Chapter 8: Problem 37

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.

Short Answer

Expert verified

The path of the planet forms a conic section, solved as \( r = \frac{1}{A \cos(\theta - \theta_0) + k} \).

Step by step solution

- Substitute Variable

Start by making the substitution \( u = \frac{1}{r} \). Then \( r = \frac{1}{u} \).

- Derivative Substitutions

Compute the necessary derivatives: \( \frac{dr}{d\theta} = \frac{dr}{du} \cdot \frac{du}{d\theta} \), and \( \frac{dr}{du} = -\frac{1}{u^2} \), so \( \frac{dr}{d\theta} = -\frac{1}{u^2} \cdot \frac{du}{d\theta} \).

- Second Derivative

Find the second derivative: \( \frac{d}{d\theta} \left( \frac{dr}{d\theta} \right) = \frac{d}{d\theta} \left( -\frac{1}{u^2} \cdot \frac{du}{d\theta} \right) = \frac{2}{u^3} \left( \frac{du}{d\theta} \right)^2 - \frac{1}{u^2} \cdot \frac{d^2u}{d\theta^2} \).

- Substitute and Simplify

Substitute these into the original equation: \( \frac{1}{r^{2}} \cdot \left( \frac{d}{d\theta} \left( \frac{1}{r^2} \cdot \alpha \right) \) where \( \alpha = -\frac{1}{u^2} \cdot \frac{du}{d\theta} \), simplify using \( r = \frac{1}{u} \): \( u^2 \left[ 2 \left( \frac{du}{d\theta} \right)^2 - u \frac{d^2u}{d\theta^2} \right] - u^3 = -ku \).

- Simplify to a Second Order Differential Equation

This simplifies to \( \frac{d^2u}{d\theta^2} + u = k \). It is a second-order differential equation.

- Solve Differential Equation

Solve the differential equation: the general solution is \( u = A \cos(\theta - \theta_0) + k \), where \( A \) and \( \theta_0 \) are constants. Substituting back \( u = \frac{1}{r} \), get \( \frac{1}{r} = A \cos(\theta - \theta_0) + k \).

- Identify Conic Section

Notice that this is the polar form of a conic section: \( r = \frac{1}{A \cos(\theta - \theta_0) + k} \), showing the path of the planet is indeed a conic section.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates

Polar coordinates offer a different way of describing points in a plane. Instead of using Cartesian coordinates \(x, y\),
we use a radius \(r\) and an angle \(\theta\).
The radius \(r\) is the distance from the origin, and the angle \(\theta\) is measured from the positive x-axis.

This system is very useful when dealing with circular or rotational problems, like planetary motion.
For instance, the distance of a planet from the sun can be effectively described using polar coordinates.

Inverse Square Law

The inverse square law describes how a force diminishes with distance.
In physics, it's often used to describe gravitational and electrostatic forces.

For planetary motion, the gravitational force between two objects is given by \( F = \frac{Gm_1m_2}{r^2} \), where \(G\), \(m_1\), and \(m_2\) are constants, and \(r\) is the distance between objects.

When dealing with planetary motion, understanding this law is essential.
It dictates that as the distance increases, the force weakens rapidly.
This relationship is vital in forming the differential equations governing planetary paths.

Second Order Differential Equation

A second order differential equation involves second derivatives.
It often appears in the study of dynamic systems, like the motion of planets.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone.
They include ellipses, parabolas, and hyperbolas.

In planetary motion, the path of a planet is a conic section.
Kepler's laws of planetary motion state that the orbit of a planet around the sun is an ellipse.
This relationship is grounded in the differential equations describing planetary motion, which, once solved, take the form of a conic section.

Understanding conic sections helps in visualizing and predicting planetary paths.
Recognizing the types of conic sections can aid in identifying whether a planet's orbit is circular, elliptical, parabolic, or hyperbolic.