You’ve probably studied maths and physics as two different subjects. You’ve also probably noticed how much they have in common. So how deep is the relationship between them and how are their histories intertwined?
The relationship between mathematics and physics has intrigued me more and more the deeper I go into the subjects. Physics is often described as ‘applied mathematics’, since maths is an essential tool for physics, so I grew curious about just how maths is used within physics since the two seem so linked.
From describing planetary motion to modelling the behaviour of electrons, maths is the language that physical laws are built with, and among the areas of maths I have encountered, calculus stands out as an essential tool in creating these laws. Calculus allows us to describe how systems change, with derivatives telling us how one quantity changes with respect to another, while integrals accumulate infinitely many tiny contributions to find total quantities. I wanted to look at how powerful calculus – and mathematics as a whole – is in uncovering the world of physics.
From the seventeenth century, the study of physics motivated important advances in mathematics. The creation of calculus was strongly linked to how essential it was for physics, as a new mathematical language was needed after new dynamics rose from Galileo Galilei and Isaac Newton’s works. As we can see from the definition of derivatives, classical mechanics is built off from maths, with velocity being how position changes with time, or similarly, position is the integral of velocity.
Since mechanics is such a fundamental aspect of physics, this shows how intrinsic calculus is to the subject. Mechanics goes further when Newton’s laws also take on a calculus form – a differential equation that when solved tells us exactly how something moves. This showed me how calculus is more than just doing maths, but a window into how nature behaves. Consider a simple harmonic oscillator, for example a mass on a spring. If you solve this differential equation you get an insight into the motion, which turns out to be periodic and repeating in a sinusoidal fashion with constant amplitude. A single equation, interpreted using calculus, conveys the entire behaviour of the system. Examples like this show why calculus is one of the most commonly used parts of maths within physics, and why it is so important.
However, calculus doesn’t just underpin mechanics. It can be seen throughout all of physics. Another example can be seen in Maxwell’s equations, which describe the behavior of electric and magnetic fields. Maxwell had taken a set of known experimental laws and unified them into a coherent set, which were in differential equation form. These show how the universe can be modelled mathematically and demonstrates why physical systems are described through calculus.
The more I study maths and physics, the more I see laws that are relationships between quantities and how solving differential equations can lead to making predictions. Many physical interpretations come from integrals or derivatives and calculus enables nature’s rules to be expressed through more than just words. Calculus paves the way for abstract ideas – such as motion, force, and energy – to be translated into equations which can then be analysed and used, hence why this area of mathematics remains so relevant to physics – it gives us the tools to understand, model and explore the workings of the universe.