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.
We have shown that the motion of a planet around the Sun, expressed in polar coordinates with the Sun at the origin, under general relativity follows the initial-value problem
,
,
,
where , , , is the gravitational constant of the universe, is the mass of the planet, is the mass of the Sun, is the constant angular momentum of the planet, is the speed of light, and is the smallest distance of the planet from the Sun during its orbit (i.e., at perihelion).
In recent posts, we used the method of undetermined coefficients to show that the general solution of the differential equation is
.
We now use the initial conditions to find the constants and . (We did this earlier when we solved the differential equation via variation of parameters, but we repeat the argument here for completeness.) From the initial condition , we obtain
,
so that
.
Next, we compute and use the initial condition :
.
Substituting these values for and , we finally arrive at the solution
.
Published by John Quintanilla
I’m a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT’s program for preparing secondary teachers of mathematics and science.
View all posts by John Quintanilla
Published
June 7, 2024February 13, 2024
