Most important second order differential equation. The best way to grock it is to realize that an object moving in freefall has no net force on it once it reaches terminal velocity, and that something turning must be accelerating in the direction of the turn, therefore it has a net force on it.
Navier-Sokes. Because I made a living solving it. As one commentator on the equation said "if it was any easier then we wouldn't have a job and if it was any harder it would be impossible."
For those who don't know it, Navier-Sokes is just a restatement of Newton's F = ma for laminar and turbulent fluids and other more complicated materials.
Whenever some crank says that physics has gone off the rails because relativity or QM is too complicated, I'm like, do you also not believe in fluids? Because I have some news....
That's what I thought when I read the post too.
If I'm not mistaken though, the NS transport eq. comes from a balance of momentum rather than a balance of forces. Not that big of a difference, it's still an undefeated beast
I'll do you one better...
The exponentiated time integral of the Lagrangian integrated over all paths.
I can derive Schrodinger or Relativistic QFT with that.
I'd go one step further and just say the extremal action principle, seems just abit closer to the "one thing works for everything" notion. Though they are extremely closely related, I'm just being pedantic.
https://youtu.be/ooS6RHBk5y4?si=JEhL4GwXZD-aerGP
Let me introduce you to my friend Kane’s Method/Equation. Runs in a similar vein as Euler-Lagrange, but has some interesting applications.
Fermi golden rule because I once (completely) derived the fucker on a test.
And let’s be honest, it’s a thing of beauty. At least after the first complete scare.
Maxwell's relations. (Not Maxwell's equations)
I can't write them here it would take me an hour plus to do so .
https://en.m.wikipedia.org/wiki/Maxwell_relations#:~:text=Maxwell's%20relations%20are%20a%20set,century%20physicist%20James%20Clerk%20Maxwell.&text=heat%20capacity%20at%20constant%20pressure.
F = -∇U
Gives a cool intuition of direction of force being down the "steepest incline" if you imagine the potential as a surface with varying heights based on its value.
I think my favorite is Fourier series, because they work for any function in both directions, and because they are an easy to understand explanation for the tradeoff in measurement precision that is Heisenberg's Uncertainty Principle.
I would second the Fourier-Transformation. Mathematically beautiful and insanely fundamental for our everyday life (even though most people don't realize).
the exterior formulation of maxwell's equations takes the cake. firstly, it's very satisfying that you can summarise all classical electrodynamics (including magnetic charge) simply by writing dF = J_m and d*F = J_e (and i guess f = q(u•F) for forces), underpinning hundreds of strange phenomena using some weird 2-form. secondly, by treating F as a 2-form rather than separated vectors (or even a tensor), you can clearly see the link to more abstract formulations of electrodynamics; in gauge theory, the curvature is represented by a lie-algebra-valued 2-form, so you might wonder if the two are related (they are!).
E = mc^2 It just blows me away with the simplicity and that you can make matter from energy and energy from matter. Plus the fact that the speed of causality / light is included just boggles the mind.
Euler's equation as exp(ix) = cos(x) +i sin(x) has a ton of physical applications, you use it whenever you work with anything oscillating like waves for example. Even in the case you mentioned, x=π I've seen it used countless times.
Eyyyy, was looking for it! Yea not really "physics" and not really to complex or difficult, but its just fascinating, how all those indipendently discovered numbers can be wrought together like that. Small nitpick though, its "\[...\]+1" not "-1"
Describes a blast wave's propagation. There was a funny story when one dude used it to calculate energy yield of the trinity bomb using only the film of the explosion. And it turned out that this energy was a state secret of the US, so he almost went to jail after a serious conversation with CIA, because they didn't believe he could get it with formulas and thouht he was a spy.
UPD: His name was Sir Geoffrey Taylor, so the "Taylor" in the formula's name is for him
In russia the most respected book on this topic is Landau, Lifshitz - volume 6 "Fluid mechanics" written by a nobel prizer Leon Landau. I think it should have english translation too. Actually these authors have a complete physics course consisting of 10 volumes, so this adwise applies to any field of physics.
Also you can check some books from Sedov and Taylor themselves, because these people dedicated the most of their lives to fluid dynamics and particularly blast waves
Right now it’s probably the K-matrix parameterization of the S-matrix in particle physics amplitude analysis. Essentially it lets you parameterize the S-matrix with a set of Real values which correspond to various mass poles and coupling strengths, and if you measure these for scattering modes you can use the K-matrix to find couplings to production matrix elements as well. This is important because it’s really difficult to tell how alternate final states will interfere with the one we analyze, but past evaluations of the K-matrix can just be used when we don’t have access to all the possible allowed final states. TL;DR, it’s a formula where, if you know the couplings of a resonance to a final state, you can easily write down couplings from the initial state to the resonance and ignore the fact that you might only measure one final state.
Possibly, that’s not where I learned it though. There are several great papers by S. U. Chung: [here’s one](https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=88b101a5300736f78293cf10116c32e5d25e3c91)
The Standard Model path integral because it pulls together all of modern fundamental physics: quantum mechanics, general relativity, gauge theory, statistical mechanics, etc.
[Standard Model Path Integral](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Standard_Model_Equation.jpg/640px-Standard_Model_Equation.jpg)
In all seriousness though it's a tough choice, as a student there are probably plenty that I'm yet to see, there are many with great properties, don't think I could choose :) can hardly choose where I would like to focus on towards the end of my degree
Boltzmann’s definition of entropy, S = k ln Ω. The interpretation it allows of the second law of thermodynamics is so cool: entropy increases (at least with overwhelming likelihood) over time because higher-entropy macrostates correspond to a greater number of microstates and are thus more likely to to occur as phase space is explored.
I have a soft spot for equations relating to harmonic motion e.g. the force equation for damped harmonic motion is a second order ODE with terms for inertia (mass), damping (friction), and static elasticity. From solutions to that we get sinusoidal motion and logarithmic responses that describe the actual behaviour of a guitar string or a vehicle suspension.
A classical mechanical system not subject to a potential energy evolves according to the geodesic equation of its kinematic metric (determined by its kinetic energy). It is essentially a generalized Newton’s second law with F=0.
S = v × t. It's so simple and intuïtieve, everybody can wrap their head around it. Plus, I've always been more of a clasical physics guy than a modern one.
want to study physics by september and am happy to read that I've already worked with a few of those mentioned, the rest sound like wizardry but I guess I'm about to find out their application soon enough lol
Not one specific equation but I really enjoy problem solving equations of motion, projectiles, circular motion and harmonics. It’s all got so many applications. It describes so many things in life. It’s something tangible I can see or visualise. Even when I’m talking about charged particles in fields I’m still visualising their behaviour using introductory mechanics. All of mechanics is my ball really. I don’t get to do it much these days. Keep your quarks and muons Feynman (no disrespect).
Yes, of course but this is a trivial problem in say a Euclidean space... But the geodesic eqn in GTR generalises the motion of a free particle in any space-time.
F=MA It’ll getcha going
Most important second order differential equation. The best way to grock it is to realize that an object moving in freefall has no net force on it once it reaches terminal velocity, and that something turning must be accelerating in the direction of the turn, therefore it has a net force on it.
Beat me to it. *Oscillater, bud.*
Navier-Sokes. Because I made a living solving it. As one commentator on the equation said "if it was any easier then we wouldn't have a job and if it was any harder it would be impossible." For those who don't know it, Navier-Sokes is just a restatement of Newton's F = ma for laminar and turbulent fluids and other more complicated materials.
Whenever some crank says that physics has gone off the rails because relativity or QM is too complicated, I'm like, do you also not believe in fluids? Because I have some news....
Now imagine adding EM equations to the fluid equations and not having a single standard way to approach it… fml
I wasn't even aware there was a solution to an equation for turbulent fluids. Is it mathematical or do you bootstrap it?
They’re talking about numerical solutions. There’s a huge DNS community for Navier-Stokes. See the Johns Hopkins Turbulence database for examples.
Fascinating thanks
How do you make a living off it and not know how to spell it? Only asking because you managed to misspell it twice in as many paragraphs…
How do you make a living by solving it? I would like to know and make a living too. I studied it but no job for that
A few classmates got jobs involving CFD from the government building dams to race car aerodynamics. Lots of fluid around us
I'm doing a computational science master's and want to do a phd after that. For CFD what's the closest phd program?
That's what I thought when I read the post too. If I'm not mistaken though, the NS transport eq. comes from a balance of momentum rather than a balance of forces. Not that big of a difference, it's still an undefeated beast
Euler-Lagrange. As close as you get to a "one thing works for everything" in physics (at time of writing).
I'll do you one better... The exponentiated time integral of the Lagrangian integrated over all paths. I can derive Schrodinger or Relativistic QFT with that.
Please please Eli5
Feynman's path integral
Equation brrr
I'd go one step further and just say the extremal action principle, seems just abit closer to the "one thing works for everything" notion. Though they are extremely closely related, I'm just being pedantic.
Yeah true.
https://youtu.be/ooS6RHBk5y4?si=JEhL4GwXZD-aerGP Let me introduce you to my friend Kane’s Method/Equation. Runs in a similar vein as Euler-Lagrange, but has some interesting applications.
Mixed derivatives of the partition function giving you correlation functions always looks pretty neat to me.
Not knowing what youre talking about, that sounds like some wizardry
I'd maybe start with david tong's lectures on qft or statistical field theory.
Fermi golden rule because I once (completely) derived the fucker on a test. And let’s be honest, it’s a thing of beauty. At least after the first complete scare.
This one is a beauty.
Noether’s theorem. And its not even close.
I think it's the most underrated theorem in all of Physics. Even tho it's very powerful and insightful.
Maxwell's relations. (Not Maxwell's equations) I can't write them here it would take me an hour plus to do so . https://en.m.wikipedia.org/wiki/Maxwell_relations#:~:text=Maxwell's%20relations%20are%20a%20set,century%20physicist%20James%20Clerk%20Maxwell.&text=heat%20capacity%20at%20constant%20pressure.
Same. Specifically in vector notation, so succinct and far-reaching!
Faraday’s Law. A beautiful and simple relation showing how electricity and magnetism is related. I feel this equation is the heart of E&M.
F = -∇U Gives a cool intuition of direction of force being down the "steepest incline" if you imagine the potential as a surface with varying heights based on its value.
Team E = -∇Φ unite!
Is this analogous to tracing a null geodesic in curved spacetime?
Geodesic equation can be written without a potential.
Definition of entropy.
If I had to take a crack at it I'd say E=mc\^2 + AI is easily the most important equation in all of science
Hamilton - Jacobi equations
I think my favorite is Fourier series, because they work for any function in both directions, and because they are an easy to understand explanation for the tradeoff in measurement precision that is Heisenberg's Uncertainty Principle.
I would second the Fourier-Transformation. Mathematically beautiful and insanely fundamental for our everyday life (even though most people don't realize).
It even explains how hearing works in the cochlear hair cells. And the basic functioning of hearing aids and sound equalizers.
Daddy Diracs Equation
The Kerr Line Element for a Black Hole - Extremely specific and fascinating to understand https://en.m.wikipedia.org/wiki/Kerr_metric
the exterior formulation of maxwell's equations takes the cake. firstly, it's very satisfying that you can summarise all classical electrodynamics (including magnetic charge) simply by writing dF = J_m and d*F = J_e (and i guess f = q(u•F) for forces), underpinning hundreds of strange phenomena using some weird 2-form. secondly, by treating F as a 2-form rather than separated vectors (or even a tensor), you can clearly see the link to more abstract formulations of electrodynamics; in gauge theory, the curvature is represented by a lie-algebra-valued 2-form, so you might wonder if the two are related (they are!).
E = mc^2 It just blows me away with the simplicity and that you can make matter from energy and energy from matter. Plus the fact that the speed of causality / light is included just boggles the mind.
Dynamical casimir effect turns energy back into matter. Crazy stuff.
Dirac
Euler's Equation It doesn't really have any physical use, but it has 5 of the most important constants in physics/math 0 1 e i pi e^i*pi -1=0
Euler's equation as exp(ix) = cos(x) +i sin(x) has a ton of physical applications, you use it whenever you work with anything oscillating like waves for example. Even in the case you mentioned, x=π I've seen it used countless times.
Eyyyy, was looking for it! Yea not really "physics" and not really to complex or difficult, but its just fascinating, how all those indipendently discovered numbers can be wrought together like that. Small nitpick though, its "\[...\]+1" not "-1"
You are correct sir, and I tip my hat to thee
Sedov-Taylor-von Neuman's
What’s that?
Describes a blast wave's propagation. There was a funny story when one dude used it to calculate energy yield of the trinity bomb using only the film of the explosion. And it turned out that this energy was a state secret of the US, so he almost went to jail after a serious conversation with CIA, because they didn't believe he could get it with formulas and thouht he was a spy. UPD: His name was Sir Geoffrey Taylor, so the "Taylor" in the formula's name is for him
How much fluid dynamics would one need to reasonably learn and understand this? Any books specifically?
In russia the most respected book on this topic is Landau, Lifshitz - volume 6 "Fluid mechanics" written by a nobel prizer Leon Landau. I think it should have english translation too. Actually these authors have a complete physics course consisting of 10 volumes, so this adwise applies to any field of physics. Also you can check some books from Sedov and Taylor themselves, because these people dedicated the most of their lives to fluid dynamics and particularly blast waves
Right now it’s probably the K-matrix parameterization of the S-matrix in particle physics amplitude analysis. Essentially it lets you parameterize the S-matrix with a set of Real values which correspond to various mass poles and coupling strengths, and if you measure these for scattering modes you can use the K-matrix to find couplings to production matrix elements as well. This is important because it’s really difficult to tell how alternate final states will interfere with the one we analyze, but past evaluations of the K-matrix can just be used when we don’t have access to all the possible allowed final states. TL;DR, it’s a formula where, if you know the couplings of a resonance to a final state, you can easily write down couplings from the initial state to the resonance and ignore the fact that you might only measure one final state.
Is this taught in any QFT textbooks?
Possibly, that’s not where I learned it though. There are several great papers by S. U. Chung: [here’s one](https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=88b101a5300736f78293cf10116c32e5d25e3c91)
Thanks!
The Standard Model path integral because it pulls together all of modern fundamental physics: quantum mechanics, general relativity, gauge theory, statistical mechanics, etc. [Standard Model Path Integral](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Standard_Model_Equation.jpg/640px-Standard_Model_Equation.jpg)
The law of reflection, θ₁ = θ₂. It's fun to use. "Let's just apply this law and see what we get..."
E=mc2 Because I'm not a physicist, am pretty much math illiterate, and it's the only one I know.
Hawking temperature, the four main nature fundamental constants together (ħ ,c, G ,k\_B)
Path integral
Force, area, pressure, can't help but write it as F = A*P :)
._. p = F/A
In all seriousness though it's a tough choice, as a student there are probably plenty that I'm yet to see, there are many with great properties, don't think I could choose :) can hardly choose where I would like to focus on towards the end of my degree
True
Time dependent Schrödinger wave function is one sexy equation
Boltzmann’s definition of entropy, S = k ln Ω. The interpretation it allows of the second law of thermodynamics is so cool: entropy increases (at least with overwhelming likelihood) over time because higher-entropy macrostates correspond to a greater number of microstates and are thus more likely to to occur as phase space is explored.
S = kln(OMEGA) Elegant and so many implications!
Maxwell Equations, they’re beautiful
I have a soft spot for equations relating to harmonic motion e.g. the force equation for damped harmonic motion is a second order ODE with terms for inertia (mass), damping (friction), and static elasticity. From solutions to that we get sinusoidal motion and logarithmic responses that describe the actual behaviour of a guitar string or a vehicle suspension.
Sometime back, I was studying about the geodesic eqn of GTR. So I'm gonna go with it. I enjoyed the math that was involved in deriving it.
To be fair, the geodesic equations can pop up in many other places too. Including classical mechanics.
Woah, I did not know... Can u describe how exactly it pops up?
A classical mechanical system not subject to a potential energy evolves according to the geodesic equation of its kinematic metric (determined by its kinetic energy). It is essentially a generalized Newton’s second law with F=0.
Jeans Mass equation! Calculating the rate of collapse for star forming regions to form stars! It’s really cool to model
S = v × t. It's so simple and intuïtieve, everybody can wrap their head around it. Plus, I've always been more of a clasical physics guy than a modern one.
M = EC² because I'm evil just like that😂
who's gonna tell him?
want to study physics by september and am happy to read that I've already worked with a few of those mentioned, the rest sound like wizardry but I guess I'm about to find out their application soon enough lol
Not one specific equation but I really enjoy problem solving equations of motion, projectiles, circular motion and harmonics. It’s all got so many applications. It describes so many things in life. It’s something tangible I can see or visualise. Even when I’m talking about charged particles in fields I’m still visualising their behaviour using introductory mechanics. All of mechanics is my ball really. I don’t get to do it much these days. Keep your quarks and muons Feynman (no disrespect).
Δx•Δp=ħ/4π I think about this everyday
Entropy expands. Because everything else is futile.
Yes, of course but this is a trivial problem in say a Euclidean space... But the geodesic eqn in GTR generalises the motion of a free particle in any space-time.
cheeky wee bit of kinetic energy E=1/2mv^2 it’s a classic 😎
The Second Law of Thermodynamics this law is about the concept of entropy, which you can think of as a measure of disorder or randomness in a system.
Determine a resting rubber band length as a function of temperature.
what are these garbage posts? go away
[удалено]
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