The differential equation for the path of a planet around the sun (or any object in an inverse square force field) is, in polar coordinates, $$ \frac{1}{r^{2}} \frac{d}{d \theta}\left(\frac{1}{r^{2}} \frac{d r}{d \theta}\right)-\frac{1}{r^{3}}=-\frac{k}{r^{2}} $$ Make the substitution \(u=1 / r\) and solve the equation to show that the path is a conic section.

The inverse square law is fundamental in understanding the motion of planets and other celestial bodies. It states that a force (such as gravity) between two masses is inversely proportional to the square of the distance between them. In formula terms, if the force is denoted as F and the distance as r, we can write: \( F \propto\frac{1} {r^2} \). This means that if the distance between two objects doubles, the gravitational force becomes four times weaker.

The inverse square law is pivotal in deriving the differential equations describing planetary motion. For example, Newton used this concept to explain why planets orbit the sun. By substituting the distances in Newton's Law of Universal Gravitation into our planetary motion equation, we align our understanding of forces with observed celestial paths.

Conic sections are the curves obtained by intersecting a plane with a cone. These include circles, ellipses, parabolas, and hyperbolas. When dealing with planetary motion, especially in polar coordinates, we often find that the paths taken by celestial bodies are conic sections.

The equation derived from our differential equation simplifies into the form of conic sections. When we say that the path of a planet is a conic section, it means it can be a circle, an ellipse, a parabola, or a hyperbola, depending on the energy and conditions of the orbit. For example, if the energy of a planetary body relative to the central mass (like the sun) is negative, the path is an ellipse.

Understanding conic sections helps us predict the behavior of orbits and plan space missions as they reflect natural laws governing gravitational forces.

A simple harmonic oscillator is a system where the restoring force is directly proportional to the displacement from equilibrium, leading to oscillatory motion. In our differential equation context, it appears when we rearrange terms to get the form \(\frac{d^2u}{d\theta^2}\ + \ u = k\)\.

This form closely resembles the simple harmonic oscillator equation: \( \frac{d^2x}{dt^2} + \omega^2 x = 0 \), where \[ \omega \] is the angular frequency. Solving it gives us sinusoidal solutions like \( x(t) = A\cos(\theta - \theta_0) + k \), indicating repetitive orbital paths.

Recognizing our planetary motion equation as a simple harmonic oscillator helps identify the sinusoidal nature of the orbit, harmonizing the planets' positions with predictable, periodic motion. The constants in the solution's general form are determined by initial conditions, aligning the theoretical paths with real-world observations.