The Beauty of Mathematical Physics
I love physics. Why? Two reasons-one, it allows us to know more about the universe, right from here, our very own planet. All you need is a paper and pen, and you can discover the most fundamental truths about reality! (okay maybe just theoretical physics) But the second reason (and it is connected to the first one) is that it is one of the most beautiful and fundamental applications of math, IMHO. In fact, physics offers a perspective into not just the way nature works, but also into human creation and ingenuity. It says a lot about our curiosity to know, to learn, and really, really makes you wonder. I can’t remember how many times I have sat alone, just thinking about how wondrously mathematics can describe natural phenomena. And not in an ad hoc way which only partly explains it, but completely explains it. In this article, I want to give a few important examples of the laws of nature, the very essence of physics, condensed beautifully in mathematical form.
Lets start with the first one,
Who hasn’t heard of this? The famous Newton’s second law is the most basic equation governing mechanics, and it can be expressed simply as a product of two simply objects-mass, and acceleration. And it makes sense-of course, the heavier an object is, the more effort that has to be applied to get it moving from rest, and the higher I want my final speed to be regardless of mass, the more force I have to apply! How wonderfully this simple equation encapsulates basic proportionality!
Now, the second stop on our journey is the Work-Energy theorem. This one shows how the work done on a body (the energy transferred to a body) is equal to the change in kinetic energy. And its just one simple line integral.
That’s it. The change in kinetic energy of a body is simply equal to this integral. But that’s not all. The fact that this is the integral of a dot product tells us that is the particle is moving in a way such that its motion is perpendicular field, no work is done! The second thing it tells us is that if F is conservative, i.e., if we find that the work done in moving a particle through a closed loop is zero, then it tells us that there is a potential associated with the field, which then leads to the law of conservation of energy! Calculus and physics are a match made in heaven.
The third and final stop is something called the wave equation governing transverse waves. As you know, a wave is a disturbance in a medium. This causes the particles in it to be displaced perpendicular to the direction of energy transfer. Now, how can such a phenomenon be mathematically modelled? Through sine waves, of course! Since the displacement is a function of both position and time, the argument of the sine should be a function of position of time, and it is! in fact this function is of the form
Now, this is not the wave equation. The wave equation is actually a differential equation(you see a lot of those in physics), that connects the displacement of particles, the position of the particle, and the wave speed.
This is the equation that I mentioned. It is so simple, yet so informative. If you don’t see how wonderful it is, just check if the function I mentioned is a valid solution to it. It is! this equation is the definition of a wave. You see something like it, then that thing is a wave! In fact, this is what Maxwell used to prove that light is a disturbance in the electromagnetic field! Light is an electromagnetic wave! It’s the fields themselves that are oscillating.
Math is beautiful and interesting to me because of these equations. I love how complex phenomena can be condensed into a few equations, and how from these equations, you can know the inner workings of the universe. These equations are simply the rules of the game that is physics, and the way to play the game is through math. Think about it! Not just mathematical and theoretical physicists, but engineers too use Newton’s laws and Maxwell’s equations all the time. Without these basic mathematical formulations, modern life wouldn’t exist! There, I said it! You wouldn’t have cars, you wouldn’t have phones, you wouldn’t have machines! According to me, it is in the marriage of physics and math that true beauty exists. I love math. I love physics. Change my mind.