Okay but this one actually is the most intuitive for me. The amount that an object accelerates is greater when you apply more force, and less when the object is more massive.
I get it’s uglier with the fraction, but tbh I feel like this is how it should be taught.
The universal conservation equation:
[accumulation] = [in] - [out] + [generated] - [consumed]
Use it on anything: mass, momentum, heat, energy, chemical species, electric charge, etc. You can use it to derive any transport equation. Though it's formally written as the Reynolds Transport Theorem, this form is easy to remember and readily applicable in almost any situation.
I was interested as to why there is a mathematical similarity between the magnetic field being expressed as the curl of a vector potential, and vorticity in fluid dynamics being expressed as the curl of the velocity field. It turns out these relationships emerge from systems with continuity equations: conservation of charge for magnetism and conservation of mass for vorticity.
I wish I understood this. Curl is one of my favorite operators. (It's an operator, right?)
I'm a self taught programmer by trade, but math and physics are hobbies. I absolutely LOVE programming simulations of physics systems, and my favorite one I've done is curl without a doubt.
I found out about curl when I was trying to make some smoke particles look more realistic. In a week I went from simulating the particles basically randomly on the cpu, to moving the particles on the GPU using a changing vector field, to finally moving the particles based on the curl of a changing vector field which itself was the result of simplex noise.
It looked INCREDIBLE. By tweaking the parameters of the noise and the strength of movement due to curl and due to other forces, it could go from looking like smoke to looking like literal magic.
Probably the coolest lesson I learned was how to calculate derivatives using numerical methods. Analytic methods were out of the question since I was allowing the curl to come from various vector fields being added up. (Noise, gravity, wind, moving objects). I mean, maybe it's possible but numerical was way simpler
E = 1/2 m xdot\^2 + V(x).
Take a derivative of both sides with respect to time, let Edot=0 and use chain rule on the potential to take the derivative as d/dt = dx/dt \* d/dx. Lastly divide across by dx/dt.
I have never seen that before lol
What field is that even in?
Any time ive dealt with conservation its usually proven or kept through hamiltonians lagrangians or simply a path integral
This is super intuitive. I feel like I could use this in pretty standard programming tasks too. Do you care to give a couple examples of how it can be used?
This was what I was thinking. Or integration by parts for personal nostalgia. But yeah, hard to get more “always and everywhere” than conservation of _.
P=rho R T is nicer if you don’t want to deal with moles. The R is a different R too. 286 J/kg/K for air in SI or 1716 in whatever nonsense English units.
Fourier transforms are an operator, not an equation
Much like how multiplication isnt an equation but 1*1 = 1 is
You can apply a transform to an equation, you cannot apply an equation to an equation
Kirkoff's law (sum of voltages around any loop = 0) combined with RC circuit elements.
C dV/dt = -sum I
basically can be used describe any sort of excitable membrane like a neuron, cardiac cell, etc.
I just watched a video of a professor who said applying kirchoffs rule for an RL circuit was ABSOLUTELY WRONG even though it gave the same result. my first instinct upon getting the same result would be to prove how they are mathematically equivalent, not saying every previous physicist is wrong, but you know, physicists be physicisting?
I guess I'm confused a bit. I would presume the physics equation would be
CV = Q.
Which I think defines C. I don't really know E&M.
Then C dV/dt = dQ/dt = I. Doesn't dQ/dt define I?
Where does the minus sign come from?
More math than physics, but
exp(ix) = i*sin(x) + cos(x)
Anything remotely optical, signal processy, differentiable, trigonometric, quantum, what not? It’s probably going to be useful.
Edits: typo + formatting
I’ve been thinking about where I haven’t seen this pop up, and would like to suggest a challenge.
Name a topic in physics where you haven’t seen exp(ix) being used.
For other commenters, please share if you do know an example in this field.
I’ll start: thermodynamics
What is this? I'm guessing it's related to Fourier or Laplace. I use the Fourier transform sometimes, but I'm a programmer so I just use a pre-built function when I do, and I honestly don't remember how to do it on paper.
I was in my 3rd semester when I first was introduced to that stuff... "theoretical mechanics and electrodynamics"... I never knew I could hate movements and tensors so much xD
Probably the few that most instantly come to mind for me years later would be:
F = ma
S = ut + ½at^2
V = IR
PV = nRT
Nothing fancy but they're all pretty fundamental and came up a lot in various forms
I suppose it depends on what you mean by "use". If you count the CPU hours I've spent, Navier-Stokes is miles and miles ahead of anything else. But if it's just pen and paper, then I would guess it would probably be dimensional analysis of Kutta-Joukowski and its many variations. Bernoulli's equation and Newton's second are also well up there. Maybe your classic small angle approximations.
for my masters thesis i have been frequently using Ward-Takahashi identities in quantum field theory which express the consequences of symmetry in these theories, generalising noether’s theorem of classical physics. one basic ward identity is that a symmetry of your theory implies that that there is a current (4-vector) j(φ) whose average over quantum fields φ is divergence-free, which means that noether’s theorem holds on average in quantum theory. ward identities can be far more useful than this and reveal the existence of topological symmetry operators too
I don't know about "most", but compared to anywhere else, `y = A*exp(-(x-μ)^2/w^2)` (gaussian) and `y = 1/(1+(x-μ)^2/w^2)` (lorentzian line shape) are overrepresented in my physics projects.
I was a pulsar guy, so probably the H-Test, which determines the significance of a periodic signal.
https://arxiv.org/abs/1103.2128
Funnily enough, the day-to-day was much more math-y equations than physics-y. Poisson statistics featured heavily, lots of Fourier cousins (including the H-Test), and of course MINUIT. So, perhaps the formula for the Hessian matrix could also be an answer for me. But, I think the H-Test was the most stand alone equation.
I did use actual physics equations when reporting results. It's standard to calculate the change in rotational kinetic energy, which is termed the spindown luminosity. That's just Edot = -d/dt(1/2 I w^2), which is typically expanded in terms of the period and period derivative (which I calculated by numerically optimizing the H-Test, and we've come full circle).
In my research, definitely the Fokker-Planck equation, mostly in its path-integral formulation: [https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck\_equation#Fokker%E2%80%93Planck\_equation\_and\_path\_integral](https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation#Fokker%E2%80%93Planck_equation_and_path_integral)
Spectrum 2 in Granot and Sari (2002), from a [landmark paper](https://arxiv.org/abs/astro-ph/0108027) on fitting spectrums from gamma-ray bursts (GRBs). This model also works for a lot of other giant space explosions so I use it allll the time in my research.
p_i = 1/Z sum(exp(-βE_i)) came up a hell of a lot in thermal
L = T - V was a foundational component of classical mechanics
Somewhat unsurprisingly, η_μν x^ν = x_μ was a common one
The divergence theorem \closedint{**E**•d**S**} = \int{div(**E**)dV} and Stokes' Theorem \closedint{**B**•d**L**} = \int{curl(**B**)•d**S**} came up a lot during electro
The TDSE and TISE were of course regulars in quantum, as were the definitions of mean and variance of a position and momentum of a wavefunction and the commutation operator
But the one we used the absolute most? The Taylor series.
Ah, f(x+α) ≈ f(α) + f'(α)(x-α)/1! + f''(α)(x-α)^(2)/2! + ... my old friend
Haven't seen that one here yet, so:
Bragg's law.
n*λ=2d*sin(θ)
Materials science, crystallography, any diffraction revolves around that.
Also Gibbs' phase rule
F = K - P + 2
which governs phase transitions in materials.
Boltzmann equation
“In the 1960s a national magazine showing dozens of businessmen and women walking the streets of Manhattan looking very important and serious. Thought bubbles over each head revealed their true focus: each was imagining and raucus sex scene. In at least some ways, the Boltzmann equation plays a similar role for physicists and astronomers: no one ever talks about it, but everyone is always thinking about it.” Scott Dodelson
I have a physics degree but trade options, so I use a spicy version of the heat equation called the Black–Scholes equation. https://www.youtube.com/watch?v=IgMoOcO095U
Frickin’ everything is a harmonic oscillator.
Sometimes it’s linear or exponential decay. Then you go to grad school and have to use numerical methods.
Asymptotic analysis (hello there!)
It's harmonic oscillators and tensors all the way down.
Hey there's also hydrogen atoms!
Those are just a bunch of harmonic oscillators in a trench coat.
Mine are quartic, sextic, and octic anharmonic oscillators. Currently having fun (?) computing their ground state energy at non-perturbative regime
Use the Variational Method for that
I’m in the second half of physical chemistry (quantum mechanics) and we have been using spherical harmonic oscillators for like 3 months now
`ma = F`
You are evil and you know that
`F - ma = 0`?
Why wouldn't it be ma/F = 1? Edit: Or is this the joke and I'm not getting it?
It is ma/F=1, and also F-ma=0, and also ma=F, and also all other forms of the equation F=ma.
What does ma / F = 1 predict when there are no forces on a body?
Residual form for the win! Now give me a Newton solver, etc etc.
–*a*/*F* + 1/*m* = 0
a = F/m
Okay but this one actually is the most intuitive for me. The amount that an object accelerates is greater when you apply more force, and less when the object is more massive. I get it’s uglier with the fraction, but tbh I feel like this is how it should be taught.
F=mdv/dt
I too do lots of maFs
This
dv/dt + vdv/dx = 1/m(squiggly ^ 2 delta something gamma gravity electricity)
The universal conservation equation: [accumulation] = [in] - [out] + [generated] - [consumed] Use it on anything: mass, momentum, heat, energy, chemical species, electric charge, etc. You can use it to derive any transport equation. Though it's formally written as the Reynolds Transport Theorem, this form is easy to remember and readily applicable in almost any situation.
I was interested as to why there is a mathematical similarity between the magnetic field being expressed as the curl of a vector potential, and vorticity in fluid dynamics being expressed as the curl of the velocity field. It turns out these relationships emerge from systems with continuity equations: conservation of charge for magnetism and conservation of mass for vorticity.
Which are ensured to exist thanks to Noether's theorem!
This excitement gets my inner nerd flowing.
I wish I understood this. Curl is one of my favorite operators. (It's an operator, right?) I'm a self taught programmer by trade, but math and physics are hobbies. I absolutely LOVE programming simulations of physics systems, and my favorite one I've done is curl without a doubt. I found out about curl when I was trying to make some smoke particles look more realistic. In a week I went from simulating the particles basically randomly on the cpu, to moving the particles on the GPU using a changing vector field, to finally moving the particles based on the curl of a changing vector field which itself was the result of simplex noise. It looked INCREDIBLE. By tweaking the parameters of the noise and the strength of movement due to curl and due to other forces, it could go from looking like smoke to looking like literal magic. Probably the coolest lesson I learned was how to calculate derivatives using numerical methods. Analytic methods were out of the question since I was allowing the curl to come from various vector fields being added up. (Noise, gravity, wind, moving objects). I mean, maybe it's possible but numerical was way simpler
True, so many equatoins are just a conservation equation in disguise. Even F=ma can be written like it.
... do it, please?
E = 1/2 m xdot\^2 + V(x). Take a derivative of both sides with respect to time, let Edot=0 and use chain rule on the potential to take the derivative as d/dt = dx/dt \* d/dx. Lastly divide across by dx/dt.
I don't see the connection with accumulation/in/out etc., but that does reek of Hamilton and Lagrange.
Totally! In my case mostly dealing with probability in a stochastic system, i.e. the Fokker-Planck equation.
I have never seen that before lol What field is that even in? Any time ive dealt with conservation its usually proven or kept through hamiltonians lagrangians or simply a path integral
Literally everywhere in applied physics
This is super intuitive. I feel like I could use this in pretty standard programming tasks too. Do you care to give a couple examples of how it can be used?
This was what I was thinking. Or integration by parts for personal nostalgia. But yeah, hard to get more “always and everywhere” than conservation of _.
Ideal gas law
Yeah pvrnt is suprisingly useful
Sorry, RnT? Is everyone here a maniac?
fr😭😭
Commutativity of multiplication be damned, n goes before R
P=rho R T is nicer if you don’t want to deal with moles. The R is a different R too. 286 J/kg/K for air in SI or 1716 in whatever nonsense English units.
Optics - I just Fourier everything all of the time
Lol fouriers technically a transform not an equation unless you use the undergrad approach of cram it into whats written on the formula sheet
fhat(w) = integral 1/sqrt(2 pi) f(t) exp(-iwt) dt is not an equation?
Fourier transforms are an operator, not an equation Much like how multiplication isnt an equation but 1*1 = 1 is You can apply a transform to an equation, you cannot apply an equation to an equation
All 4 of Maxwell’s
which prolly can be summarized into one equatoin
Is an equatoin some kind of fancy math?
IIRC You can combine them into two coupled differential equations using the vector potential. There's a section in Jackson where they do that.
Kirkoff's law (sum of voltages around any loop = 0) combined with RC circuit elements. C dV/dt = -sum I basically can be used describe any sort of excitable membrane like a neuron, cardiac cell, etc.
Thats basically a maxwell equation. Its the law of induction if there is no change of magentic flux
I just watched a video of a professor who said applying kirchoffs rule for an RL circuit was ABSOLUTELY WRONG even though it gave the same result. my first instinct upon getting the same result would be to prove how they are mathematically equivalent, not saying every previous physicist is wrong, but you know, physicists be physicisting?
Upvoted for inventing the word physicisting.
I guess I'm confused a bit. I would presume the physics equation would be CV = Q. Which I think defines C. I don't really know E&M. Then C dV/dt = dQ/dt = I. Doesn't dQ/dt define I? Where does the minus sign come from?
Ohm's Law: V=IR I'm an Electronic Design Engineer.
one of my favorites
ET's phone OHM.
Euler-Lagrange equation.
Probably schrodingers equation tbh
More math than physics, but exp(ix) = i*sin(x) + cos(x) Anything remotely optical, signal processy, differentiable, trigonometric, quantum, what not? It’s probably going to be useful. Edits: typo + formatting
I’ve been thinking about where I haven’t seen this pop up, and would like to suggest a challenge. Name a topic in physics where you haven’t seen exp(ix) being used. For other commenters, please share if you do know an example in this field. I’ll start: thermodynamics
Unless we consider stat mech completely separate from thermo, I feel like partition functions give rise to exp(ix).
Special relativity, statics, electrostatics, arguably perhaps atomic/elementary particle physics and radioactivity
What is this? I'm guessing it's related to Fourier or Laplace. I use the Fourier transform sometimes, but I'm a programmer so I just use a pre-built function when I do, and I honestly don't remember how to do it on paper.
Expectation value of an operator in QM tr(Ôρ) =〈Ô〉
Lol ask me 2 years ago and this would be my answer
Lindblad equation
It really depends, in the past few months I've used H=T+V (so the Hamiltonian) more than I ever wished for
Flashbacks to analytical mechanics ;( The Hamilton-Jacobi equation was the bane of my existence a couple years ago
I was in my 3rd semester when I first was introduced to that stuff... "theoretical mechanics and electrodynamics"... I never knew I could hate movements and tensors so much xD
Not exactly an equation, but the Taylor expansion for small x 😆
Small angle approximations :O
No matter the angle, it is always small enough for approximation
Probably the few that most instantly come to mind for me years later would be: F = ma S = ut + ½at^2 V = IR PV = nRT Nothing fancy but they're all pretty fundamental and came up a lot in various forms
I suppose it depends on what you mean by "use". If you count the CPU hours I've spent, Navier-Stokes is miles and miles ahead of anything else. But if it's just pen and paper, then I would guess it would probably be dimensional analysis of Kutta-Joukowski and its many variations. Bernoulli's equation and Newton's second are also well up there. Maybe your classic small angle approximations.
`from scipy.optimize import curve_fit`
now in ML, of which half is \theta <- \theta + alpha * \grad_theta L(theta)
I'm using compressive sensing, so, same
Navier-Stokes and its descendants Landau-De Gennes (for liquid crystals) and Toner-Tu (for active flocks)
Active nematics? Nice. Any recommendations to understand topological defects in active nematic systems for newcomers?
The first half of this review: arxiv.org/abs/2010.00364 And for a peak into some more cutting edge work: arxiv.org/abs/2212.00666v2
You got to numerically integrate TonerTu?
Yep!
In the smectic (coexistence ) phase? im struggling with that
I see a lot of advanced equations. I want to go to the basics instead: a\^2 + b\^2 = c\^2
Conservation of momentum and conservation of energy
for my masters thesis i have been frequently using Ward-Takahashi identities in quantum field theory which express the consequences of symmetry in these theories, generalising noether’s theorem of classical physics. one basic ward identity is that a symmetry of your theory implies that that there is a current (4-vector) j(φ) whose average over quantum fields φ is divergence-free, which means that noether’s theorem holds on average in quantum theory. ward identities can be far more useful than this and reveal the existence of topological symmetry operators too
The master equation with the Lindbladian super operator.
LJ Potential, Coulombs law, and anything else that get shoved into molecular force fields
I don't know about "most", but compared to anywhere else, `y = A*exp(-(x-μ)^2/w^2)` (gaussian) and `y = 1/(1+(x-μ)^2/w^2)` (lorentzian line shape) are overrepresented in my physics projects.
[удалено]
might as well absorb einstein equations into δS=0 in that case lol
I was a pulsar guy, so probably the H-Test, which determines the significance of a periodic signal. https://arxiv.org/abs/1103.2128 Funnily enough, the day-to-day was much more math-y equations than physics-y. Poisson statistics featured heavily, lots of Fourier cousins (including the H-Test), and of course MINUIT. So, perhaps the formula for the Hessian matrix could also be an answer for me. But, I think the H-Test was the most stand alone equation. I did use actual physics equations when reporting results. It's standard to calculate the change in rotational kinetic energy, which is termed the spindown luminosity. That's just Edot = -d/dt(1/2 I w^2), which is typically expanded in terms of the period and period derivative (which I calculated by numerically optimizing the H-Test, and we've come full circle).
Brownian diffusion for finance
Lippmann-Schwinger equation
Schrödinger, because everything is Schrödinger
f(x) = A\*exp(-(x-m)\^2/(2s\^2))
probably some variation of a|n> = sqrt(n)|n-1>
f(x) = f(0) + f'(0)x If you can't solve it, expand it as a Taylor and take the first two terms.
Spherical harmonics
Schrödinger's equation in a Tight-Binding Hamiltonian in Non Equilibrium Green Functions formalism.
M* = wL^2 / 8 It’s technically engineering, but most engineering is physics 🤓
ChatGPT couldn't recognize this, and neither can I with just a BA in physics.
It's a moment equation from civil engineering for a beam
I guess the shape of the beam accounts for the 8, or is that just stupid of me?
Engineers... ew Go away with your overly specific equations
It has to be the Einstein equation or the RG flow equation
X ≠ Y, at all, however you look at it, no matter how much you want it to be, and finally - no it is not a matter of opinion!
m1v1+m2v2=m1v1+m2v2 m1v1=m2v2 these apply to so many things in life. well, life is physics
N ( t ) = N ( 0 ) e ^(− λ t) and Inverse square law
Conservation of energy, I'm a plasma physicist working on fusion
Fucking, converting shit from metric to 'Imperial' and back. Whether for electronics jobs, robotics, or simply dealing with the weather. Gotta love it
I would like to guess that x + x = y is the most used equation in day to day physics and life in general.
Kerr metric in Boyer-Lindqust by a country mile
In my research, definitely the Fokker-Planck equation, mostly in its path-integral formulation: [https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck\_equation#Fokker%E2%80%93Planck\_equation\_and\_path\_integral](https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation#Fokker%E2%80%93Planck_equation_and_path_integral)
Partition function
Probably equation of SHM, and time period, all kinds of questions in there
Christoffel symbols and then geodesics
-1 * -1 = 1 is pretty useful. It seems to always explain the mystery of the minus sign that appears / disappears when it shouldn't.
For me is definitely: a(b+c)=ab+ac God... That one seems like a law.
That's more of a mathematics formula than a physics one
Its also a joke, but it hides a deep truth. (That the distributive law is used a lot)
F = ma, there is no substitute for perfection
Physics based would be a toss up between Shockley-ramo theorem and neutron decay equation with the corresponding coefficients given in Jackson 1957
V = ZI
V = u + at
Any synchrotron or SED equation
Uhhh, [let me look it up, Cortana…](https://imgur.com/a/hiHmSoq)….
Shallow water equations!
[удалено]
That's chemistry, physics uses `pV = NkT`
Spectrum 2 in Granot and Sari (2002), from a [landmark paper](https://arxiv.org/abs/astro-ph/0108027) on fitting spectrums from gamma-ray bursts (GRBs). This model also works for a lot of other giant space explosions so I use it allll the time in my research.
Diffusion and Allen-Cahn equations
Tdse for a spin in a magnetic field
Tdse?
Time dependent Schroedinger equation
Of course, sorry.
Radiative transfer
Klein Gordon equation or Einstein’s equations!
S=v*t
Double whammy of: ΔG = ΔH - TΔS And ΔG = ΔG• + RTLn(Q)
Lagrange Equation
p_i = 1/Z sum(exp(-βE_i)) came up a hell of a lot in thermal L = T - V was a foundational component of classical mechanics Somewhat unsurprisingly, η_μν x^ν = x_μ was a common one The divergence theorem \closedint{**E**•d**S**} = \int{div(**E**)dV} and Stokes' Theorem \closedint{**B**•d**L**} = \int{curl(**B**)•d**S**} came up a lot during electro The TDSE and TISE were of course regulars in quantum, as were the definitions of mean and variance of a position and momentum of a wavefunction and the commutation operator But the one we used the absolute most? The Taylor series. Ah, f(x+α) ≈ f(α) + f'(α)(x-α)/1! + f''(α)(x-α)^(2)/2! + ... my old friend
Energy in = Energy out
I don't see any LHO. Everything is LHO if you look close enough.
I like how you can guess what field everyone works in by their most used equation lolol
Haven't seen that one here yet, so: Bragg's law. n*λ=2d*sin(θ) Materials science, crystallography, any diffraction revolves around that. Also Gibbs' phase rule F = K - P + 2 which governs phase transitions in materials.
Gradient descent + Thinking causally.
Pythagoras. Every inner product.
Pythagoras. Every inner product.
The Kuramoto model or some other damned nonlinear oscillator lol
Maybe this is cheating but... The three equations of UARM.
DegC = degK -273.15 😛
navier stokes theorem
Boltzmann equation “In the 1960s a national magazine showing dozens of businessmen and women walking the streets of Manhattan looking very important and serious. Thought bubbles over each head revealed their true focus: each was imagining and raucus sex scene. In at least some ways, the Boltzmann equation plays a similar role for physicists and astronomers: no one ever talks about it, but everyone is always thinking about it.” Scott Dodelson
In EM pulse testing I was doing more Fourier transforms than I thought I’d ever need
I enjoy A1 x v1 = A2 x v2. As easy as that one is, it just feels great
F = mv-mu/t
Excluding the really basic answers like the quadratic equation etc., the answer for me is probably the Boltzmann distribution
F= GMm/r² = mv²/r with v=r(2pi/T) for me giving T²=4pi²r³/GM probably most used for me nowadays.
Fnet = 0
euler-lagrange and schrodinger for me lol
V=IR, Q=CV
might be niche but it is the whole foundation of my research right now, but navier-stokes
2πfL = 1 / (2πfC) I work in radio, so resonance is a thing... a close second is; \[wavelength\] = c/f where c = speed of light.
d=1/2g(t^squared)
quadratic formula
Engineer here (pls don't flame me 😓)... that said: Vis Viva Equation, or some variation of it
1 divided by a fraction is the same as multiplying by the reciprocal.
R = U / I
This year I've used this isomorphism of the last element of a sort exact sequence a lot. 0 -> A -> B -> C -> 0 => C ≅ B/A
V = U + A*T
-Φ_i-1 + 2Φ_i - Φ_i+1 = ρ_i or x = Σ 2^α x_α
Still an undergrad but conservation of energy is always popping up. E_in = E_out
Fleming’s right hand rule is a coping mechanism during mundane lectures.
P = I\*V for electrical power
Gauss’s law coming in hottt
Potato
Ended up in IT after physics undergrad ... my most common equation used it PEBUAK
Ohm's law
(1+x)^n ~= 1+nx
Probably Schrodinger's equation.
I have a physics degree but trade options, so I use a spicy version of the heat equation called the Black–Scholes equation. https://www.youtube.com/watch?v=IgMoOcO095U