For example, we have differential equations that model, for example, resonance, vibration, heat transfer, the motion of a spring, etc. How were these equations first calculated and derived, and to such a degree that we can be sure of their accuracy in modeling real life phenomena? What is the process for actually deriving them like?
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You make observations.
Then you come up with mathematical equations that produce the same results.
Then you continue to make observations and see how the results of your equations compare with real life occurances.
You do an experiment. Measure the results.
Change one variable. Measure the results. Repeat several times.
Change another variable. Measure the results. Repeat several times.
Now you have a rough idea about how much each parameter affect the results.
For example you apply a known force to a spring and measure its deflection. Apply different force, measure deflection. Repeat several times and you have data relating force to deflection.
Apply the same loads to similar springs with different diameters. Now you have data with diameter as a variable.
Repeat with different materials, different temperatures etc.
If you are interested in how to obtain equations from experimental data, look up numerical analysis.
Physicists, and chemists, and most scientists in one way or another, follow the fundamental idea that correlations (observable relationships of some sort) exist for a reason. They start with the simplest possible situations, where some factor X is behaving in a direct fashion with regard to some other factor Y (everything else which might matter or could matter is considered to remain constant, so only thinking of the very special case of X and Y and nothing else).
Correlation is not proof of causation, but it is pretty good evidence that something is happening that relates the two factors together in some real fashion, and the question is only one of what, not if.
There are several ways to come up with a mathematical representation of that presumed process: you can envisage the process (imagine it acts like two perfect spheres hitting each other, perhaps) and then generate the equations which would describe the paths of such objects if it were to happen, then deal with the fun of derivatives (slight changes to the conditions) and run with it; OR you could look at a series of test results and figure out what sort of relationship the two parameters are displaying (first order linear, or what?) and then play with the equations.
We mostly do not start from scratch anymore given that the main (principle) basic behaviors have already been determined and modeled/defined.
Sometimes it is just basic common sense or a basic presumption, like total energy is heat plus work. That is, energy change manifests on one way or the other (could be a third way but we don’t know about it so we start with the idea that change in energy causes change in heat and/or does some work; worry about a third concern if we find that the model fails in certain conditions).
Fundamentally, though, you look at a process, a system, and think about what might be happening, and once you have a hypothesis about what is happening, create equations which represent that hypothetical process mathematically. Then you test reality to see if it mirrors what your hypothesis-based math says it ought to.
Your question is kind of like asking how a writer comes up with a story. It comes from thinking. Some thoughts make sense or “work”, some do not. Over time, the brain gets used to finding answers that work (answers that do not work are a waste of time and frustrating, so you learn to not go that way).
No modern scientist (at least none I have ever met) is working in a knowledge vacuum. We have been shown, by the work of the old-time greats, the pioneers, how to play the game. Now, if you are asking how the first folks came up with their ideas, I cannot answer. Genius has its own way of showing itself.
They just understand math better.
The best analogy I can think of is language.
It’s fairly easy to learn some phrase in a foreign language. You may even learn that there’s some word in the phrase that you can substitute in to change the meaning a bit.
But you’re almost certainly using the phrase wrong. You’re probably using it in the wrong context, you’re probably mispronouncing words, you may not even be remembering the phrase correctly and you certainly can’t figure out how to create a new phrase in that language.
How do people create new phrases in a language? They learn the language. They learn the patterns of how different ideas are expressed. They learn how to put those components together to express more complex ideas.
It’s the same thing with math. You may learn some formula, plug in values and get an answer but the people who create those formulas think of math as a language.
I’m not a physicist but I used to study economics and I can give you an example of why my thought process was.
The formulas generally describe some relationship. Variables are basically nouns so I’ll need at least two variables, maybe more if I think there are more factors involved in some relationship. Next I start to think about how they’re related. If they move in the same direction I can just put them on opposite sides of the “=” sign. If they move in opposite directions I’ll add a “-” to represent that. I’ll also look at the nature of the relationship; depending on how sensitive one factor is to an other I might add a placeholder for a coefficient or an exponent to represent that. That takes care of a lot the simple models. If I suspect that terms interact with each other I multiply them together.
After all that I would run regressions of my data on that proposed formula. That will get me empirical values for the coefficients and some other metrics that can tell me how likely my formula is to match the data.
I can also provide an example of how I think about physics equations. Eg F=MA. You can just plug values into that but what does it really mean? That formula says that there are 3 things that matter when it comes this relationship; force, mass and acceleration. For one thing that means the formula leaves out a lot of things; we don’t care about color, texture, shape, time of day, or phase of the moon. It also describes the relationship between F,M and A; for any given mass if we want more acceleration we need more force, if we want to accelerate something at a certain rate we’ll need more force as the mass gets bigger, if we’re applying a constant amount of force it less massive objects will accelerate more. That’s all pretty intuitive. It also tells us that the relationship between mass and acceleration is scalable rather than constant. That is, I can’t generally add something to mass and and subtract the same amount from acceleration and hope to get the same force, but if I double mass and halve acceleration I’ll have the same force.
You start in simple broad observations. I’ve had to derive a bunch of these and they can start really simply. The starting point for mechanical resonance, mechanical vibrations and the motion for a spring is Newton’s second law. if I push on something, it moves. These specific ones are all related and are in the same family.
Testing and observation is vital to making sure it models real world stuff. If it doesn’t reflect the real world, then it’s back to the drawing board.
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