Hi everyone!
Basically what the title asks. I was wondering if there was some sort of one to one pairing between certain phenomena and math concepts. For example, I’ve heard that multiplication is an indication of interaction between two variables.
Another good example of what I mean is Maxwells equations. How was he able to figure out all the details to comprehensively describe electromagnetism? How did he know what math tool to use to depict a real world phenomena?
How can one read these equations and discern what would happen in reality?
Thanks for your time yall!
In: Physics
You play around with things until you get something that seems to work.
A good example of this (if slightly historically inaccurate) is [epicycles](https://en.wikipedia.org/wiki/Deferent_and_epicycle). Observations of the planets showed that they moved in slightly weird paths – sometimes moving backwards, sometimes getting further away. The solution to this was to add in “epicycles” – or cycles within the cycles (as you can see in the top right diagram on that page). This leads to a [wonderfully complicated](https://en.wikipedia.org/wiki/File:Cassini_apparent.jpg) but largely accurate model of the Solar System. It works as a mathematical model, and could predict the motion of the planets.
Of course it is largely “wrong.” The Earth isn’t the centre of the Solar System; the problem they were dealing with was not that there were circles on circles in the motion of the planets, but that the planets *including the Earth* were moving around the Sun. We can still use the epicycle model to track the position of planets relative to the Earth (the maths is fine, mostly), but we’re better off using the heliocentric model, with at least Newtonian gravity, to understand what is really going on.
You asked about [Maxwell’s equations](https://en.wikipedia.org/wiki/History_of_Maxwell%27s_equations), and they are a lovely example of how science works, step by step, to produce useful results. Maxwell didn’t come up with them alone. Their current form was first expressed by Heaviside in the 1880s, alongside Gibbs and Hertz; originally they were known as the Hertz-Heaviside equations, or the Maxwell-Hertz equations. Heaviside’s versions are slightly different to Maxwell’s, but built on them, using new notation and new mathematical tools developed in part to help with these physics problems.
Maxwell’s versions are scattered through a paper published 20 years earlier. But they are also built on earlier work. Of the four modern “Maxwell’s equations” they are Gauss’s law (applied to electric fields), Gauss’s law applied to magnetism, Faraday’s law of induction (also called the Maxwell-Faraday equation), and Ampère’s circuital law (with Maxwell’s addition).
Maxwell’s contribution was adding in an extra term to that last one, which was needed to make them all fit together neatly – which he also did – to create a coherent and consistent system of equations. Maxwell was building on the work of Coulomb, Ampère, Faraday, Lenz and any number of other physicists and mathematicians of the 18th and 19th centuries. There is a quote from Newton, referring back to Descartes, that if he had “seen further it is by standing on the shoulders of giants”, but the reality is all science is collaborative; we do what we do by building on the work of brilliant, but normal scientists and mathematicians who helped us on the path.
> How can one read these equations and discern what would happen in reality?
By understanding what all the maths means, and plugging in values.
Maxwell’s equations – in their modern form – relate the electric field **E** and magnetic field **B** (or **H** if we’re using an H-field), with the local charge density *ρ* and local current density **J**.
Like any mathematical equation we take the things we know, plug them in, and rearrange to get something useful. For example, if we plug in *ρ = 0* and **J** = 0 (i.e. no current or charge), we can combine all four equations to get a thing called the wave equation; we get rippling, self-propelling electric and magnetic fields, that move at a fixed speed. We can use the equations to predict things; given certain values for *ρ* and **J**, we should get certain values for **E** and **B**, or in other combinations.
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