7.1 Why computational physics?

Now that you are familiar with the basics of programming and can write simple programs, you can begin to solve physics problems that have no ``closed-form'' solutions. The fact that a problem has no closed-form solution does not mean that it doesn't have a solution, only that that solution can't be written as any combination of elementary functions. Most (all?) of the interesting problems in the physical sciences fit into this category. Some of the problems that cannot be solved in closed-form are surprisingly simple. For example, the time dependence of the angular displacement a simple pendulum, $\theta(t)$, cannot be written as any combination of elementary functions<sup>7.1</sup> and neither can the orbits of three (or more) celestial bodies. This may seem a little odd to you if you're used to the ``nice, pretty'' answers of PHY 161 and PHY 262 but nice, pretty answers are actually the exception rather than the rule. Simple examples from math may be more familiar. Consider the equation $x=\cos x$. Can you solve it? How do you isolate $x$? This amazingly simple-looking equation cannot be simplified, that is, we cannot isolate $x$ and write it in terms of elementary functions. And yet the solution exists and is easy to find. Make sure your calculator is in radians mode, enter 0.7, and then repeatedly strike the cos key. Soon, you'll find that the answer stops changing. The equation $x=\cos x$ is solved for $x\approx 0.739085133215161$. If we rewrite the equation as $x-\cos x = 0$ and then plot it (ezplot('x-cos(x)',[0 1]), we can see graphically that there is indeed an answer and that it occurs near $x=0.7$.

Figure: $x-\cos x$.

Just because we can't write the solution in closed-form doesn't mean that it somehow isn't as good. You might argue that we actually can't write our solution down exactly so in that sense it's only approximate but we can determine it to as many digits as we want to. Consider $y=\arccos(-1)=\pi$. We can't write down $\pi$ exactly either but we don't consider our answer anything but exact. In the same way that $x-\cos x = 0$ doesn't have a pretty answer, most physics problems don't either--and the non-existence of a pretty answer does not imply that the system we're studying is exceptionally complex. Consider the two examples we just mentioned, a simple pendulum and a ``three-body'' problem, these are exceptionally simple cases and yet they have no closed-form solutions. When we can't solve a problem ``with paper and pencil'' we must resort to ``numerical'' solutions (like we did for $x=\cos x$ above). The easiest way to implement methods for finding solutions to problems that must be solved ``numerically'' is to use a computer. Thus, computational physics implies a computer-based approach to solving physics problems. At this stage, I should also mention that computational physics isn't just solving physics problems by computer but also includes computer simulations. Computer simulations are more than just solving a difficult equation numerically. Computer simulations allow us to make models of a physical system and then watch the evolution of the model almost as if we were performing an experiment in a real laboratory. Computer simulations have come to play a significant role in physics research. Some significant discoveries in physics have been made by performing ``computer experiments'' in ``virtual laboratories.'' Until the recent past, physics was divided into two major fields, theoretical physics and experimental physics. But recently, computational physics has become an important third field in physics. In the rest of this chapter, you will be introduced to the major techniques of computational physics.