Problem 9 - A History of Mathematics: An Introduction

Question

Show that the differential equation $$ \frac{d^{2} r}{d t^{2}}-r\left(\frac{d \theta}{d t}\right)^{2}=-\frac{k}{r^{2}} $$ derived by assuming that the component \(a_{r}\) of the force from Exercise 6 is inversely proportional to \(r^{2}\) is equivalent to the differential equation Hermann derived using the inverse square property of the central force.

Step-by-Step Solution

Answer

In conclusion, the given differential equation and the one derived using the inverse square property of the central force are equivalent, provided that the mass is assumed as constant with the value of 1. The equivalence of the differential equations demonstrates that Hermann's derivation was indeed valid for the central force having an inverse square nature.

Step 1: Write down the given differential equation using the latex formula

The given differential equation is: $$\frac{d^{2} r}{d t^{2}}-r\left(\frac{d \theta}{d t}\right)^{2}=-\frac{k}{r^{2}}$$

Step 2: Derive the differential equation using the inverse square property of the central force

The central force acting on a particle in a plane polar coordinate system is: $$F_r = -\frac{k}{r^2}$$ Now, let's find the radial and angular components of the acceleration. The radial acceleration \(a_r\) is the time derivative of the radial component of the velocity, i.e., $$a_r = \frac{d^2r}{dt^2} - r \left(\frac{d\theta}{dt}\right)^2 \Rightarrow \frac{d^2r}{dt^2} - r \left(\frac{d\theta}{dt}\right)^2 = a_r$$ Now, we will use Newton's second law, which states that the force acting on the particle equals the mass times the acceleration: $$F_r = m \cdot a_r$$ Substitute the expressions for \(F_r\) and \(a_r\) in the previous equation: $$-\frac{k}{r^2} = m \left(\frac{d^2r}{dt^2} - r \left(\frac{d\theta}{dt}\right)^2\right)$$ Dividing both sides by mass, we obtain the following differential equation: $$\frac{d^2r}{dt^2} - r \left(\frac{d\theta}{dt}\right)^2 = -\frac{k}{mr^2}$$

Step 3: Compare the derived equation and the given equation to show equivalence

Comparing the derived equation: $$\frac{d^2r}{dt^2} - r \left(\frac{d\theta}{dt}\right)^2 = -\frac{k}{mr^2}$$ with the given equation: $$\frac{d^{2} r}{d t^{2}}-r\left(\frac{d \theta}{d t}\right)^{2}=-\frac{k}{r^{2}}$$ We see that both the equations are the same if we choose a constant \(m = 1\). Therefore, the given differential equation derived by assuming that the component \(a_{r}\) of the force is inversely proportional to \(r^2\) is equivalent to the differential equation Hermann derived using the inverse square property of the central force.

Key Concepts

Inverse Square Law

The inverse square law is a cornerstone in physics which posits that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity. In other words, if the distance is doubled, the force (or intensity) decreases by a factor of four, and so on.

This concept is fundamental in understanding gravitational and electrostatic forces, where the force between two masses or charges diminishes as the square of the distance between them increases. Mathematically, the law can be described by the equation: \[ F = \frac{k}{r^2} \] where \( F \) represents the force, \( k \) is a constant that depends on the type of force and the nature of the objects involved, and \( r \) is the distance from the source.

The inverse square law is not only crucial in comprehending gravitational and electrostatic interactions but also in explaining phenomena such as the intensity of light radiating from a source, and the strength of a sound as it travels through a medium.

Central Force

A central force is a force that is directed along the line joining the particle and a fixed point (the center), and its magnitude only depends on the distance of the particle from the center. This type of force is always directed radially and does not have any tangential component.

When dealing with the dynamics of orbital motion, central forces play a crucial role, such as in the case of planets orbiting a star where gravity provides the central force. An essential characteristic of central forces is that they conserve the angular momentum of the particle which is orbiting. One of the most famed examples of a central force follows the inverse square law, as seen in Newton's law of universal gravitation.

For a particle under the influence of a central force, the motion can be described using plane polar coordinates, where the force can be written as: \[ F_r = -\frac{k}{r^2} \] Here, \( F_r \) is the radial component of the force, and \( -k \) is a constant associated with the particular force, such as the gravitational constant.

Plane Polar Coordinates

Plane polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a fixed point (often called the pole or the origin) and an angle from a fixed direction (the polar axis).

In plane polar coordinates, a position vector is typically described by \( r \) (the radial distance from the origin) and \( \theta \) (the angle between the position vector and the polar axis). This system is particularly useful for problems with circular or spherical symmetry, such as those involving central forces.

Radial and Angular Components

For a moving particle, the velocity and acceleration can be decomposed into radial and angular components. The radial component changes the distance of the particle from the origin, while the angular component changes its angle. This distinction proves especially helpful in analyzing motions influenced by a central force, allowing for the differential equation that outlines the motion to be effectively broken down and solved.

Newton's Second Law

Newton's second law of motion is a fundamental principle in classical mechanics that describes the relationship between the force applied to an object, its mass, and the acceleration it experiences. In its simplest form, the law states that the force (\( F \)) applied to an object is equal to the product of its mass (\( m \)) and the acceleration (\( a \)) it undergoes: \[ F = m \times a \]

This relationship is at the heart of classical dynamics and allows us to calculate one variable if the other two are known. In the context of differential equations for motion under a central force, Newton's second law provides the mathematical framework to relate the force acting on an object to its motion.

By expressing Newton's second law in terms of plane polar coordinates, as seen in our original exercise, we can create a link between the concepts of inverse square law, central force, and the motion described in these specific coordinates. This approach is critical in solving problems related to orbital dynamics and understanding how objects move under the influence of gravity.