### Expressions Central to Physics

If an expression is named, then it should be included here.

Expression name domain description
$$F = m a$$ Second Law of Motion Classical Mechanics
• $$F$$ is PDG symbol4202, Force
• $$m$$ is PDG symbol5156, mass
• $$a$$ is PDG symbol9140, acceleration
$$E = m c^2$$ Energy-mass equivalence special theory of relativity wikidata:Q35875
• $$E$$ is PDG symbol4931, Energy
• $$m$$ is PDG symbol5156, mass
• $$c$$ is PDG symbol4567, speed of light
$$\Delta x \Delta \rho \geq \frac{\hbar}{2}$$ Uncertainty principle Quantum Mechanics
• $$\Delta x$$ is PDG symbol,
• $$\Delta \rho$$ is PDG symbol,
• $$\hbar$$ is PDG symbol1054, Reduced Planck's constant
$$i \hbar \frac{\partial}{\partial t} | \psi(t) \rangle = \hat{H} | \psi(t) \rangle$$ Schrödinger equation Quantum Mechanics
• $$\hbar$$ is PDG symbol1054, Reduced Planck's constant
• $$t$$ is PDG symbol1467, time
• $$\psi(t)$$ is PDG symbol9329, wave function
• $$\hat{H}$$ is PDG symbol6799, Hamiltonian
$$\vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$$ Maxwell-Faraday equation Electrodynamics
• $$\vec{E}$$ is PDG symbol4326, electric field
• $$\vec{B}$$ is PDG symbol2069, magnetic field
• $$t$$ is PDG symbol1467, time
$$(i \partial\!\!\!\big / -m) \psi = 0$$ Dirac equation Partical Physics
Dirac developed an equation that explained spin number as a consequence of the union of quantum mechanics and special relativity. The equation also predicted the existence of anti-matter, previously unsuspected and unobserved, and which was experimentally discovered in 1932.
•  is PDG symbol,
$$\Delta S \geq 0$$ where $$S = k_{Boltzmann} \ln W$$ Law of entropy Thermodynamics
when energy changes from one form to another form, or when matter moves freely, the disorder in a closed system increases.
•  is PDG symbol,
$$G_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}$$ Einstein field equations
The expression on the left hand side of the equation represents the curvature of spacetime. The expression on the right is the energy density of spacetime. The equation dictates how energy determines he curvature of space and time.
•  is PDG symbol,
$$\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} c^2$$ wave equation
•  is PDG symbol,
$$E = h f$$ Planck's equation quantum mechanics
• $$E$$ is PDG symbol4931, Energy
• $$h$$ is PDG symbol4413, Planck's constant
• $$f$$ is PDG symbol4201, frequency
$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} - \nu \,\nabla^2 \mathbf{u} = - \nabla w + \mathbf{g}$$ Navier-Stokes fluid dynamics describes the motion of viscous fluid substances. mathematically expresses conservation of momentum and conservation of mass for Newtonian fluids.
•  is PDG symbol,
• $$t$$ is PDG symbol1467, time
$$\vec{\nabla} \times \vec{B} = \mu_0 \vec{J}$$ and $$\vec{\nabla} \times \vec{H} = \vec{J}_f$$ Ampère's circuital law
• $$\vec{B}$$ is PDG symbol2069, magnetic field
•  is PDG symbol,
$$P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2$$ Bernoulli's equation
• $$\rho$$ is PDG symbol,
 Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy of equations
•  is PDG symbol,
 Bessel's differential equation
•  is PDG symbol,
$$\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}$$ Boltzmann equation
•  is PDG symbol,
$$\Delta E = \xi \frac{1}{2} \rho \left( v_1 - v_2 \right)^2$$ Borda–Carnot equation
• $$\Delta E$$ is PDG symbol,
• $$\xi$$ is PDG symbol,
• $$\rho$$ is PDG symbol,
• $$v_1$$ is PDG symbol,
• $$v_2$$ is PDG symbol,
$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu\frac{\partial^2 u}{\partial x^2}$$ Burgers' equation
• $$u$$ is PDG symbol,
• $$t$$ is PDG symbol1467, time
• $$x$$ is PDG symbol,
• $$\nu$$ is PDG symbol,
$$\frac{\Delta p}{L} = f_{\mathrm{D}} \cdot \frac{\rho }{2} \cdot \frac{\langle v \rangle^2}{D}$$ Darcy–Weisbach equation
• $$\Delta p$$ is PDG symbol,
• $$L$$ is PDG symbol,
• $$f_{\mathrm{D}}$$ is PDG symbol,
• $$\rho$$ is PDG symbol,
• $$v$$ is PDG symbol,
• $$D$$ is PDG symbol,
$$f = \left( \frac{c \pm v_{\rm{receiver}}}{ c \pm v_{\rm{source}} } \right) f_0$$ Doppler equations
• $$f$$ is PDG symbol,
• $$c$$ is PDG symbol,
• $$v_{\rm{receiver}}$$ is PDG symbol,
• $$v_{\rm{source}}$$ is PDG symbol,
• $$f_0$$ is PDG symbol,
$$N = R_{*} f_{p} n_e f_1 f_i f_c L$$ Drake equation (aka Green Bank equation)
• $$N$$ is PDG symbol,
• $$R_{*}$$ is PDG symbol,
• $$f_{p}$$ is PDG symbol,
• $$n_e$$ is PDG symbol,
• $$f_1$$ is PDG symbol,
• $$f_i$$ is PDG symbol,
• $$f_c$$ is PDG symbol,
• $$L$$ is PDG symbol,
 Euler equations (fluid dynamics)
•  is PDG symbol,
 Euler's equations (rigid body dynamics)
•  is PDG symbol,
$$T^{\mu\nu} \, = (e+p)u^\mu u^\nu+p g^{\mu\nu}$$ Relativistic Euler equations
• $$T^{\mu\nu}$$ is PDG symbol,
• $$e$$ is PDG symbol,
• $$p$$ is PDG symbol,
• $$u^\mu$$ is PDG symbol,
• $$p$$ is PDG symbol,
• $$g^{\mu\nu}$$ is PDG symbol,
$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\mathbf{q}}} \right) = \frac{\partial L}{\partial \mathbf{q}}$$ Euler–Lagrange equation
•  is PDG symbol,
$$\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}$$ Faraday's law of induction
• $$\mathcal{E}$$ is the electromotive force
• $$\Phi_B$$ is the magnetic flux.
$$\frac{\partial}{\partial t} p(x, t) = -\frac{\partial}{\partial x}\left[\mu(x, t) p(x, t)\right] + \frac{\partial^2}{\partial x^2}\left[D(x, t) p(x, t)\right]$$ Fokker–Planck equation
describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion
 Fresnel equations
•  is PDG symbol,
 Friedmann equations
•  is PDG symbol,
$$\Phi_E = \frac{Q}{\varepsilon_0}$$ Gauss's law for electricity Electrodynamics
The net electric flux through any hypothetical closed surface is equal to $$\frac{1}{\varepsilon _{0}}$$ times the net electric charge within that closed surface.
$$\vec{\nabla} \cdot \vec{g} = -4 \pi G \rho$$ Gauss's law for gravity equivalent to Newton's law of universal gravitation
• $$G$$ is the universal gravitational constant,
• $$\rho$$ is the mass density at each point.
$$\vec{\nabla} \cdot \vec{B} = 0$$ Gauss's law for magnetism
•  is PDG symbol,
$$\left( \frac{\partial \left( \frac{G} {T} \right) } {\partial T} \right)_p = - \frac {H} {T^2}$$ Gibbs–Helmholtz equation
•  is PDG symbol,
$$\left(-\frac{\hbar^2}{2m}{\partial^2\over\partial\mathbf{r}^2} + V(\mathbf{r}) + {4\pi\hbar^2a_s\over m}\vert\psi(\mathbf{r})\vert^2\right)\psi(\mathbf{r})=\mu\psi(\mathbf{r})$$ Gross–Pitaevskii equation
•  is PDG symbol,
 Hamilton–Jacobi–Bellman equation
•  is PDG symbol,
$$\nabla^2 f = -k^2 f$$ Helmholtz equation
•  is PDG symbol,
$$J(\phi) = A \cos^2 \phi + B \cos\,\phi + C$$ Karplus equation
•  is PDG symbol,
$$M = E - e \sin E$$ Kepler's equation
• $$M$$ is the mean anomoly
• $$E$$ is the eccentric anomoly
• $$e$$ is the eccentricity
 Kepler's laws of planetary motion
•  is PDG symbol,
$$U(P) = \frac{1}{4\pi} \int_{S} \left[ U \frac{\partial}{\partial n} \left( \frac{e^{iks}}{s} \right) - \frac{e^{iks}}{s} \frac{\partial U}{\partial n} \right]dS$$ Kirchhoff's diffraction formula
•  is PDG symbol,
$$\left( \nabla^2 - \frac{m^2 c^2}{\hbar^2} \right) \psi(\vec{r}) = 0$$ Klein–Gordon equation
a relativistic wave equation, related to the Schrödinger equation
$$\partial_t \phi + \partial^3_x \phi - 6\, \phi\, \partial_x \phi =0\$$ Korteweg–de Vries equation
•  is PDG symbol,
$$\frac{d \vec{M}}{d t}=-\gamma \left(\vec{M} \times \vec{H}_{\mathrm{eff}} - \eta \vec{M}\times\frac{d \vec{M}}{d t}\right)$$ Landau–Lifshitz–Gilbert equation
• $$\vec{M}$$ is PDG symbol,
• $$t$$ is PDG symbol1467, time
• $$\gamma$$ is PDG symbol,
• $$\vec{H}_{\mathrm{eff}}$$ is PDG symbol,
• $$\eta$$ is PDG symbol,
$$\frac{1}{\xi^2} \frac{d}{d\xi} \left({\xi^2 \frac{d\theta}{d\xi}}\right) + \theta^n = 0$$ Lane–Emden equation
• $$\xi$$ is PDG symbol,
• $$\theta$$ is PDG symbol,
$$m \frac{d \vec{v}}{dt} = -\lambda \vec{v} + \eta(t)$$ Langevin equation
• $$m$$ is PDG symbol,
• $$\vec{v}$$ is PDG symbol,
• $$t$$ is PDG symbol1467, time
• $$\lambda$$ is PDG symbol,
• $$\eta$$ is PDG symbol,
$$\frac{\mathbf{d}\varepsilon_1}{\sigma'_1}=\frac{\mathbf{d}\varepsilon_2} {\sigma'_2}=\frac{\mathbf{d}\varepsilon_3}{\sigma'_3}=\mathbf{d}\lambda$$ Levy–Mises equations
•  is PDG symbol,
$$\dot\rho=-{i\over\hbar}[H,\rho]+\sum_{n,m = 1}^{N^2-1} h_{nm}\left(A_n\rho A_m^\dagger-\frac{1}{2}\left\{A_m^\dagger A_n, \rho\right\}\right)$$ Lindblad equation
•  is PDG symbol,
$$\vec{F} = q \vec{E} + q\vec{v}\times \vec{B}$$ Lorentz equation
The electromagnetic force on a charge $$q$$ is a combination of a force in the direction of the electric field \vec{E} proportional to the magnitude of the field and the quantity of charge, and a force at right angles to the magnetic field \vec{B} and the velocity \vec{v} of the charge, proportional to the magnitude of the field, the charge, and the velocity.
• $$\vec{F}$$ is PDG symbol,
• $$q$$ is PDG symbol,
• $$\vec{E}$$ is PDG symbol,
• $$\vec{v}$$ is PDG symbol,
• $$\vec{B}$$ is PDG symbol,
 Maxwell's equations
•  is PDG symbol,
 Maxwell's relations
•  is PDG symbol,
 Newton's laws of motion
•  is PDG symbol,
 Reynolds-averaged Navier–Stokes equations
•  is PDG symbol,
 Prandtl–Reuss equations
•  is PDG symbol,
$$h_f = \frac{L}{D} (aV + bV^2)$$ Prony equation
• $$h_f$$ is PDG symbol,
• $$L$$ is PDG symbol,
• $$D$$ is PDG symbol,
• $$a$$ is PDG symbol,
• $$V$$ is PDG symbol,
• $$b$$ is PDG symbol,
 Rankine–Hugoniot equation
•  is PDG symbol,
$$\hat{F} \hat{C} = \hat{S} \hat{C} \hat{\epsilon}$$ Roothaan equations
• $$\hat{F}$$ is PDG symbol,
• $$\hat{C}$$ is PDG symbol,
• $$\hat{S}$$ is PDG symbol,
• $$\hat{\epsilon}$$ is PDG symbol,
 Saha ionization equation
•  is PDG symbol,
$$\frac{S}{k_{\rm B} N} = \ln \left[ \frac VN \left(\frac{4\pi m}{3h^2}\frac UN\right)^{3/2}\right]+ {\frac 52}$$ Sackur–Tetrode equation
• $$S$$ is PDG symbol,
• $$k_{\rm B}$$ is PDG symbol,
• $$N$$ is PDG symbol,
• $$V$$ is PDG symbol,
• $$m$$ is PDG symbol,
• $$h$$ is PDG symbol,
• $$U$$ is PDG symbol,
$$\left[ \Delta - \lambda^2 \right] u(\vec{r}) = - f(\vec{r})$$ screened Poisson equation
•  is PDG symbol,
 Schwinger–Dyson equation
•  is PDG symbol,
 Sellmeier equation
•  is PDG symbol,
 Stokes–Einstein relation
•  is PDG symbol,
$$\delta v = v_e \ln \frac{m_0}{m_f} = I_{\rm{sp}} g_0 \ln \frac{m_0}{m_f}$$ Tsiolkovsky rocket equation
• $$\delta v$$
$$PV = nRT$$ Van der Waals equation
• $$P$$
• $$V$$
• $$n$$
• $$R$$
• $$T$$
$$\frac{\partial f_{\alpha}}{\partial t} + \mathbf {v}_{\alpha} \cdot \frac{\partial f_{\alpha}}{\partial \mathbf {x}}+ \frac{q_{\alpha}\mathbf {E}}{m_{\alpha}} \cdot \frac{\partial f_{\alpha}}{\partial \mathbf {v}} = 0$$ Vlasov equation
•  is PDG symbol,
$$\vec{v} = \frac{d \vec{x}}{d t} = g(t)$$ Wiener equation
•  is PDG symbol,
$$\frac{\partial\psi}{\partial t} +\nabla\cdot\left( \psi{\mathbf u}\right) =0$$ Advection equation
•  is PDG symbol,
 Barotropic vorticity equation
•  is PDG symbol,
$$\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = \sigma$$ Continuity equation
•  is PDG symbol,
$$\frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla \cdot \big[ D(\phi,\mathbf{r}) \ \nabla\phi(\mathbf{r},t) \big]$$ Diffusion equation
•  is PDG symbol,
$$F_{\rm d} = \frac{1}{2} \rho\, u^2\, c_{\rm d}\, A$$ Drag equation
• $$F_{\rm d}$$ is the drag force
• $$\rho$$ is the mass density of the fluid
• $$u$$ is the flow velocity relative to the object
• $$A$$ is the reference area
• $$c_{\rm d}$$ is the drag coefficient
 Equation of motion
•  is PDG symbol,
 Equation of state
•  is PDG symbol,
 Equation of time
•  is PDG symbol,
 Heat equation
•  is PDG symbol,
$$p V = n R T$$ Ideal gas equation
• $$p$$
• $$V$$
• $$n$$
• $$R$$
• $$T$$
 Ideal MHD equations
•  is PDG symbol,
 Mass–energy equivalence equation
•  is PDG symbol,
 Primitive equations
•  is PDG symbol,
 Relativistic wave equations
•  is PDG symbol,
$$v^2 = GM \left({ 2 \over r} - {1 \over a}\right)$$ Vis-viva equation astrodynamics
• $$v$$ is the relative speed of the two bodies
• $$r$$ is the distance between the two bodies
• $$a$$ is the length of the semi-major axis
• $$G$$ is the gravitational constant
• $$M$$ is the mass of the central body
$$\frac{D\boldsymbol\omega}{Dt} = \frac{\partial \boldsymbol \omega}{\partial t} + (\mathbf u \cdot \nabla) \boldsymbol \omega$$ Vorticity equation
•  is PDG symbol,
$$\sum_i I_i = 0$$ Kirchoff’s Current Law Electrical circuits At every node of an electrical circuit, current $$I$$ sums to zero.
$$dU = dQ + dW$$ First Law of Thermodynamics Thermodynamics
•  is PDG symbol,
$$\lambda = \sqrt{1 - (v^2/c^2)}$$ Lorentz Transformations
•  is PDG symbol,
 Snell's law optics
•  is PDG symbol,
$$\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$$ Gauss Lens Formula, aka thin lens equation optics u = object distance; v = image distance; f = Focal length of the lens
$$\frac{1}{f} = (n-1)\left( \frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)d}{nR_1R_2} \right)$$ Lensmaker's equation optics The focal length of a lens in air, where
• $$f$$ is the focal length of the lens,
• $$n$$ is the refractive index of the lens material,
• $$R_{1}$$ is the radius of curvature (with sign, see below) of the lens surface closer to the light source,
• $$R_{2}$$ is the radius of curvature of the lens surface farther from the light source, and
• $$d$$ is the thickness of the lens (the distance along the lens axis between the two surface vertices).
$$\lambda = h/p$$ De Broglie Wavelength quantum mechanics
• $$\lambda$$ is the De Broglie Wavelength
• $$h$$ is Planck’s Constant
• $$p$$ is momentum of the particle.
$$2 a \sin \theta = n \lambda$$ Bragg’s Law of Diffraction optics
• $$a$$ is Distance between atomic planes
• $$n$$ is Order of Diffraction
• $$\theta$$ is Angle of Diffraction
• $$\lambda$$ is Wavelength of incident radiation
$$\mathcal{L}_{SM} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + ...$$ Lagrangian for the Standard model subatomic physics
•  is PDG symbol,
$$F = G \frac{m_1m_2}{r^2}$$ Newton's law of universal gravitation classical mechanics
• $$F$$ is the gravitational force acting between two objects,
• $$m1$$ and $$m2$$ are the masses of the objects,
• $$r$$ is the distance between the centers of their masses, and
• $$G$$ is the gravitational constant.
$$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{-2\pi i x \xi}\,dx$$ Fourier Transform
•  is PDG symbol,
$$\left[M\frac{\partial }{\partial M}+\beta(g)\frac{\partial }{\partial g}+n\gamma\right] G^{(n)}(x_1,x_2,\ldots,x_n;M,g)=0$$ Callan-Symanzik equation Quantum field theory
•  is PDG symbol,
$$F_{\rm{spring}} = k x$$ Hooke's Law Classical Mechanics
•  is PDG symbol,
$$\lambda_{\rm{peak}} = \frac{b}{T}$$ Wien's displacement law
• $$\lambda_{peak}$$ Wavelength
• $$T$$ is absolute temperature
• $$b$$ is Wien's displacement constant
$$j^* = \sigma T^4$$ Stefan–Boltzmann law
• $$j$$ is PDG symbol,
• $$\sigma$$ is PDG symbol,
• $$T$$ is PDG symbol,
$$F = K \frac{q_1 q_2}{r^2}$$ Coulomb's law Electrodynamics quantifies the amount of force between two stationary, electrically charged particles.
• $$F$$ is PDG symbol,
• $$K$$ is PDG symbol,
• $$q$$ is PDG symbol,
• $$r$$ is PDG symbol,
$$P_1V_1 = P_2V_2$$ Boyle's law fluid mechanics
•  is PDG symbol,
$$V_1T_2 = V_2T_1$$ Charles's law
• $$V$$
• $$T$$

•  is PDG symbol,

•  is PDG symbol,

•  is PDG symbol,

•  is PDG symbol,

•  is PDG symbol,

•  is PDG symbol,

•  is PDG symbol,

•  is PDG symbol,