Section 21.1 Introduction to the Classical Central Force Problem
In this chapter, we will examine a mathematically tractable and physically useful problem - that of two bodies interacting with each other through a central force.
Definition 21.1. Central Force.
A central force is a force between two objects that has two characteristics:
A central force depends only on the separation distance between the two bodies,
A central force points along the line connecting the two bodies.
The most common examples of this type of force are those that have \(\frac{1}{r^2}\) behavior, specifically the Newtonian gravitational force between two point (or spherically symmetric) masses and the Coulomb force between two point (or spherically symmetric) electric charges. Clearly both of these examples are idealizations - neither ideal point masses or charges nor perfectly spherical mass or charge distributions exist in nature, except perhaps for elementary particles such as electrons. However, deviations from ideal behavior are often small and can be neglected to within a reasonable approximation. (Power series to the rescue!) Also, notice the difference in length scale: the archetypal gravitational example is planetary motion - at astronomical length scales; the archetypal Coulomb example is the hydrogen atom - at atomic length scales.
Our ultimate goal is to solve the equations of motion for two masses \(m_1\) and \(m_2\) subject to a central force acting between them. When you considered this problem in introductory physics, you assumed that one of the masses was so large that it effectively remained at rest while all of the motion belonged to the other object. This assumption works fairly well for the Earth orbiting around the Sun or for a satellite orbiting around the Earth, but in general we are going to have to solve for the motion of both objects.
The two solutions to the central force problem - classical behavior exemplified by the gravitational interaction (addressed in this chapter) and quantum behavior exemplified by the Coulomb interaction (addressed in
Chapter 22) - are quite different from each other. By studying these two cases together in parallel, we will be able to explore the strong similarities and the important differences between classical and quantum physics.
Two of the unifying themes of this topic are the conservation laws:
Conservation of Energy
Conservation of Angular Momentum
The classical and quantum systems we will explore both have versions of these conservation laws, but they come up in the mathematical formalisms in different ways.
In the classical mechanics case, we will obtain the equations of motion in three equivalent ways,
using Newton’s second law,
using Lagrangian mechanics,
using energy conservation.
so that you will be able to compare and contrast the methods. The Newtonian approach is the most straightforward and naive, but it suggests changes of coordinates that inform the other methods. The Lagrangian and energy conservation approaches are slightly more sophisticated in that they exploit more of the symmetries from the beginning.
We will also consider forces that depend on the distance between the two bodies in ways other than \(\frac{1}{r^2}\) and explore the kinds of motion they produce.
