows the differential equation
,
where and is a certain constant. Deriving this governing differential equation will require some principles from physics. If you’d rather skip the physics and get to the mathematics, we’ll get to solving this differential equations in the next post.
From Newton’s second law, the gravitational force on the planet as it orbits the Sun satisfies
,
where the force and the acceleration are vectors. When written in polar coordinates, this becomes
,
where is a unit vector pointing away from the origin and is a unit vector perpendicular to that points in the direction of increasing .
Furthermore, from Newton’s Law of Gravitation, if the Sun is located at the origin, then the gravitational force on the planet is
,
where is the mass of the sun, is the mass of the planet, and is the gravitational constant of the universe (which is a constant, no matter what Q from Star Trek: The Next Generation says).
Since these are the same force, the components must be the same. (Also, the component must be zero, but we won’t need to use that fact.) Therefore,
,
or
.
In a previous post, we showed that
,
where is a constant, and
.
Substituting, we find
.
So, substituting and , we finally obtain the governing equation
.
This is the governing differential equation of planetary motion under Newtonian mechanics. For now, it’s not obvious why we chose as the constant on the right-hand side instead of just , but the reason for this choice will become apparent in future posts.
In the next few posts, we use differential equations (or, if you’d prefer, just calculus) to show that Newtonian mechanics predicts that planets orbit the Sun in ellipses.