Confirming Einstein’s Theory of General Relativity With Calculus, Part 6j: Rationale for Method of Undetermined Coefficients VII

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 $(r,\theta)$ with the Sun at the origin, under general relativity follows the initial-value problem

$u''(\theta) + u(\theta) = \displaystyle \frac{1}{\alpha} + \frac{\delta}{\alpha^2} + \frac{\delta \epsilon^2}{2\alpha^2} + \frac{2\delta \epsilon \cos \theta}{\alpha^2} + \frac{\delta \epsilon^2 \cos 2\theta}{2\alpha^2}$ ,

$u(0) = \displaystyle \frac{1}{P}$ ,

$u'(0) = 0$ ,

where $u = \displaystyle \frac{1}{r}$ , $\displaystyle \frac{1}{\alpha} = \frac{GMm^2}{\ell^2}$ , $\delta = \displaystyle \frac{3GM}{c^2}$ , $G$ is the gravitational constant of the universe, $m$ is the mass of the planet, $M$ is the mass of the Sun, $\ell$ is the constant angular momentum of the planet, $c$ is the speed of light, and $P$ is the smallest distance of the planet from the Sun during its orbit (i.e., at perihelion).

Let me summarize the partial results that we’ve found in the past few posts.

1. The general solution of the associated homogeneous differential equation

$u''(\theta) + u(\theta) = 0$

$u_0(\theta) = c_1 \cos \theta + c_2 \sin \theta$ .

2. One particular solution of the nonhomogeneous differentiatial equation

$u''(\theta) + u(\theta) = \displaystyle \frac{1}{\alpha} + \frac{\delta}{\alpha^2} + \frac{\delta \epsilon^2}{2\alpha^2}$

$u_1(\theta) = \displaystyle \frac{1}{\alpha} + \frac{\delta}{\alpha^2} + \frac{\delta \epsilon^2}{2\alpha^2}$ .

3. One particular solution of the nonhomogeneous differential equation

$u''(\theta) + u(\theta) = \displaystyle \frac{2\delta \epsilon \cos \theta}{\alpha^2}$

$u_2(\theta) = \displaystyle \frac{\delta \epsilon}{\alpha^2} \theta \sin \theta$ .

4. One particular solution of the nonhomogeneous differential equatio

$u''(\theta) + u(\theta) = \displaystyle \frac{\delta \epsilon^2 \cos 2\theta}{2\alpha^2}$

$u_3(\theta) = \displaystyle -\frac{\delta \epsilon^2}{6\alpha^2} \cos 2\theta$ .

To solve the original differential equation, we will simply add these four solutions together:

$u(\theta) = u_0(\theta) + u_1(\theta) + u_2(\theta) + u_3(\theta)$

$= c_1 \cos \theta + c_2 \sin \theta + \displaystyle \frac{1}{\alpha} + \frac{\delta}{\alpha^2} + \frac{\delta \epsilon^2}{2\alpha^2} +\frac{\delta \epsilon}{\alpha^2} \theta \sin \theta- \frac{\delta \epsilon^2}{6\alpha^2} \cos 2\theta$ .

It’s a straightforward exercise to show that this new function satisfies the original differential equation:

$u''(\theta) + u(\theta) = u_0''(\theta) + u_1''(\theta) + u_2''(\theta) + u_3''(\theta) + u_0(\theta) + u_1(\theta) + u_2(\theta) + u_3(\theta)$

$= [u_0''(\theta) +u_0(\theta)]+ [u_1''(\theta) + u_1(\theta)]+[u_2''(\theta) + u_2(\theta)] + [u_3''(\theta) + u_3(\theta)]$

$= 0 + \displaystyle \left( \frac{1}{\alpha} + \frac{\delta}{\alpha^2} + \frac{\delta \epsilon^2}{2\alpha^2} \right) + \frac{2\delta \epsilon \cos \theta}{\alpha^2} + \frac{\delta \epsilon^2 \cos 2\theta}{2\alpha^2}$

$= \displaystyle \frac{1}{\alpha} + \frac{\delta}{\alpha^2} + \frac{\delta \epsilon^2}{2\alpha^2} + \frac{2\delta \epsilon \cos \theta}{\alpha^2} + \frac{\delta \epsilon^2 \cos 2\theta}{2\alpha^2}$ ,

as required.

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Confirming Einstein’s Theory of General Relativity With Calculus, Part 6j: Rationale for Method of Undetermined Coefficients VII

Published by John Quintanilla

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