Approximation

Motivation

The mathematical tool of approximation plays into our interest in applying math to physics by helping us answer a number of important questions.
• What's the dominant physics?
• How can we calculate the dominant physics when it's too complex to handle analytically?
• How can we know how good an approximate answer is?

What's the dominant physics?

When we attempt to describe some aspects of the "blooming, buzzing confusion"* we live in, we always have to make choices. We focus on what we believe to be critical issues and ignore other elements that we consider unimportant or irrelevant. This is not just true of our daily lives: when we want to describe the world mathematically we also have to make choices. If we want to describe a thrown ball we typically make the choices to:

• treat the ball as a rigid body, ignoring its deformability
• treat the earth as if it were flat, so we can take the gravitational field as a constant
• ignore air resistance (if the ball is small enough and the speed in air is not too large)
• ignore the coriolis effect of the rotating earth
• ignore special relativistic effects
• ignore quantum mechanics

We could go on for a long time, mentioning a multitude of small effects that might be important, depending on how accurate an answer we needed. Indeed, the real art of being a good physicist is "finding the physics" -- deciding what is important to pay attention to and what it is OK to ignore. The classic "physicist joke" relies on this insight.

When a physical factor is important or dominant, we try to treat it as correctly as we can; exactly, if possible. The less important effects, we either ignore or we use approximation method in mathematics to treat "as carefully as is appropriate for what we want to know." I believe that Einstein had something like this in mind when he said: Physics should be made as simple as possible, but not simpler.

How can we calculate the dominant physics when it's too complex to handle analytically?

Often, when dealing with "real world physics," we find that it is often impossible to solve our problems in terms of the simple kind of analytic answers (i.e., given in terms of functions that we can easily evaluate) we typically see in undergraduate physics. In these cases, we may have to use powerful computer technologies to generate answers that are approximate.

These "approximate" answers can, in fact, be as good as the "analytic" answers we are used to generating. For example, we may be perfectly happy to write our analytic answers in terms of sines or cosines, but these are transcendental functions. That is, they cannot be exactly generated for any arbitrary value (only for some special cases). We are presently accustomed to pushing a button and having a calculator or computer spit out 8 or 10 decimal values for a sine or cosine (as a previous generation was accustomed to looking them up in a paper table). But generating those decimal values (or creating the previous generation's numerical tables) requires both an approximation process and a way of controlling our errors.

How can we know how good an approximate answer is?

One of the most important tools in the mathematics of approximation is the controlling and understanding of error. That is, if we have done some approximate numerical calculation, can we get bounds on how big the mistake we might be making is? These studies are important components of mathematical approximation tools.

This plays quite well with how physics looks at the world. In physics (or in science in general), we never make exact statements, but we make approximate statements that have controlled errors. Thus, if we say, "The world is a sphere," that statement is wrong. But if we say, "There is a point inside the earth such that if you measured the distances to the surface in all different directions, all of your results would agree to better than 1%," we have a result that can be seen to be correct and that we can expect to remain correct for millions of years (barring some unforseen catastrophe).

To get some idea of how we bound results in approximations, read the chapter on series and convergence in Boas (chapter 1).

* William James, describing how he thought a newborn infant must perceive the world.

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