The Equations Physics Cannot Solve

In introductory high school physics classes, students are commonly taught that solving a physics problem is just a matter of substituting numbers into equations and finding exact answers using algebraic methods to solve for unknown quantities. While this method of teaching is certainly very useful for introducing the concept that equations in physics can solve real-world problems, it has a major flaw. That is, it gives students the impression that finding an accurate numerical answer is always possible, and that this process of problem solving is all there is to the science of physics. In reality, there is rarely ever a realistic scenario in which a tidy, numerical answer can be found. Instead, what is actually more important to a physicist is finding relations between variables, such as velocity and position, upon which calculations can be made. In certain situations, exact solutions which can be used to describe certain phenomena do not even exist. One surprisingly simple example of this fact is the motion of a pendulum, which has no precise equation describing how it moves. One may note the use of words like “exact” and “accurate” when referring to finding a solution. This is because there are certain algorithms which can produce approximate results with minimal error.

To understand how physicists can create models which accurately describe real life phenomena, we must first take a detour to understand a fundamental mathematical concept which has applications in nearly every field of research: differential equations. The motivation behind creating these differential equations is that, in some cases, it is far easier to describe how a system changes rather than its exact state at a certain point in time. Take, for instance, the idea of compound interest, where the rate at which you gain interest depends on the amount of money you have at that point in time. This simple statement about how the rate at which money increases leads to a more complex relation of it being exponential with time. Another example illustrating this idea of how simple descriptions of change result in more nuanced solutions involving time is the motion of a projectile. Any object on earth accelerates downward at a constant rate due to gravity. Now, acceleration itself is the rate of change of velocity, which in turn is the rate of change of position. Combining all this information together we find that the path a projectile takes is parabolic, a relatively intricate answer from a simple constant variable.

The examples above showed simple instances where we can actually arrive at a final solution from a differential equation. However, this is not always the case, especially when external factors begin to make the system more complicated. The classic example of this is the oscillating pendulum. This may seem strange since nearly every introductory high school physics class gives an “exact” equation which describes its movement as a perfect sinusoidal wave. However, if a student were to record a real pendulum’s movement, they would find that this equation they were taught does not provide correct results. This is because the true motion of a pendulum is not a perfect wave, but instead a more complex function. Moreover, this model completely neglects air resistance, which chips away at the pendulum’s energy causing it to slow down until it eventually stops.

In fact, there exists no exact equation to describe a pendulum’s movement with time. Understandably, one may now wonder if a differential equation is of any use if an answer cannot be found. However, numerical results can still be calculated using special computational algorithms.

By using a brute-force method of solving differential equations, approximate answers can be found with very minimal error. One way of solving numerically would be to simulate taking tiny steps in time and finding the velocity and acceleration of the pendulum after each of these time-steps. Then, after each step, approximate the angle using relatively simple calculations. Actually trying to do this by hand would be completely impractical, since for an answer to be fairly accurate, each time step would have to be around 0.01 seconds. Thus, simulating a pendulum for just ten seconds requires one thousand separate iterations. Luckily, performing this many calculations is as simple as writing a few lines of code when using a computer. Shown below is a graph of six simulated pendulums, each with a different level of air resistance, which has been made using a simple program written in the Python programming language.

Differential equations appear all throughout the fields of physics, chemistry, biology, and more. Population dynamics, for instance, is a great example of how a group of two simple equations can lead to fascinating outcomes that can predict how predator-prey populations evolve. Furthermore, more complex problems in physics involving three dimensional objects, such as heat flow in a metallic object, can be solved using computational algorithms.

While many real life scenarios involve the use of differential equations, many lack exact analytical solutions. However, numerical methods still offer valuable insight into how a system behaves under a set of rules defined in a single equation. With the introduction of powerful computers which are able to perform thousands of calculations a second, numerical algorithms extend the impact of differential equations far beyond the limitations of pure algebraic manipulation.