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Laws reflect scientific knowledge that experiments have repeatedly verified (and never [[Falsifiability|falsified]]). Their accuracy does not change when new theories are worked out, but rather the scope of application, since the equation (if any) representing the law does not change. As with other scientific knowledge, they do not have absolute certainty (as mathematical [[theorems]] or [[Identity (mathematics)|identities]] do), and it is always possible for a law to be overturned by future observations. A law can usually be formulated as one or several statements or [[equation]]s, so that it can be used to predict the outcome of an experiment, given the circumstances of the processes taking place.
 
Laws differ from [[hypotheses]] and [[postulates]], which are proposed during the [[Scientific method|scientific process]] before and during validation by experiment and observation. [[Hypotheses]] and [[postulates]] are not laws since they have not been verified to the same degree and may not be sufficiently general, although they may lead to the formulation of laws. A law is a more solidified and formal statement, distilled from repeated experiment. Laws are narrower in scope than [[Scientific theory|scientific theories]], which may contain one or several laws.<ref>[{{cite web|url=http://ncse.com/evolution/education/definitions-fact-theory-law-scientific-work |title=Definitions from |publisher=the NCSE] |date= |accessdate=2019-03-18}}</ref> Science distinguishes a law or theory from facts.<ref>{{cite journal |url=http://dels.nas.edu/resources/static-assets/materials-based-on-reports/reports-in-brief/role_of_theory_final.pdf | title=The Role of Theory in Advancing 21st Century Biology: Catalyzing Transformative Research |publisher = The National Academy of Sciences |year =2007 |journal=Report in Brief |}}</ref> Calling a law a [[scientific fact|fact]] is [[ambiguous]], an [[overstatement]], or an [[equivocation]].<ref name=gouldfact>{{cite journal | url = http://www.stephenjaygould.org/library/gould_fact-and-theory.html | first = Stephen Jay | last = Gould | authorlink = Stephen Jay Gould | title = Evolution as Fact and Theory | journal = Discover | volume = 2 | issue = 5 | date = 1981-05-01 | pages = 34–37}}</ref> Although the nature of a scientific law is a question in [[philosophy]] and although scientific laws describe nature mathematically, scientific laws are practical conclusions reached by the [[scientific method]]; they are intended to be neither laden with [[ontology|ontological]] commitments nor statements of logical [[wikt:absolute#Noun|absolutes]].
 
According to the [[unity of science]] thesis, ''all'' scientific laws follow fundamentally from physics. Laws which occur in other sciences ultimately follow from [[physical law]]s. Often, from mathematically fundamental viewpoints, [[universal constant]]s emerge from a scientific law.
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! scope="col" widthstyle="150width:150px;" | Physics, conserved quantity
! scope="col" widthstyle="140width:140px;" | Conserved quantity ''q''
! scope="col" widthstyle="140width:140px;" | Volume density ''ρ'' (of ''q'')
! scope="col" widthstyle="140width:140px;" | Flux '''J''' (of ''q'')
! scope="col" widthstyle="10width:10px;" | Equation
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| [[Hydrodynamics]], [[fluid]]s <br />
|| '''j''' = [[probability current]]/flux
| <math> \frac{\partial |\Psi|^2}{\partial t}=-\nabla \cdot \mathbf{j} </math>
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! scope="col" widthstyle="width:600px;" colspan="2"| '''Laws of motion'''
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|colspan="2" |'''[[Principle of least action]]:'''
<math> \mathcal{S} = \int_{t_1}^{t_2} L \,\mathrm{d}t \,\!</math>
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| rowspan="2" scope="col" widthstyle="width:300px;" |'''The [[Euler–Lagrange equation]]s are:'''
 
:<math> \frac{\mathrm{d}}{\mathrm{d} t} \left ( \frac{\partial L}{\partial \dot{q}_i } \right ) = \frac{\partial L}{\partial q_i} </math>
 
:<math> p_i = \frac{\partial L}{\partial \dot{q}_i}\quad \dot{p}_i = \frac{\partial L}{\partial {q}_i} </math>
|width style="width:300px;"| '''Hamilton's equations'''
:<math> \dfrac{\partial \mathbf{p}}{\partial t} = -\dfrac{\partial H}{\partial \mathbf{q}} </math><br /><math> \dfrac{\partial \mathbf{q}}{\partial t} = \dfrac{\partial H}{\partial \mathbf{p}} </math>
 
:<math>H \left(\mathbf{q}, \frac{\partial S}{\partial\mathbf{q}}, t\right) = -\frac{\partial S}{\partial t}</math>
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| colspan="2" scope="col" widthstyle="width:600px;" | '''Newton's laws'''
 
'''[[Newton's laws of motion]]'''
 
in which '''F'''<sub>E</sub> = resultant external force (due to any agent not part of system). Body ''i'' does not exert a force on itself.
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| scope="col" widthstyle="width:300px;"|'''[[Einstein field equations]] (EFE):'''
:<math>R_{\mu \nu} + \left ( \Lambda - \frac{R}{2} \right ) g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}\,\!</math>
 
where Λ = [[cosmological constant]], ''R<sub>μν</sub>'' = [[Ricci curvature tensor]], ''T<sub>μν</sub>'' = [[Stress–energy tensor]], ''g<sub>μν</sub>'' = [[metric tensor]]
| scope="col" widthstyle="width:300px;"|'''[[Geodesic equation]]:'''
:<math>\frac{{\rm d}^2x^\lambda }{{\rm d}t^2} + \Gamma^{\lambda}_{\mu \nu }\frac{{\rm d}x^\mu }{{\rm d}t}\frac{{\rm d}x^\nu }{{\rm d}t} = 0\ ,</math>
where Γ is a [[Christoffel symbol]] of the [[Christoffel symbols#Christoffel symbols of the second kind (symmetric definition)|second kind]], containing the metric.
 
where ''m'' is the [[rest mass]] of the particlce and γ is the [[Lorentz factor]].
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| scope="col" widthstyle="width:300px;"|'''[[Newton's law of universal gravitation]]:'''
For two point masses:
 
 
:<math> \mathbf{g} = G \int_{V} \frac{\mathbf{r} \rho \mathrm{d}{V}}{\left | \mathbf{r} \right |^3}\,\!</math>
| scope="col" widthstyle="width:300px;"| '''[[Gauss' law for gravity]]:'''
 
An equivalent statement to Newton's law is:
:<math>\nabla\cdot\mathbf{g} = 4\pi G\rho \,\!</math>
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| colspan="2" scope="col" widthstyle="width:600px;"|'''Kepler's 1st Law:''' Planets move in an ellipse, with the star at a focus
:<math>r = \frac{l}{1+e \cos\theta} \,\!</math>
 
is the [[Eccentricity (mathematics)|eccentricity]] of the elliptic orbit, of semi-major axis ''a'' and semi-minor axis ''b'', and ''l'' is the semi-latus rectum. This equation in itself is nothing physically fundamental; simply the [[Polar coordinate system|polar equation]] of an [[ellipse]] in which the pole (origin of polar coordinate system) is positioned at a focus of the ellipse, where the orbited star is.
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| colspan="2" widthstyle="width:600px;"|'''Kepler's 2nd Law:''' equal areas are swept out in equal times (area bounded by two radial distances and the orbital circumference):
:<math>\frac{\mathrm{d}A}{\mathrm{d}t} = \frac{\left | \mathbf{L} \right |}{2m} \,\!</math>
where '''L''' is the orbital angular momentum of the particle (i.e. planet) of mass ''m'' about the focus of orbit,
:<math>T^2 = \frac{4\pi^2}{G \left ( m + M \right ) } a^3\,\!</math>
where ''M'' is the mass of the central body (i.e. star).
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!colspan="2"|'''[[Laws of thermodynamics]]'''
|- valign="top"
| scope="col" widthstyle="width:150px;"|'''[[First law of thermodynamics]]:''' The change in internal energy d''U'' in a closed system is accounted for entirely by the heat &delta;''Q'' absorbed by the system and the work &delta;''W'' done by the system:
:<math>\mathrm{d}U=\delta Q-\delta W\,</math>
'''[[Second law of thermodynamics]]:''' There are many statements of this law, perhaps the simplest is "the entropy of isolated systems never decreases",
:<math>\Delta S \ge 0</math>
meaning reversible changes have zero entropy change, irreversible process are positive, and impossible process are negative.
| rowspan="2" widthstyle="width:150px;"| '''[[Zeroth law of thermodynamics]]:''' If two systems are in [[thermal equilibrium]] with a third system, then they are in thermal equilibrium with one another.
:<math>T_A = T_B \,, T_B=T_C \Rightarrow T_A=T_C\,\!</math>
 
:<math>\mathrm{d} U = T \mathrm{d} S - P \mathrm{d} V + \sum_i \mu_i \mathrm{d}N_i \,\!</math>
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| colspan="2" widthstyle="width:500px;"|'''[[Onsager reciprocal relations]]:''' sometimes called the ''Fourth Law of Thermodynamics''
:<math> \mathbf{J}_{u} = L_{uu}\, \nabla(1/T) - L_{ur}\, \nabla(m/T) \!</math>;
:<math> \mathbf{J}_{r} = L_{ru}\, \nabla(1/T) - L_{rr}\, \nabla(m/T) \!</math>.
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| scope="col" widthstyle="width:300px;"|'''[[Maxwell's equations]]'''
 
'''[[Gauss's law]] for electricity'''
'''[[Ampère's circuital law]] (with Maxwell's correction)'''
:<math>\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t} \ </math>
| scope="col" widthstyle="width:300px;"| '''[[Lorentz force]] law:'''
: <math>\mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)</math>
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| colspan="2" scope="col" widthstyle="width:600px;"| '''[[Quantum electrodynamics]] (QED):''' Maxwell's equations are generally true and consistent with relativity - but they do not predict some observed quantum phenomena (e.g. light propagation as [[EM wave]]s, rather than [[photons]], see [[Maxwell's equations]] for details). They are modified in QED theory.
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| widthstyle="width:300px;"| '''[[Quantum mechanics]], [[Quantum field theory]]'''
 
'''[[Schrödinger equation]] (general form):''' Describes the time dependence of a quantum mechanical system.
 
The [[Hamiltonian quaternions|Hamiltonian]] (in quantum mechanics) ''H'' is a [[self-adjoint operator]] acting on the state space, <math>| \psi \rangle </math> (see [[Dirac notation]]) is the instantaneous [[quantum state vector]] at time ''t'', position '''r''', ''i'' is the unit [[imaginary number]], ''ħ'' = ''h''/2π is the reduced [[Planck's constant]].
| rowspan="2" scope="col" widthstyle="width:300px;"|'''[[Wave-particle duality]]'''
 
'''[[Planck constant|Planck–Einstein law]]:''' the [[energy]] of [[photon]]s is proportional to the [[frequency]] of the light (the constant is [[Planck's constant]], ''h'').
:<math> i\hbar \frac{\partial}{\partial t}\psi = -\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi </math>
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| colspan="2" widthstyle="width:600px;"| '''[[Pauli exclusion principle]]:''' No two identical [[fermion]]s can occupy the same quantum state ([[boson]]s can). Mathematically, if two particles are interchanged, fermionic wavefunctions are anti-symmetric, while bosonic wavefunctions are symmetric:
 
<math>\psi(\cdots\mathbf{r}_i\cdots\mathbf{r}_j\cdots) = (-1)^{2s}\psi(\cdots\mathbf{r}_j\cdots\mathbf{r}_i\cdots)</math>
 
where '''r'''<sub>''i''</sub> is the position of particle ''i'', and ''s'' is the [[spin (physics)|spin]] of the particle. There is no way to keep track of particles physically, labels are only used mathematically to prevent confusion.
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