Confirming Einstein’s Theory of General Relativity With Calculus, Part 4a: Angular Momentum

Calculus, Differential Equations, Physics, Precalculus 2 Minutes

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 part of the series, we will show that if the motion of a planet around the Sun is expressed in polar coordinates (r,\theta), with the Sun at the origin, then under Newtonian mechanics (i.e., without general relativity) the motion of the planet follows the differential equation

u''(\theta) + u(\theta) = \displaystyle \frac{1}{\alpha},

where u = 1/r and \alpha 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 a few posts.

One principle from physics that we’ll need is the Law of Conservation of Angular Momentum. Mathematically, this is expressed by

mr^2 \displaystyle \frac{d\theta}{dt} = \ell,

where \ell is a constant. Of course, this can be written as

\displaystyle \frac{d\theta}{dt} = \displaystyle \frac{\ell}{mr^2};

this will be used a couple times in the derivation below.

As we’ll soon see, we will need to express the second derivative \displaystyle \frac{d^2 r}{d t^2} in a form that depends only on \theta. To do this, we use the Chain Rule to obtain

r' = \displaystyle \frac{dr}{dt}

= \displaystyle \frac{dr}{d\theta} \cdot \frac{d\theta}{dt}

= \displaystyle \frac{\ell}{mr^2} \frac{dr}{d\theta}

= \displaystyle - \frac{\ell}{m} \frac{d}{d\theta} \left( \frac{1}{r} \right).

This last step used the Chain Rule in reverse:

\displaystyle \frac{d}{d\theta} \left( \frac{1}{r} \right) = \frac{d}{dr} \left( \frac{1}{r} \right) \cdot \frac{dr}{dt} = -\frac{1}{r^2} \cdot \frac{dr}{dt}.

To examine the second derivative \displaystyle \frac{d^2 r}{d t^2}, we again use the Chain Rule:

\displaystyle \frac{d^2 r}{d t^2} = \displaystyle \frac{dr'}{dt}

= \displaystyle \frac{dr'}{d\theta} \cdot \frac{d\theta}{dt}

= \displaystyle \frac{\ell}{mr^2} \frac{dr'}{d\theta}

= \displaystyle \frac{\ell}{mr^2} \frac{d}{d\theta} \left[ \frac{dr}{dt} \right]

= \displaystyle \frac{\ell}{mr^2} \frac{d}{d\theta} \left[ - \frac{\ell}{m} \frac{d}{d\theta} \left( \frac{1}{r} \right) \right]

= \displaystyle - \frac{\ell^2}{m^2r^2} \frac{d}{d\theta} \left[ \frac{d}{d\theta} \left( \frac{1}{r} \right) \right]

= \displaystyle - \frac{\ell^2}{m^2r^2} \frac{d^2}{d\theta^2} \left( \frac{1}{r} \right) .

While far from obvious now, this will be needed when we rewrite Newton’s Second Law in polar coordinates.

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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.

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  1. Pingback: Confirming Einstein’s General Theory of Relativity with Calculus: Index – Mean Green Math

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