Mathematical descriptions of physical laws

For worded descriptions and criteria of physical laws, see Physical law.

For the nomenclature used see Physical quantity and Defining equation (physics).

Physical laws are often summarized by a single equation, or at least a small set of equations. This article tabulates many of the important bands of physics where such laws occur.

General principles

Conservation and continuity

All conserved quantities can be written as a conservation law (also as continuity equation). The formalism of how conservation laws can be used is given below. See also Noether's theorem.

Let

so that

S = Closed surface of region to calculate flux, arbitrary but fixed for calculation, R = Region of space bounded by S containing total amount of quantity Q, which occupies the closed system, A = surface or cross-sectional area through which flows pass through, $\scriptstyle{\mathbf{\hat{n}}} \,\!$ = unit normal to A, A = vector area, V = Volume of closed system, r = position vector, t = time, q = any conserved quantity as a function of space and time within the closed system (since it may flow throughout, but the total amount in the system is still constant and independent of space and time), Q = total amount of q for system (constant), I = current of q, J = current density of q, $\scriptstyle{\mathbf{\hat{t}}} \,\!$ = unit vector in direction of J, ρ = volume density of q T = time taken for all of z to pass a boundary (including a point), Σ = Total generation (positive) or removal (negative) of q per unit time by the sources and sinks in the volume V, σ = Total generation (positive) or removal (negative) of q per unit volume per unit time by the sources and sinks in the volume V N = number of discrete constituents of the system, e.g. particles possessing momenta

$q = q \left ( \mathbf{r}, t \right ) \,\!$ $\Sigma = \Sigma \left( t \right ) \,\!$, $\mathbf{A}= A \mathbf{\hat{n}} \,\!$ $\rho = \frac{{\rm d} q}{{\rm d} V} \,\!$, $\sigma = \frac{{\rm d} \Sigma}{{\rm d} V} \,\!$, $I = \frac{{\rm d} q}{{\rm d} t} \,\!$ As a scalar $J\cos\theta = \frac{{\rm d}^2 q}{{\rm d} A {\rm d} t} \,\!$ or vector $\mathbf{J}\cdot\mathbf{\hat{t}} = \frac{{\rm d}^2 q}{{\rm d} A {\rm d} t} \,\!$ (θ = angle between J and normal to A)

Then we have the following equations.

Summary of conservative formalisms

The following appear trivial and self-evident, but often calculations for conservative situations are applied in one of these forms.

Equation type	Discrete equation	Continuum equation
Constancy equation	All changes $\Delta Q = 0 \,\!$	Differential changes $\mathrm{d} Q = 0 \,\!$
System equation/s	$\sum_{i=1}^{N_1} q_i = \sum_{j=1}^{N_2} q_j = Q\,\!$	$\oint_V \rho \mathrm{d} V = Q\,\!$
Time derivatives, Current	$\sum_{i=1}^{N_1} \left (\frac{{\rm d} q_i}{{\rm d} t} \right ) = \sum_{j=1}^{N_2} \left ( \frac{{\rm d} q_j}{{\rm d} t} \right ) \,\!$ $\sum_{i=1}^{N_1} I_i = \sum_{j=1}^{N_2} I_j \,\!$	$\int_0^T I \mathrm{d} t = Q\,\!$
Time–area derivatives, Current density, flux	$\sum_{i=1}^{N_1} \left (\frac{{\rm d}^2 q_i}{{\rm d} A {\rm d} t} \right ) = \sum_{j=1}^{N_2} \left ( \frac{{\rm d}^2 q_j}{{\rm d} A {\rm d} t} \right ) \,\!$ $\sum_{i=1}^{N_1} J_i = \sum_{j=1}^{N_2} J_j \,\!$ $\sum_{i=1}^{N_1} \mathbf{J}_\mathrm{i} = \sum_{j=1}^{N_2} \mathbf{J}_\mathrm{j} \,\!$	$\oint_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} = 0\,\!$ by the divergence theorem; $\oint_V \nabla \cdot \mathbf{J} \mathrm{d} V = 0\,\!$ $\nabla \cdot \mathbf{J} = 0\,\!$

Summary of classical continuity equations

The general continuity equation can be written in differential or integral form:

$\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = \sigma\,$ $\frac{{\rm d}q}{{\rm d}t} + \iint\limits_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \mathbf{J} \cdot {\rm d}\mathbf{A} = \Sigma$

In the table below, the fluxes, flows and continuity equations have been collected for comparison. Also alternative forms using currents have been included; often these forms are used in introductory transport mechanics since they are simply statements which relate current and flux, simply by conservation of the quantity as it flows. The nomenclature is very similar throughout the equations, to make clear the nature of transport much of the nomenclature been re-stated.

Phyics, conserved quantity	Conserved quantity q	Volume density ρ (of q)	Flux J (of q)	Vector DE	Conservative equation
Hydrodynamics, fluid	m = mass (kg)	ρ = volume mass density (kg m-3)	ρ u, where u = velocity field of fluid (m s-1)	$\nabla \cdot (\rho \mathbf{u}) + \frac{\partial \rho}{\partial t} = 0 \,\!$	$j_\mathrm{m} = \rho_1 \mathbf{A}_1 \cdot \mathbf{u}_1 = \rho_2 \mathbf{A}_2 \cdot \mathbf{u}_2 \,\!$ jm = mass current at the cross-section (kg s-1)
Electromagnetism, electric charge	q = electric charge (C)	ρ = volume electric charge density (C m-3)	J = electric current density (A m-2)	$\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0 \,\!$	$I = \mathbf{A}_1 \cdot \mathbf{J}_1 = \mathbf{A}_2 \cdot \mathbf{J}_2 \,\!$ I = electric current at the cross-section (A)
Thermodynamics, energy	E = energy (J)	u = volume energy density (J m-3)	q = heat flux (W m-2)	$\nabla \cdot \mathbf{q} + \frac{\partial u}{\partial t} = 0 \,\!$	$P = \mathbf{A}_1 \cdot \mathbf{J}_1 = \mathbf{A}_2 \cdot \mathbf{J}_2 \,\!$ P = thermal current at the cross-section (W)
Quantum mechanics, probability	p = Pr(x,t) = probability distribution	P = P(x,t) = probability density function (m-3)	j = probability current/flux	$\nabla \cdot \mathbf{j} + \frac{\partial P}{\partial t} = 0 \,\!$

Principle of least action

Main article: Principle of least action

A system always minimizes the action associated with all parts of the system. Various minimized action formulations are given, all of course equivalent.

Formulation	Nomenclature	Equations
Statement	$\mathcal{S} \,\!$ = action	$\delta \mathcal{S} = 0 \,\!$
Maupertuis'-Euler	t = time, t1, t2 = initial and final times, T = Kinetic energy	$\mathcal{S} = \int_{t_1}^{t_2} 2T \mathrm{d}t = \int_{q_1}^{q_2} \mathbf{p} \cdot \mathrm{d}\dot{\mathbf{q}} \,\!$ $2T=\mathbf{p} \cdot \mathrm{d}\dot{\mathbf{q}} \,\!$
Jacobi's	s = arc length, C = curve, T = Kinetic energy V = Potential energy E = Total mechanical energy	$\mathcal{S} = \int_C \sqrt{2T} \mathrm{d}s = \int_C \sqrt{2\left(E-V\right)} \mathrm{d}s \,\!$ $E = T \left(\dot{\mathbf{q}} \right ) + V\left(\mathbf{q}\right ) \,\!$
Lagrange	L = Lagrangian	$\mathcal{S} = \int_{t_1}^{t_2} L \mathrm{d}t \,\!$ $L = L\left ( \mathbf{q}, \dot{\mathbf{q}}, t\right ) \,\!$

Classical mechanics

Galilean frame transforms

For classical (Galileo-Newtonian) mechanics, the transformation law from one inertial or accelerating (including rotation) frame (reference frame travelling at constant velocity - including zero) to another is the Galilean transform.

Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative accelerations.

Motion of entities		Inertial frames	Accelerating frames
Translation V = Constant relative velocity between two inertial frames F and F'. A = (Variable) relative acceleration between two accelerating frames F and F'.		Relative position $\mathbf{r}' = \mathbf{r} + \mathbf{V}t \,\!$ Relative velocity $\mathbf{v}' = \mathbf{v} + \mathbf{V} \,\!$ Equivalent accelerations $\mathbf{a}' = \mathbf{a}$	Relative accelerations $\mathbf{a}' = \mathbf{a} + \mathbf{A}$ Apparent/ficticous forces $\mathbf{F}' = \mathbf{F} - \mathbf{F}_\mathrm{app}$
As measured in frame F: r = position v = velocity a = acceleration F = force	As measured in frame F': r' = position v' = velocity a' = acceleration F' = force Fapp = apparent forces
Rotation Ω = Constant relative angular velocity between two frames F and F'. Λ = (Variable) relative angular acceleration between two accelerating frames F and F'.		Relative angular position $\theta' = \theta + \Omega t \,\!$ Relative velocity $\boldsymbol{\omega}' = \boldsymbol{\omega} + \boldsymbol{\Omega} \,\!$ Equivalent accelerations $\boldsymbol{\alpha}' = \boldsymbol{\alpha}$	Relative accelerations $\boldsymbol{\alpha}' = \boldsymbol{\alpha} + \boldsymbol{\Lambda}$ Apparent/ficticous torques $\boldsymbol{\tau}' = \boldsymbol{\tau} - \boldsymbol{\tau}_\mathrm{app}$
As measured in frame F: θ = angular position, angle ω = angular velocity α = angular acceleration τ = torque	As measured in frame F': r' = angular position ω' = angular velocity α' = angular acceleration τ' = torque τapp = apparent torque

Laws of classical mechanics

The following^[1][2] general approaches to classical mechanics are summarized below in the order of establishment. They are equivalent formulations, Newton's is very commonly used due to simplicity, but Hamilton's and Lagrange's equations are more general, and their range can extend into other branches of physics with suitable modifications.

Laws	Equation
Newton's laws	The laws can be summarized by two equations: $\mathbf{F} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} \,\!$ $\mathbf{F}_\mathrm{ij}=-\mathbf{F}_\mathrm{ji} \,\!$ For a dynamical system the two equations (effectively) combine into one: $\frac{\mathrm{d}\mathbf{p}_\mathrm{i}}{\mathrm{d}t} = \mathbf{F}_{E} + \sum_{\mathrm{i} \neq \mathrm{j}} \mathbf{F}_\mathrm{ij} \,\!$
p = momentum of body, Fij = force ON body i BY body j, Fij = force ON body j BY body i, FE = resultant external force (due to any agent not part of system). Body i does not exert a force on itself.
Euler-Lagrange equations	Euler-Lagrange equation system $p_\alpha = \frac{\partial L}{\partial \dot{q}_\alpha}$ $\dot{p}_\alpha = \frac{\partial L}{\partial {q}_\alpha}$ Euler-Lagrange equation $\frac{\mathrm{d}}{\mathrm{d} t} \left ( \frac{\partial L}{\partial \dot{q}_\alpha } \right ) = -\frac{\partial L}{\partial q_{\alpha}}$
qi = generalized coordinates, pi = generalized momenta
Hamilton's equation system	$\frac{\partial \mathbf{p}}{\partial t} = -\frac{\partial H}{\partial \mathbf{q}}$ $\frac{\partial \mathbf{q}}{\partial t} = \frac{\partial H}{\partial \mathbf{p}}$ The Hamiltonian as a function of generalized coordinates and momenta has the general form: $H = \left [ \mathbf{q}(t), \mathbf{p}(t), t \right ]$
Appell's equation	$\frac{\partial S}{\partial \alpha_i} = Q_i$ where: $S = \dfrac{1}{2} \sum_{i=1}^{N}m_i \mathbf{a}_\mathrm{i}\cdot\mathbf{a}_\mathrm{i}$
Qi = generalized force αi = generalized acceleration ai = acceleration of particle

Laws of thermodynamics

Main article: Laws of thermodynamics

Main laws

Property/effect	Equation
Zeroth law of thermodynamics (systems in thermal equilibrium)	$(T_A = T_B) \and (T_B=T_C) \Rightarrow T_A=T_C\,\!$
First law of thermodynamics	$\Delta U = \Delta Q + \Delta W \,\!$ Internal energy increase $\Delta U > 0 \,\!$, decrease $\Delta U < 0 \,\!$ Heat energy transferred to system $\Delta Q > 0 \,\!$, from system $\Delta Q < 0 \,\!$ Work done transferred to system $\Delta W > 0 \,\!$ by system $\Delta W < 0 \,\!$
Second Law of Thermodynamics	$\Delta S \geqslant 0 \quad \forall \Delta t > 0\,\!$
Third Law of Thermodynamics	$S = S_\mathrm{structural} + CT\,\!$

Extensions of the laws

Property or effect	Equation
Fundamental thermodynamic relation (systems in thermal equilibrium)	$\mathrm{d} U = T \mathrm{d} S - P \mathrm{d} V + \sum_i \mu_i \mathrm{d}N_i \,\!$
Onsager reciprocal relations	$\mathbf{J}_{u} = L_{uu}\, \nabla\frac{1}{T} - L_{u\rho}\, \nabla\frac{\mu}{T} \!$ $\mathbf{J}_{\rho} = L_{\rho u}\, \nabla\frac{1}{T} - L_{\rho\rho}\, \nabla\frac{\mu}{T}. \!$
u = Energy Density (J m−3) J = Energy Flux Density (W m−2) L = Reciprocity Coefficients (K kg m−3)

Maxwell's relations

Main article: Maxwell's relations

The potentials as functions of their natural variables are:

$U(S,V)\,$ = Internal energy
$H(S,P)\,$ = Enthalpy
$A(T,V)\,$ = Helmholtz free energy
$G(T,P)\,$ = Gibbs free energy

$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V = \frac{\partial^2 U }{\partial S \partial V}$ $\left(\frac{\partial T}{\partial P}\right)_S = +\left(\frac{\partial V}{\partial S}\right)_P = \frac{\partial^2 H }{\partial S \partial P}$ $+\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V = - \frac{\partial^2 A }{\partial T \partial V}$ $-\left(\frac{\partial S}{\partial P}\right)_T = \left(\frac{\partial V}{\partial T}\right)_P = \frac{\partial^2 G }{\partial T \partial P}$

Laws of gravitation

Modern laws

ETF and GEM Equations

Main article: Einstein field equations (ETF), Gravitomagnetism (GEM)

In a relatively flat spacetime due to weak gravitational fields (by General Relativity), the following gravitational analogues of Maxwell's equations can be found, to describe an analogous Gravitomagnetic Field. They are well established by the theory, but have yet to be verified by experiment.^[3]

Laws	Nomenclature	Equations
Einstein Tensor Field (ETF) Equations	Λ = Cosmological Constant Rμν = Ricci Curvature Tensor Tμν = Stress-Energy Tensor gμν = The Metric Tensor Gμν = Einstein Tensor	$R_{\mu \nu} + \left ( \Lambda - \frac{R}{2} \right ) g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}\,\!$ $G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}\,\!$
GEM Equations	g = Gravitational Field ξ = Gravitomagnetic Field	$\nabla \cdot \mathbf{g} = -4 \pi G \rho \,\!$ $\nabla \cdot \boldsymbol{\xi} = \mathbf{0} \,\!$ $\nabla \times \mathbf{g} = -\frac{\partial \boldsymbol{\xi} } {\partial t} \,\!$ $\nabla \times \boldsymbol{\xi} = \frac{1}{c^2} \left [ -4 \pi G\mathbf{j}_\mathrm{m} + \frac{\partial \mathbf{g}} {\partial t} \right ] \,\!$
Gravitomagnetic Lorentz Force	Ω = Gravitational Torsion Field	$\mathbf{F} = m \left( \mathbf{g} + \mathbf{v} \times 2 \boldsymbol{\xi} \right)$ $2 \boldsymbol{\xi} = \mathbf{\Omega} \,\!$

Classical laws

Main article: Kepler's Laws

It can be found that Kepler's Laws, though originally discovered from planetary observations (also due to Tycho Brahe), are true for any central forces.^[4]

For Kepler's 1st law, the equation is nothing physically fundamental; simply the polar equation of an ellipse where the pole (origin of polar coordinate system) is positioned at a focus of the ellipse, centred on the central star.

e = (elliptic) eccentricity
a = elliptic semi-major axes = planet aphelion
b = elliptic semi-minor axes = planet perihelion

$e = \sqrt{1-\left ( \frac{b}{a} \right )^2} \,\!$

Property or effect	Equation
Newton's law of universal gravitation	$\mathbf{F} = \frac{G m_1 m_2}{\left | \mathbf{r} \right |^2} \mathbf{\hat{r}} \,\!$ For a non uniform mass distribution this becomes: $\mathbf{g} = G \oint_{V} \frac{\mathbf{r} \rho \mathrm{d}{V}}{\left | \mathbf{r} \right |^3}\,\!$
ρ = Local mass density of body V = Volume of body
Gauss' law for gravity	$\oint_S \mathbf{g} \cdot \mathrm{d}\mathbf{A} = 4\pi G M_\mathrm{enc} \,\!$
Kepler's 1st Law Planets move in an ellipse, with the star at a focus	$\mathbf{r} = \frac{a}{1+e \cos\theta} \mathbf{\hat{r}} \,\!$
Kepler's 2nd Law	$\frac{\mathrm{d}A}{\mathrm{d}t} = \frac{\left | \mathbf{L} \right |}{2m} \,\!$
Kepler's 3rd Law	$T^2 = \frac{4\pi^2}{G \left ( m + M \right ) }r^3\,\!$

Laws of electromagnetism

Maxwell's equations

Main article: Maxwell's equations

No Magnetic Monopoles

Name	Differential form
Gauss's law for electrostatics	$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$
Gauss's law for magnetostatics	$\nabla \cdot \mathbf{B} = \mathbf{0}$
Maxwell–Faraday law (Faraday's law of induction)	$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$
Maxwell-Ampère circuital law (Ampere's Law with Maxwell's correction)	$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t} \$
Lorentz force law	$\mathbf{F}=q_e\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)$

Monopoles Inclusion

To introduce monopoles mathematically, magnetic pole strength q_m (AKA magnetic charge or monopole charge, all synonymous) is defined, treating magnetic poles analogously to electric charges - see section below. Pole srength can be quantified into volume densities, currents and current densities just like electric charge.

The mathematical convention is: north charge N; q_m > 0, south charge S; q_m < 0, neutral charge; q_m = 0 (equivalent to no net magnetic charge, or none at all).

From the SI system two units are possible, Wb (Weber) and A m (Ampere-metre). Dimensional analysis leads to the conversion:

$q_m \left ( \mathrm{Wb} \right ) = \mu_0 q_m \left ( \mathrm{A \cdot m} \right )$

Maxwell's Equations would become one of the columns in the table below, at least theoretically. Subscripts e are electric charge quantities; subscripts m are magnetic charge quantities. They are consistent if no magnetic monopoles exist, since the monopole quantities are then zero and the equations reduce to the original form of Maxwell's equations.

Name	Weber (Wb)	Ampere meter (A m) convention
Gauss's law for electrostatics	$\nabla \cdot \mathbf{E} = \frac{\rho_e}{\epsilon_0}$	$\nabla \cdot \mathbf{E} = \frac{\rho_e}{\epsilon_0}$
Gauss's law for magnetostatics	$\nabla \cdot \mathbf{B} = \rho_m$	$\nabla \cdot \mathbf{B} = \mu_0\rho_m$
Faraday's law of induction	$-\nabla \times \mathbf{E} = \frac{\partial \mathbf{B}} {\partial t} + \mathbf{j}_m$	$-\nabla \times \mathbf{E} = \frac{\partial \mathbf{B}} {\partial t} + \mu_0\mathbf{j}_m$
Ampère's law	$\nabla \times \mathbf{B} = \mu_0 \mathbf{J}_e + \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t}$	$\nabla \times \mathbf{B} = \mu_0 \mathbf{J}_e + \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t}$
Lorentz force equation	$\mathbf{F}=q_e\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right) + \frac{q_m}{\mu_0} \left(\mathbf{B}-\frac{1}{c^2}\mathbf{v}\times\mathbf{E}\right)$	$\mathbf{F}=q_e\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right) + q_m\left(\mathbf{B}-\frac{1}{c^2}\mathbf{v}\times\mathbf{E}\right)$

Pre-Maxwell laws

These laws are not fundamental, since they can be derived from Maxwell's Equations. Coulomb's Law can be found from Gauss' Law (electrostatic form) and the Biot-Savart Law can be deduced from Ampere's Law (magnetostatic form). Lenz' Law and Faraday's Law can be incorporated into the Maxwell-Faraday equation. Nonetheless they are still very effective for simple calculations.

Name	Equation
Coulomb's Law	$\mathbf{E} = \frac{\mathbf{F}}{Q} = \frac{q}{4\pi\epsilon_0 \left | \mathbf{r} \right |^2} \mathbf{\hat{r}} \,\!$
Biot-Savart law	$\mathbf{B} = \frac{\mu_0}{4\pi} \int_C \frac{I \left ( \mathrm{d} \mathbf{l} \times \mathbf{r} \right )}{\left | \mathbf{r} \right |^3}\,\!$
Lenz's law	The magnetic flux set up by the current in a conductor tends to oppose that current.

Kirchoff's Laws

emf loop rule around any closed circuit $\sum_i V_i = \sum_i I_i R_i = 0 \,\!$ Current law at junctions $I_\mathrm{in} = I_\mathrm{out} \,\!$

Parallels between classical gravitation and electromagnetism

Below the mathematics can only be incorporated into magnetism if magnetic monopoles exist. If they do not, then as stated in the above Laws of electromagnetism section the pole strength is zero and monopole quantities disappear.

Let:

α = Mass, electric charge or magnetic monopole strength,
Λ = Gravitational, electric field or magnetic field,
F = Force,
Π = Mass moment, electric dipole due to charges, magnetic moment due to current or monopoles,
d = Pole separation for electric/magnetic dipoles,
r = Position vector,
X = Proportionality constants related to other physical constants,
Y = Physical constant, gravitational constant G, vacuum permittivity ε₀, vacuum permeability μ₀,
Ф_Λ = Field flux,
n, m = any from +1, 0, −1

and using the same notation as before in the General principles section above, we have the following.

Property or effect	Equation
Constants	$X = \left ( 4 \pi \right )^n Y^m \,\!$
Quantification: volume densities, currents, current densities	$\rho = \frac{{\rm d}\alpha}{{\rm d}V} \,\!$ $I = \frac{{\rm d}\alpha}{{\rm d}t} \,\!$ $\mathbf{J} = \mathbf{\hat{t}}\frac{{\rm d}I}{{\rm d}A} \,\!$
Force due to field	$\mathbf{F} = \alpha\boldsymbol{\Lambda}\,\!$ $\mathbf{F} = X \frac{\alpha_1\alpha_2}{\left | \mathbf{r}\right |^2}\mathbf{\hat{r}} \,\!$ For an array of point masses, charges or poles: $\mathbf{F} = X \sum_{i \neq j} \frac{\alpha_i\alpha_j}{\left | \mathbf{r}_{\rm i} - \mathbf{r}_{\rm j} \right |^2}\mathbf{\hat{r}}_{\rm ij} \,\!$ where the unit vector rij corresponds to the separation between a pair of point masses charges or poles: $\mathbf{\hat{r}}_{\rm ij} = \frac{\mathbf{r}_{\rm ij}}{\left | \mathbf{r}_{\rm ij} \right |} = \frac{\mathbf{r}_{\rm i} - \mathbf{r}_{\rm j}}{\left | \mathbf{r}_{\rm i} - \mathbf{r}_{\rm j} \right |} \,\!$
Mass moments or electric/magnetic dipole moments	$\boldsymbol{\Pi} = 2^n \mathbf{d} \alpha \,\!$
Mass moment density, electric or magnetic dipole moment densities (polarization/ magnetization)	$\mathbf{H} = \frac{{\rm d}\langle \boldsymbol{\Pi} \rangle}{{\rm d}V} \,\!$ The average moment is simply: $\langle \boldsymbol{\Pi} \rangle = \frac{1}{N}\sum_{i = 1}^N \boldsymbol{\Pi}_i \,\!$
Field flux	$\Phi_\Lambda = \oint_S \boldsymbol{\Lambda} \cdot {\rm d}\mathbf{S} \,\!$
"Gaussian law"	Equivalent forms (by the divergence theorem) are: $\Phi_\Lambda = X \alpha \,\!$ $\nabla \cdot \boldsymbol{\Lambda} = X \rho \,\!$

Radiation laws

Photonics

Subscripts 1 and 2 refer to initial and final optical media respectivley.

Name	Equations
Law of reflection	$\theta_1 = \theta_2 \,\!$
θ = angle of refraction relative to the interface normal
Law of refraction, Snell's law	$\frac{n_1}{n_2} = \frac{\sin \theta_2}{\sin \theta_1} \,\!$
n = refractive index of medium
Angle of total polarisation	$\tan \theta_B = n_2/n_1\,\!$
θB = Reflective polarization angle, Brewster's angle
intensity from polarized light, Malus' law	$I = I_0\cos^2\theta\,\!$
I0 = Initial intensity, I = Transmitted intensity, θ = Polarization angle between polarizer transmission axes and electric field vector
Bragg's law (solid state diffraction)	$\frac{\delta}{2\pi} \lambda = 2d \sin\theta \,\!$ For constructive interference: $\delta/2\pi = n \,\!$ For destructive interference: $\delta/2\pi = n/2 \,\!$ where $n \in \mathbf{N}\,\!$
d = lattice spacing δ = phase differance between two waves
Huygen-Fresnel-Kirchhoff principle	$A \mathbf ( \mathbf{r} ) = \frac{-i}{2\lambda} \iint_\mathrm{aperture} \frac{e^{i \mathbf{k} \cdot \left ( \mathbf{r} + \mathbf{r}_0 \right ) }}{ \left | \mathbf{r} \right |\left | \mathbf{r}_0 \right |} \left [ \cos \alpha_0 - \cos \alpha \right ] \mathrm{d}S \,\!$ where $\mathbf{r}_0 \cdot \mathbf{\hat{n}} = \left | \mathbf{r}_0 \right | \cos \alpha_0 \,\!$ $\mathbf{r} \cdot \mathbf{\hat{n}} = \left | \mathbf{r} \right | \cos \alpha \,\!$ $\left | \mathbf{r} \right |\left | \mathbf{r}_0 \right | \ll \lambda \,\!$
r0 = position from source to aperture, incident on it r = position from aperture diffracted from it to a point α0 = incident angle with respect to the normal, from source to aperture α = diffracted angle, from aperture to a point S = imaginary surface bounded by aperture $\mathbf{\hat{n}}\,\!$ = unit normal vector to the aperture
Kirchhoff's diffraction formula	$A \left ( \mathbf{r} \right ) = - \frac{1}{4 \pi} \iint_\mathrm{aperture} \frac{e^{i \mathbf{k} \cdot \mathbf{r}_0}}{\left | \mathbf{r}_0 \right |} \left[ i \left | \mathbf{k} \right | U_0 \left ( \mathbf{r}_0 \right ) \cos{\alpha} + \frac {\partial A_0 \left ( \mathbf{r}_0 \right )}{\partial n} \right ] \mathrm{d}S$

These ratios are sometimes also used, following simply from other definitions of refractive index, wave phase velocity, and the luminal speed equation;

$\frac{n_1}{n_2} = \frac{v_2}{v_1} = \frac{\lambda_2}{\lambda_1} = \sqrt{\frac{\epsilon_1 \mu_1}{\epsilon_2 \mu_2}} \,\!$

where:
ε = permittivity of medium,
μ = permeability of medium,
λ = wavelength of light in medium,
v = speed of light in media.

Radiation

Name	Equations
Radioactive decay	Statistical decay of a radionuclide: $\frac{\mathrm{d} N}{\mathrm{d} t} = - \lambda N$
N = N(t) = Number of atoms at time t t = Time (s) λ = Decay constant (s-1)
Stefan–Boltzmann law	$I = \sigma T^4 \,\!$
I = Intensity (W m-2)
Wien's displacement law	$\lambda_{\mathrm{max}}T = b \,\!$
b = Wein constant (m K)
Planck's law	$I \left ( \nu, T \right ) = \frac{ 2h \nu^3}{c^2 \left ( e^{h\nu/kT}-1 \right )}$ $I \left ( \lambda, T \right ) = \frac{ 2h c^3}{\lambda^5 \left ( e^{hc/\lambda kT}-1 \right )}$ $I \left ( \lambda, T \right ) = I \left ( \nu, T \right ) \left | \frac{\mathrm{d}\nu}{\mathrm{d}\lambda} \right |$
I = Specific radiative intensity (W m–2 Hz–1 sr–1)

Quantum mechanics

Wavefunction equations

Name	Equation
Schrödinger equation	General classical form for all wavefunctions ψ = ψ(r, t): $\hat{H} \Psi = \hat{E} \Psi$
Dirac equation	For elementary spin-1/2 particles: $\left(\boldsymbol{\beta} mc^2 + c \sum_{k = 1}^3 \boldsymbol{\alpha}_k p_k \right) \Psi = i \hbar \frac{\partial}{\partial t} \Psi$ Dirac matrices: $\boldsymbol{\alpha}_i^2=\boldsymbol{\beta}^2=\mathbf{I}_4$ $\boldsymbol{\alpha}_i\boldsymbol{\alpha}_j + \boldsymbol{\alpha}_j\boldsymbol{\alpha}_i = 0 \,$ $\boldsymbol{\alpha}_i\boldsymbol{\beta} + \boldsymbol{\beta}\boldsymbol{\alpha}_i = 0 \,$
Klein–Gordon equation	General form for all relativistic wavefunctions ψ = ψ(r, t): $\left(-\frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \nabla^2\right)\Psi = \left ( \frac{m_0c}{\hbar} \right )^2 \Psi$ Can be obtained by inserting the quantum operators into the momentum-energy invariant of relativistic mechanics: $\frac{E^2}{c^2} - p^2 = \left ( m_0c \right )^2$

Commonly used corollaries of Schrödinger's equation are obtained in this way: - A free particle corresponds to zero potential energy.

Wave–particle duality

Property/effect	Equation
Planck–Einstein equation	$E = hf = hc/\lambda\,\!$ $E = \hbar \omega = \hbar c k\,\!$
de Broglie wavelength	$\lambda p = h\,\!$ $\mathbf{p} = \hbar \mathbf{k}\,\!$
Heisenberg's uncertainty principle	$\Delta r \Delta p \ge \frac{\hbar}{2} \,\!$ $\Delta E \Delta t \ge \frac{\hbar}{2} \,\!$

Special relativity

Main article: 4-vector

Fundamental invariance and unification of space-time and energy-mass-momentum are given below.^[5] Both 4-vectors are very analogous, leading to analogous expressions using only the 4-vector dot product. The relativistic analogue to the classical Galilean transformation is the Lorentz transformation.

Lorentz frame tranforms

Let V = any 4-vector.

A boost B(v) in any arbitrary direction at velocity v = (v_x, v_y, v_z), or equivalently β = (β_x, β_y, β_z), without rotation, is given by:

$\mathbf{V}' = B\left ( \mathbf{v} \right )\mathbf{V}$

where

$B\left ( \mathbf{v} \right ) = \begin{bmatrix} \gamma&-\beta_x\,\gamma&-\beta_y\,\gamma&-\beta_z\,\gamma\\ -\beta_x\,\gamma&1+(\gamma-1)\dfrac{\beta_x^2}{\beta^2}&(\gamma-1)\dfrac{\beta_x \beta_y}{\beta^2}&(\gamma-1)\dfrac{\beta_x \beta_z}{\beta^2}\\ -\beta_y\,\gamma&(\gamma-1)\dfrac{\beta_y \beta_x}{\beta^2}&1+(\gamma-1)\dfrac{\beta_y^2}{\beta^2}&(\gamma-1)\dfrac{\beta_y \beta_z}{\beta^2}\\ -\beta_z\,\gamma&(\gamma-1)\dfrac{\beta_z \beta_x}{\beta^2}&(\gamma-1)\dfrac{\beta_z \beta_y}{\beta^2}&1+(\gamma-1)\dfrac{\beta_z^2}{\beta^2}\\ \end{bmatrix}$

$\beta_x = \frac{v_x}{c},$ $\beta_y = \frac{v_y}{c},$ $\beta_z = \frac{v_z}{c},$ $\beta = \frac{v}{c}=\frac{|\mathbf{v}|}{c},$, and $\gamma = \frac{1}{\sqrt{1-\beta^2}}.$

$\mathbf{V} = \left ( \begin{matrix} V_0 \\ V_1 \\ V_2 \\ V_3 \\ \end{matrix} \right ), \quad \mathbf{V}' = \left ( \begin{matrix} V'_0 \\ V'_1 \\ V'_2 \\ V'_3 \\ \end{matrix} \right ) \,\!$

4-vectors and frame-invariant results

Property/effect	3-vectors	4-vectors	Invariant result
Time and proper time			$\frac{\mathrm{d} \tau}{\mathrm{d} t} = \frac{1}{\gamma}$
Space-time	$\mathbf{r} \cdot \mathbf{r} \equiv r^2 \equiv x_1^2 + x_2^2 + x_3^2 \,\!$		$\mathbf{R} \cdot \mathbf{R} = \left ( c \tau \right )^2 \,\!$ $\begin{align} & \left ( c t \right )^2 - \left ( x_1^2 + x_2^2 + x_3^2 \right ) \\ & = \left ( c t \right )^2 - r^2 \\ & = \chi^2 = \left ( c\tau \right )^2 \end{align} \,\!$
τ = Proper time χ = Proper length 3-Position: r = (x1, x2, x3) 4-Position: R = (ct, x1, x2, x3)
Velocity transform	$\mathbf{u} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} \,\!$	$\mathbf{U} =\frac{\mathrm{d}\mathbf{R} }{\mathrm{d} \tau} = \gamma \left( c, \mathbf{u} \right)$	$\mathbf{U} \cdot \mathbf{U} = c^2 \,\!$
3-Velocity: u = (u1, u2, u3) 4-Velocity: U = (u0, u1, u2, u3)
Momentum-energy invariance	$\mathbf{p} = m\mathbf{u} \,\!$ $\mathbf{p} \cdot \mathbf{p} \equiv p^2 \equiv p_1^2 + p_2^2 + p_3^2 \,\!$	$\mathbf{P} = m \mathbf{U} \,\!$	$\mathbf{P} \cdot \mathbf{P} = \left ( m c \right )^2 \,\!$ $\begin{align} & \left ( \frac{E}{c} \right )^2 - \left ( p_1^2 + p_2^2 + p_3^2 \right ) \\ & = \left ( \frac{E}{c} \right )^2 - p^2 \\ & = \left ( mc \right )^2 \end{align} \,\!$ which leads to: $E^2 = \left ( pc \right )^2 + \left ( mc^2 \right )^2 \,\!$
E = total energy m = invariant mass 3-Momentum: p = (p1, p2, p3) 4-Position: P = (E/c, p1, p2, p3)
Acceleration transform	$\mathbf{a} = \frac{\mathrm{d}\mathbf{u}}{\mathrm{d}t} \,\!$	$\mathbf{A} =\frac{\mathrm{d}\mathbf{U} }{\mathrm{d} \tau} = \gamma \left( c\frac{\mathrm{d}\gamma}{\mathrm{d}t}, \frac{\mathrm{d}\gamma}{\mathrm{d}t} \mathbf{u} + \gamma \mathbf{a} \right)$	$\mathbf{A} \cdot \mathbf{A} = 0 \,\!$
3-Acceleration: a = (a1, a2, a3) 4-Acceleration: A = (a0, a1, a2, a3)
Force transform	$\mathbf{f} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} \,\!$	$\mathbf{F} =\frac{\mathrm{d}\mathbf{P} }{\mathrm{d} \tau} = \gamma m \left( c\frac{\mathrm{d}\gamma}{\mathrm{d}t}, \frac{\mathrm{d}\gamma}{\mathrm{d}t} \mathbf{u} + \gamma \mathbf{a} \right)$	$\mathbf{F} \cdot \mathbf{F} = 0 \,\!$
3-Acceleration: f = (f1, f2, f3) 4-Acceleration: F = (f0, f1, f2, f3)

Particle Physics

Name	Equations
Strong force	$\begin{align} \mathcal{L}_\mathrm{QCD} & = \bar{\psi}_i\left(i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \\ & = \bar{\psi}_i (i \gamma^\mu \partial_\mu - m )\psi_i - g G^a_\mu \bar{\psi}_i \gamma^\mu T^a_{ij} \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \,,\\ \end{align} \,\!$
Electroweak interaction	:$\mathcal{L}_{EW} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y.\,\!$ $\mathcal{L}_g = -\frac{1}{4}W_a^{\mu\nu}W_{\mu\nu}^a - \frac{1}{4}B^{\mu\nu}B_{\mu\nu}\,\!$ $\mathcal{L}_f = \overline{Q}_i iD\!\!\!\!/\; Q_i+ \overline{u}_i^c iD\!\!\!\!/\; u^c_i+ \overline{d}_i^c iD\!\!\!\!/\; d^c_i+ \overline{L}_i iD\!\!\!\!/\; L_i+ \overline{e}^c_i iD\!\!\!\!/\; e^c_i \,\!$ $\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2\,\!$ $\mathcal{L}_y = - y_{u\, ij} \epsilon^{ab} \,h_b^\dagger\, \overline{Q}_{ia} u_j^c - y_{d\, ij}\, h\, \overline{Q}_i d^c_j - y_{e\,ij} \,h\, \overline{L}_i e^c_j + h.c.\,\!$
Quantum electrodynamics	$\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\;,\,\!$

See also

• List of elementary physics formulae

References

1. ^ Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1 (VHC Inc.) 0-89573-752-3
2. ^ Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, ISBN 07-084018-0
3. ^ Gravitation and Inertia, I. Ciufolini and J.A. Wheeler, Princeton Physics Series, 1995, ISBN 0-691-03323-4
4. ^ 2.^ Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, ISBN 07-084018-0
5. ^ Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Manchester Physics Series, John Wiley & Sons, 2009, ISBN 9-780470-014608