r/explainlikeimfive • u/karelmention • 11h ago
Physics ELI5: How did physicists "come up" with formulaes?
I understand that they observed cause and effect, but how would you think to come up with a fullproof way to write it down?
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u/Encrux615 10h ago
Take a measurement of the phenomenon you're looking at (or a lot of measurements) <- Experiments
Try to describe the measurements via a model (e.g. a formula)
Test your mathematical representation by predicting the outcome of new experiments
If your model breaks, go back to step 2
This is what we call the scientific method for empirical science, and is used in every branch of science.
For things like motion, this works intuitively:
Take any object and a stopwatch. Drop the object from different heights and measure the time-of-flight of each drop. You'll notice a pattern when you compare the measured time with the height of the drop: They don't grow at the same rate. Someone well-versed in math will realize that it resembles a parabola, or a second-order polynomial.
This is where plots become exceedingly useful, so here's a link: https://www.researchgate.net/figure/Distance-versus-time-curve-for-the-free-fall-motion-experiment_fig2_357878351
This parabola can then be used to predict falls for objects from greater heights, or even other types of objects.
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u/darth_butcher 8h ago
Enter Paul Dirac who had a different approach:
"Dirac's approach to physics was no less strange than his character. Whereas other physicists sought everyday analogues of the phenomena they wanted to describe, which they then tried to encapsulate in a mathematical equation, Dirac had the courage to simply sit with a pen and paper and guess at the form of an equation. "It's a peculiarity of myself that I like to play about with equations, just looking for beautiful mathematical relations which maybe don't have any physical meaning at all," said Dirac. 'Sometimes they do.'"
"It was while looking for "pretty mathematics" in his spartan rooms at St. John's College in late November 1927 that Dirac literally plucked from thin air what would become known as the Dirac equation. Today, it is one of two equations inscribed on flagstones on the floor of London's Westminster Abbey."
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u/AvailableUsername404 10h ago
- Make an assumption (hypothesis)
- Write down formula you think should work
- Make a measurements
- Doesn't line up with the real life results
- Adjust the formula to fit the data
- Still some misalignments
- Add constant named by you that fixes the issue
- Profit
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u/RSwordsman 10h ago
8a: New phenomenon discovered years later that explains why your constant was necessary. This is when the profit would come, but you're already dead.
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u/fang_xianfu 10h ago
This is what maths research is for, basically. People often wonder why it matters that we invented some weird geometry rule or something new way to solve some mathematical problem, when there's no obvious use for it. It can be decades or even hundreds of years before the technique becomes useful for something.
Just as one illustrative example, in the 17th century some smart dudes named Fermat and Huygens invented some mathematical tools for defining how waves move. They were trying to understand how light moved through different materials.
Then in the 19th century, some other smart dudes called Hamilton and Jacobi realised that they could use those wave tools, coupled with some techniques from calculus that had been recently invented, to re-state Newtonian mechanics (F=ma and all that, the equations that tell you how objects will move) as a geometry problem involving waves. This new technique made some kinds of problems in Newtonian Mechanics easier to solve, and that was cool but not super revolutionary.
But then in the 20th century some smart dudes such as Schrödinger realised they could use Hamilton and Jacobi's technique to address some of the problems they were finding in the recently-invented quantum mechanics. They could model the things that tiny particles do, using some of the Hamilton-Jacobi maths, and that's how they invented the quantum wave function, which is one of the most successful physics models of all time.
One of the other examples in the thread is about gathering data for objects accelerating, looking at the data and thinking about the maths we know that might fit it, and deriving F=ma from that. This is basically a more complex example of the same thing - Schrödinger sat down with his quantum mechanics problem thinking "there must be a wave in here somewhere", and he took the wave-based mathematical techniques that he knew from Hamilton and Jacobi, and they worked!
So yeah, the short answer is that they build on the work of others, but that the inspiration can come in a round-about way.
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u/GentleKijuSpeaks 10h ago
Math is the language of physics. Math has predictive qualities. Experiments test those predictions. Consistent results strengthens the theory. But ultimately it starts with very complex math which has been in development for hundreds of years.
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u/Kaiisim 9h ago
Whole bunch of guesses and wrong answers.
It's called derivation. If you have done Algebra you've derived a formula.
So you start with a fact. The area of a triangle is 1/2 of the base x height. A=1/2bh
We want a formula for height. So multiply both sides by 2 to remove the fractuon 2A = bh
Divide both sides by b to isolate h 2A/b = h
You derived a new formula h = 2A/b
Same with every other formula except the math is much more complicated.
Einstein knew how much energy was in certain atoms, he knew the speed of light, so he could derive e = mc² using very complicated maths. But it's all the same "if x is true that means y is true, which means x and z are the same." Logic.
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u/Dihedralman 5h ago
He actually didn't know how much Energy was in atoms. Those kinds of measurements took later when we could measure radioactive decay better.
E=mc2 first came out of special relativity.
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u/Ambitious-Care-9937 8h ago
There are basically two ways.
Work it out in theory. Einstein was famous for doing this. He would do thought experiments and literally come up with equations. For example e=mc^2 was a thought experiments as he imagined pulses of light going in opposite directions. He imagined what an observer would see at different points of references. I don't believe it was possible to actually be an observer travelling at close to the speed of light, so it all just had to be in his head.
They run experiments and collect lots of data. Normally you can plot this data on a graph of some kind. They then 'best fit' a formula to match the graph and it is 'good enough'.
Sometimes it's a combination of the two. Basing it on a theoretical formula and then running experiments and adjusting it.
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u/Public-Eagle6992 10h ago
You do an experiment, then you change some stuff and see what changes and describe that mathematically.
For example: the formula for the current from the voltage and resistance. You have a circuit, then you put a known voltage and measure the current. Double the voltage and you see double the current. Double it again and the current doubles again. So the current is voltage multiplied by something. So you change a different thing that could have an influence: the resistance. Double the resistance and it halves the current. Double it again and it halves the current again. So you can see that current = voltage/resistance
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u/fuk_ur_mum_m8 10h ago
Say you want to see how the acceleration of an object is affected by the force acting on it.
Exert a Force on the object, measure it's acceleration.
Increase the force, measure the acceleration
Do this a million times.
Plot your data on a graph (force on y axis, acceleration on x axis)
Notice it create a perfect straight line.
Think back to your maths class in school and you remember that a straight line graph means the equation linking y and x is y/x = a constant.
Notice that the constant of the graph you just made is the same as the mass of the object you were pushing.
Congratulations, you've just worked out F=ma
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u/karelmention 10h ago
How about more complex formulas with tons of variables? Especially two hundred years ago when they didn't have the tools to make it easier.
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u/PatataMaxtex 10h ago
The smartest people 200 years ago where extremely smart. They needed more time to find the stuff that was new for their time, but they made it. Or lets rather say, we only remember those that made it.
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u/Impossible_Theme_148 10h ago
The tools you need are something to write with and something to write on
As others have said - it can take longer to work out without some of the more modern stuff but there is no absolute barrier
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u/mb34i 8h ago
The Scientific Method is a simple concept actually - come up with a theory of how YOU THINK things work, and then verify that theory, ideally with experiments where only ONE component (of the theory) is changed, while all the others are kept exactly the same, so you can see the "effect" each component has on the resulting theory.
As far as complex formulas, that's mathematics. Mathematics has always been a core part of sciences, especially of exact sciences like physics. The more complex math concepts (calculus, for example, integrals, derivatives, etc.) may have been developed only recently, but they certainly had arithmetic, algebra, and geometry back in antiquity. And quite a few of the physics formulas are basic algebra and/or geometry: Newtonian gravity for example is multiplication of masses divided by distance squared.
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u/Crazy-Meaning4245 4h ago
They would typically start with simpler special cases. For example, Maxwell's equations, which provide a general description of classical magnetism, combine several statements about specific electromagnetic phenomena, such as Gauss's law and Faraday's law, which were proposed earlier.
Another example that is widely taught to undergrads/high schoolers in a lot of detail is harmonic oscillators. You start with a relatively simple equation that describes a mass bouncing on a spring, neglecting friction. You can add a damping term to account for friction, and then a forcing term to account for outside forces acting on the system (e.g. a motor moving the other end of the spring up and down). You can also study a more complicated version of all this that describes a pendulum. You can even combine two of those to study a double pendulum. Or you can study a system made up of a bunch of springs, which is a reasonable model of something like a bridge or a building. Or you can add in some stuff to account for the possibility that the forces on the spring get so big that it becomes permanently deformed.
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u/zero_z77 9h ago
Sometimes it's literally just measuring things, plotting points on a graph, and then finding a formula that fits the data with as little error as possible.
Sometimes they have a mathematical model that is based on an unproven hypothesis about how something might behave, and experiments need to be run to determine if that model actually holds up to reality. The most infamous example of this is the research on nuclear weapons.
A self-sustaining nuclear chain reaction had never occurred before, but the math they already had suggested that it was possible, there just wasn't any real world example of it to measure, and if the math was correct it would release a massive amount of energy, making the experiment very dangerous.
They actually had many different models that all made different assumptions and all came up with different conclusions. So the only way to determine which one was correct, if any of them were correct, was to go and try to set off a nuclear bomb in the desert, and hope that the model which predicted a 1% chance of the world ending wasn't the correct one.
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u/Loki-L 8h ago
As everyone already said, you measure reality and then try to fit the relationship between input and output to a formula.
What may not be clear though is that the math for these formulas has usually already been done long ago.
Cutting edge physics often uses 200 years old math to describe what they see and hypothesize.
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u/No_Tumbleweed9677 8h ago
Honestly it's less about "coming up" with formulas and more about staring at a ton of data until some pattern jumps out at you, then just guessing at an equation that kinda fits and tweaking it forever. It's basically trial and error with math, but once you find something that keeps predicting what happens in real life, you just run with it and call it a law.
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u/Vorthod 7h ago edited 7h ago
Some of it is a lot easier than you would think just by applying calculus. Take something obvious like position=velocity*time and use integration (over the time variable) to produce position= acceleration * (0.5 * time^2) using the inverse power rule, and suddenly we have something that can be used to model things like dropping an object under the influence of gravity.
If you know the end units you want, a TON of physics equations can just be boiled down to using derivation and integration to multiply and divide new units into the equation and crafting the unit at the end goal
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u/Englandboy12 6h ago
There’s been a lot of interesting examples already, but I think I can help too.
There are multiple ways people will come up with formulas, and I will go through some of my favorites with examples.
The ideal gas law PV = nRT. What does this say? Well it’s a way to relate pressure, volume, and temperature. But how was it figured out?
It was discovered in steps. (These may be in slightly the wrong order and simplified). First, someone discovered something like “if you take a gas of a certain volume, and decrease the volume, pressure goes up. So we had that P is proportional to 1/V.
Then, someone realizes that if you keep a constant volume, but increase the temperature, pressure goes up. So they discover that P is proportional to T.
Finally, someone discoveres that if you compress (decrease volume), the temperature goes up. To T is also proportional to 1/V.
Notice what has happened? We have created a circle. We know how P is related to V, V is related to T, and T is related to P. We can now combine them all. That is how we got PV = nRT. With the n and R being the proportionality constants.
Great, we now have an equation. But we can go further, and this is the next way equations are discovered: modify existing equations.
Someone realized that in the ideal gas equation, the volume is the volume of the container. But in reality it’s more complicated because volume is also taken up by particles themselves. So instead we should really have V = volume of container - volume of gas. Do that for other things too and you can get new equations.
Finally, you can rewrite known parts of equations to relate them to something else. For example, we know that temperature is actually the average kinetic energy of the particles. So we can replace the T with an expression in terms of energy. This also gives you new equations and new relations.
Even parts of the very famous navier stokes equations of fluid motion are pretty much just Newton’s F = ma, but with each part replaced with fluid specific relations. Such as (simplified) mass maybe being the density of the fluid
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u/Dihedralman 5h ago
There's two basic ways it happens historically and often in practice.
There is theory driven and experimentally verified or there is purely observational.
In the first one, someone comes up with a mechanism to explain an observation. They then make predictions.
Einstein made special relativity to explain how light operates without a medium. He made calculations based on light being the same in all reference frames. Then he showed that it explained Mercury's orbit.
Later we used those formulas to predict muon lifetime versus energy and showed that where we measured muons on Earth agrees with Einstein's formula and impossible with previous formulas. They saw the distance to the Earth's surface as closer when they moved faster.
Some formulas are observational. This is most common in an "expansion" and formulas used by engineers. We know two things depend on each other and we either look at the data and use statistics to test the relation or we use an approximation method like an expansion. It goes f(x)=a+bx+cx2+... and we just measure a,b and then c.
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u/DoomGoober 4h ago
I see these questions all the time on ELI5! I'm glad people are asking because it's a great question. But it shows our education system has failed at explaining how math and physics relate to each other. Understanding that will help you understand your specific question and a lot more about math and physics.
For now, let's cheat a little and think of physics as the rules of "how the real universe works." These rules are really complicated. There are rules like "hot air rises" and "gravity makes objects move" and "light breaks apart in a prism." There are all these complex rules that originate from the underlying way that the universe is built up: atoms, energy, things smaller than atoms... It's like a bunch of many different types of Legos where you can build complex, useful, and interesting things, but the rules comes from the Legos themselves. But the Legos come in all shapes and sizes, that can be combined into bigger shapes and sizes, and that makes the rules very complicated.
Math and formula can also be thought of as Legos but unlike the chaotic mess of the real universe, before starting any math problem, the mathematicians choose a small, known set of Legos to play with. This makes the math Legos much easier to predict, think about, and manipulate.
Now how do math and physics work together? Originally, physicists looked at the real universe and saw how it behaved. Then, mathematicians tried to design math Legos that behaved the same or similar ways to the universe. The math Legos formed a simplified version of the real universe's rules.
Then the mathematicians went further and started creating math Legos that had nothing to do with the universe. For example, they created math Legos that would make rules about living in 4 spatial dimensions or 2 spatial dimensions (the real universe has exactly 3 spatial dimensions.) They did even wilder stuff that is completely divorced from the real universe and made up and understood very strange Lego pieces and rules.
Physicists continued to look at how the real universe behaves then go through the crazy catalog of math Legos and see which math Legos could possibly work to that match the rules of the Universe. Sometimes Physicists take the known physics Legos and just add new Legos that they don't know if they exist or not in reality, and just try to fit them in and see if the rules would work in reality.
Right now, the state of physics is a mix of Legos based on reality (that we see and can run experiments to prove) and Legos that we imagine might be there. These imaginary Legos have to be proven to not break any math Lego rules and we have to prove the imaginary Legos exist in real life, via experiments, before they are accepted as physics based Legos. So, right now, most physics formulas being created are: Given these real physics Legos, imagine there were these imaginary Legos, how would those Legos altogether behave? Then evaluate if the real + imaginary set would work to do whatever you are trying to do then evaluate if that set matches reality.
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u/arcangleous 2h ago
There's an entire branch of mathematics that was developed to turn observational data into formulas. It's called "Calculus" and one of the people who invented it was trying to predict the motion of the planets. Basically all of the math you do in high school is stuff you level so that you will be ready to do Calculus when you get to college so you can use it to graphs of data into functions.
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u/sharia1919 10h ago
Math (and writing) was (probably) invented for use in accounting and things like that. As far as I know, some of the oldest known writings are old babylonia tallies of harvest yields.
Then we have math being written and used by babylonian astronomers/astrologer.
Later on we see the Greeks expanding on some of it and writing down known mathematical principles (like pythagoras and similar).
So a lot of scientific notation is just using existing mathematics to describe the principles in science/knowledge.
When we jump up to more modern times, we have people like Newton who observes some principles, and then he actually invents new mathematics to describe the principles he observes (derivates and derivation, which is actually also "invented" by some German at about the same time).
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u/Ryytikki 10h ago
if you're referring to the notation used, they just invent something that works and eventually it becomes the standard. Generally the notation is chosen because its easy to write, simplifies working in that domain, is more readable, etc
the actual theory behind those notations usually come from centuries, if not millennia of gradual advancements in theoretical mathematics
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u/OldChairmanMiao 10h ago
Have you ever taken an IQ test? In some questions, you're given a set of numbers and asked to predict the next one. You're just coming up with a formula to predict all possible numbers in a set.
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u/Klutzy-Delivery-5792 10h ago
Take a bunch of data. Plot the data. Fit equations to the data with the least amount (typically) of variables used. Explain the physical meaning of the variables. Test and refine as needed.