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{{About|the scientific and mathematical aspects of scientific laws|the philosophy of scientific laws|Scientific law}}
{{Refimprove|date=February 2008}}
The '''laws of science''', '''scientific laws''', or '''scientific principles''' are statements that describe or predict a range of ] as they appear in ].<ref>{{OED|law of nature}}</ref> The term "law" has diverse usage in many cases: approximate, accurate, broad or narrow theories, in all ] disciplines (], ], ], ], ] etc.). Scientific laws summarize and explain a large collection of facts determined by ], and are tested based on their ability to predict the results of future experiments. They are developed either from facts or through ], and are strongly supported by ] ]. It is generally understood that they reflect causal relationships fundamental to reality, and are discovered rather than invented.<ref name="McComas2013">{{cite book|author=William F. McComas|title=The Language of Science Education: An Expanded Glossary of Key Terms and Concepts in Science Teaching and Learning|url=https://books.google.com/books?id=aXzGBAAAQBAJ|date=30 December 2013|publisher=Springer Science & Business Media|isbn=978-94-6209-497-0|page=58}}</ref>

Laws reflect scientific knowledge that experiments have repeatedly verified (and never ]). 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 ] or ] 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 ]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 ] and ], which are proposed during the ] before and during validation by experiment and observation. These 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 ], which may contain one or several laws.<ref></ref> Unlike hypotheses, theories and laws may be simply referred to as ].<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 ] and although scientific laws describe nature mathematically, scientific laws are practical conclusions reached by the ]; they are intended to be neither laden with ] commitments nor statements of logical ].

According to the ] thesis, ''all'' scientific laws follow fundamentally from physics. Laws which occur in other sciences ultimately follow from ]s. Often, from mathematically fundamental viewpoints, ]s emerge from scientific laws.

== Conservation laws ==

===Conservation and symmetry===
{{main article|Symmetry (physics)}}

Most significant laws in science are ]. These fundamental laws follow from homogeneity of space, time and ], in other words ''symmetry''.

*''']:''' Any quantity which has a continuous differentiable symmetry in the action has an associated conservation law.
* ] was the first law of this type to be understood, since most macroscopic physical processes involving masses, for example collisions of massive particles or fluid flow, provide the apparent belief that mass is conserved. Mass conservation was observed to be true for all chemical reactions. In general this is only approximative, because with the advent of relativity and experiments in nuclear and particle physics: mass can be transformed into energy and vice versa, so mass is not always conserved, but part of the more general conservation of mass-energy.
*''']''', ''']''' and ''']''' for isolated systems can be found to be ], translation, and rotation.
*''']''' was also realized since charge has never been observed to be created or destroyed, and only found to move from place to place.

===Continuity and transfer===

Conservation laws can be expressed using the general ] (for a conserved quantity) can be written in differential form as:

:<math>\frac{\partial \rho}{\partial t}=-\nabla \cdot \mathbf{J} </math>

where ρ is some quantity per unit volume, '''J''' is the ] of that quantity (change in quantity per unit time per unit area). Intuitively, the ] (denoted ∇•) of a ] is a measure of flux diverging radially outwards from a point, so the negative is the amount piling up at a point, hence the rate of change of density in a region of space must be the amount of flux leaving or collecting in some region (see main article for details). In the table below, the fluxes, flows for various physical quantities in transport, and their associated continuity equations, are collected for comparison.

:{| class="wikitable" align="center"
|-
! scope="col" width="150" | Physics, conserved quantity
! scope="col" width="140" | Conserved quantity ''q''
! scope="col" width="140" | Volume density ''ρ'' (of ''q'')
! scope="col" width="140" | Flux '''J''' (of ''q'')
! scope="col" width="10" | Equation
|-
| ], ]s <br />
| ''m'' = ] (kg)
| ''ρ'' = volume ] (kg m<sup>−3</sup>)
| ''ρ'' '''u''', where<br/ >
'''u''' = ] of fluid (m s<sup>−1</sup>)
| <math> \frac{\partial \rho}{\partial t} = - \nabla \cdot (\rho \mathbf{u}) </math>
|-
| ], ]
| ''q'' = electric charge (C)
| ''ρ'' = volume electric ] (C m<sup>−3</sup>)
| '''J''' = electric ] (A m<sup>−2</sup>)
| <math> \frac{\partial \rho}{\partial t} = - \nabla \cdot \mathbf{J} </math>
|-
| ], ]
| ''E'' = energy (J)
| ''u'' = volume ] (J m<sup>−3</sup>)
| '''q''' = ] (W m<sup>−2</sup>)
| <math> \frac{\partial u}{\partial t}=- \nabla \cdot \mathbf{q} </math>
|-
| ], ]
|| ''P'' = ('''r''', ''t'') = ∫|Ψ|<sup>2</sup>d<sup>3</sup>'''r''' = ]
|| ''ρ'' = ''ρ''('''r''', ''t'') = |Ψ|<sup>2</sup> = ] (m<sup>−3</sup>),<br />
Ψ = ] of quantum system
|| '''j''' = ]/flux
| <math> \frac{\partial |\Psi|^2}{\partial t}=-\nabla \cdot \mathbf{j} </math>
|-
|}

More general equations are the ] and ], which have their roots in the continuity equation.

==Laws of classical mechanics==

===Principle of least action===

{{Main|Principle of least action}}

All of classical mechanics, including ], ], ], etc., can be derived from this very simple principle:

:<math> \delta \mathcal{S} = \delta\int_{t_1}^{t_2} L(\mathbf{q}, \mathbf{\dot{q}}, t) dt = 0 </math>

where <math> \mathcal{S} </math> is the ]; the integral of the ]

:<math> L(\mathbf{q}, \mathbf{\dot{q}}, t) = T(\mathbf{\dot{q}}, t)-V(\mathbf{q}, \mathbf{\dot{q}}, t)</math>

of the physical system between two times ''t''<sub>1</sub> and ''t''<sub>2</sub>. The kinetic energy of the system is ''T'' (a function of the rate of change of the ] of the system), and ] is ''V'' (a function of the configuration and its rate of change). The configuration of a system which has ''N'' ] is defined by ] '''q''' = (''q''<sub>1</sub>, ''q''<sub>2</sub>, ... ''q<sub>N</sub>'').

There are ] conjugate to these coordinates, '''p''' = (''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p<sub>N</sub>''), where:

:<math>p_i = \frac{\partial L}{\partial \dot{q}_i}</math>

The action and Lagrangian both contain the dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the ] in the ], i.e. the curve '''q'''(''t''), parameterized by time (see also ] for this concept).

The action is a '']'' rather than a '']'', since it depends on the Lagrangian, and the Lagrangian depends on the path '''q'''(''t''), so the action depends on the ''entire'' "shape" of the path for all times (in the time interval from ''t''<sub>1</sub> to ''t''<sub>2</sub>). Between two instants of time, there are infinitely many paths, but one for which the action is stationary (to the first order) is the true path. The stationary value for the ''entire continuum'' of Lagrangian values corresponding to some path, ''not just one value'' of the Lagrangian, is required (in other words it is ''not'' as simple as "differentiating a function and setting it to zero, then solving the equations to find the points of ] etc", rather this idea is applied to the entire "shape" of the function, see ] for more details on this procedure).<ref>Feynman Lectures on Physics: Volume 2, R.P. Feynman, R.B. Leighton, M. Sands, Addison-Wesley, 1964, {{isbn|0-201-02117-X}}</ref>

Notice ''L'' is ''not'' the total energy ''E'' of the system due to the difference, rather than the sum:

:<math>E=T+V</math>

The following<ref>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</ref><ref>Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, {{isbn|0-07-084018-0}}</ref> 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.

:{| class="wikitable" align="center"
|-
!scope="col" width="600px" colspan="2"| '''Laws of motion'''
|-
|colspan="2" |''']:'''

<math> \mathcal{S} = \int_{t_1}^{t_2} L \,\mathrm{d}t \,\!</math>
|- valign="top"
|rowspan="2" scope="col" width="300px" |'''The ]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>

Using the definition of generalized momentum, there is the symmetry:

:<math> p_i = \frac{\partial L}{\partial \dot{q}_i}\quad \dot{p}_i = \frac{\partial L}{\partial {q}_i} </math>
|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>

The Hamiltonian as a function of generalized coordinates and momenta has the general form: <br />
:<math>H (\mathbf{q}, \mathbf{p}, t) = \mathbf{p}\cdot\mathbf{\dot{q}}-L</math>
|-
|]
:<math>H \left(\mathbf{q}, \frac{\partial S}{\partial\mathbf{q}}, t\right) = -\frac{\partial S}{\partial t}</math>
|- style="border-top: 3px solid;"
|colspan="2" scope="col" width="600px" | '''Newton's laws'''

''']'''

They are low-limit solutions to ]. Alternative formulations of Newtonian mechanics are ] and ] mechanics.

The laws can be summarized by two equations (since the 1st is a special case of the 2nd, zero resultant acceleration):

:<math> \mathbf{F} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}, \quad \mathbf{F}_{ij}=-\mathbf{F}_{ji} </math>

where '''p''' = momentum of body, '''F'''<sub>''ij''</sub> = force ''on'' body ''i'' ''by'' body ''j'', '''F'''<sub>''ji''</sub> = force ''on'' body ''j'' ''by'' body ''i''.

For a dynamical system the two equations (effectively) combine into one:

:<math> \frac{\mathrm{d}\mathbf{p}_\mathrm{i}}{\mathrm{d}t} = \mathbf{F}_{E} + \sum_{\mathrm{i} \neq \mathrm{j}} \mathbf{F}_\mathrm{ij} \,\!</math>

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.
|-
|}

From the above, any equation of motion in classical mechanics can be derived.

;Corollaries in mechanics

*]
*]

;Corollaries in ]

Equations describing fluid flow in various situations can be derived, using the above classical equations of motion and often conservation of mass, energy and momentum. Some elementary examples follow.

*]
*]
*]
*]
*]
*]

==Laws of gravitation and relativity==

===Modern laws===

;]

Postulates of special relativity are not "laws" in themselves, but assumptions of their nature in terms of ''relative motion''.

Often two are stated as "the laws of physics are the same in all ]" and "the ] is constant". However the second is redundant, since the speed of light is predicted by ]. Essentially there is only one.

The said posulate leads to the ] – the transformation law between two ]s moving relative to each other. For any ]

:<math>A' =\Lambda A </math>

this replaces the ] law from classical mechanics. The Lorentz transformations reduce to the Galilean transformations for low velocities much less than the speed of light ''c''.

The magnitudes of 4-vectors are invariants - ''not'' "conserved", but the same for all inertial frames (i.e. every observer in an inertial frame will agree on the same value), in particular if ''A'' is the ], the magnitude can derive the famous invariant equation for mass-energy and momentum conservation (see ]):

:<math> E^2 = (pc)^2 + (mc^2)^2 </math>

in which the (more famous) ] ''E'' = ''mc''<sup>2</sup> is a special case.

;]

General relativity is governed by the ]s, which describe the curvature of space-time due to mass-energy equivalent to the gravitational field. Solving the equation for the geometry of space warped due to the mass distribution gives the ]. Using the geodesic equation, the motion of masses falling along the geodesics can be calculated.

;]

In a relatively flat spacetime due to weak gravitational fields, gravitational analogues of Maxwell's equations can be found; the '''GEM equations''', to describe an analogous '']''. They are well established by the theory, and experimental tests form ongoing research.<ref name="Gravitation and Inertia">Gravitation and Inertia, I. Ciufolini and J.A. Wheeler, Princeton Physics Series, 1995, {{isbn|0-691-03323-4}}</ref>

:{| class="wikitable" align="center"
|- valign="top"
|scope="col" width="300px"|'''] (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 = ], ''R<sub>μν</sub>'' = ], ''T<sub>μν</sub>'' = ], ''g<sub>μν</sub>'' = ]
|scope="col" width="300px"|''']:'''
:<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 ] of the ], containing the metric.
|- style="border-top: 3px solid;"
|colspan="2"| '''GEM Equations'''

If '''g''' the gravitational field and '''H''' the gravitomagnetic field, the solutions in these limits are:

:<math> \nabla \cdot \mathbf{g} = -4 \pi G \rho \,\!</math>
:<math> \nabla \cdot \mathbf{H} = \mathbf{0} \,\!</math>
:<math> \nabla \times \mathbf{g} = -\frac{\partial \mathbf{H}} {\partial t} \,\!</math>
:<math> \nabla \times \mathbf{H} = \frac{4}{c^2}\left( - 4 \pi G\mathbf{J} + \frac{\partial \mathbf{g}} {\partial t} \right) \,\!</math>

where ρ is the ] and '''J''' is the mass current density or ].
|-
|colspan="2"| In addition there is the '''gravitomagnetic Lorentz force''':
:<math>\mathbf{F} = \gamma(\mathbf{v}) m \left( \mathbf{g} + \mathbf{v} \times \mathbf{H} \right) </math>

where ''m'' is the ] of the particlce and γ is the ].
|-
|}

===Classical laws===

{{Main|Kepler's laws of planetary motion|Newton's law of gravitation}}

Kepler's Laws, though originally discovered from planetary observations (also due to ]), are true for any '']s''.<ref>2.^ Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, {{isbn|0-07-084018-0}}</ref>

:{| class="wikitable" align="center"
|- valign="top"
|scope="col" width="300px"|''']:'''
For two point masses:

:<math>\mathbf{F} = \frac{G m_1 m_2}{\left | \mathbf{r} \right |^2} \mathbf{\hat{r}} \,\!</math>

For a non uniform mass distribution of local mass density ''ρ'' ('''r''') of body of Volume ''V'', this becomes:

:<math> \mathbf{g} = G \int_{V} \frac{\mathbf{r} \rho \mathrm{d}{V}}{\left | \mathbf{r} \right |^3}\,\!</math>
|scope="col" width="300px"| ''']:'''

An equivalent statement to Newton's law is:

:<math>\nabla\cdot\mathbf{g} = 4\pi G\rho \,\!</math>
|- style="border-top: 3px solid;"
|colspan="2" scope="col" 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>

where
:<math> e = \sqrt{1- (b/a)^2} </math>
is the ] 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 ] of an ] in which the pole (origin of polar coordinate system) is positioned at a focus of the ellipse, where the orbited star is.
|-
|colspan="2" 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,
|-
|colspan="2"|'''Kepler's 3rd Law:''' The square of the orbital time period ''T'' is proportional to the cube of the semi-major axis ''a'':
:<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).
|-
|}

==Thermodynamics==

:{| class="wikitable" align="center"
|-
!colspan="2"|''']'''
|-valign="top"
|scope="col" width="150px"|''']:''' 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>
''']:''' 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" width="150px"| ''']:''' If two systems are in ] 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>

''']:'''
:As the temperature ''T'' of a system approaches absolute zero, the entropy ''S'' approaches a minimum value ''C'': as ''T''&nbsp;&rarr;&nbsp;0, ''S''&nbsp;&rarr;&nbsp;''C''.
|-
| For homogeneous systems the first and second law can be combined into the ''']''':
:<math>\mathrm{d} U = T \mathrm{d} S - P \mathrm{d} V + \sum_i \mu_i \mathrm{d}N_i \,\!</math>
|- style="border-top: 3px solid;"
|colspan="2" width="500px"|''']:''' 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>.
|-
|}

*]
*]
*], combines a number of separately developed gas laws;
**]
**]
**]
**], into one
:now improved by other ]
*] (of partial pressures)
*]
*]
*]

==Electromagnetism==

] give the time-evolution of the ] and ] fields due to ] and ] distributions. Given the fields, the ] law is the ] for charges in the fields.

:{| class="wikitable" align="center"
|- valign="top"
|scope="col" width="300px"|''']'''

'''] for electricity'''
:<math> \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} </math>
''']'''
:<math>\nabla \cdot \mathbf{B} = 0 </math>
''']'''
:<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
'''] (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" width="300px"| '''] law:'''
: <math>\mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)</math>
|- style="border-top: 3px solid;"
|colspan="2" scope="col" width="600px"| '''] (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 ]s, rather than ], see ] for details). They are modified in QED theory.
|-
|}

These equations can be modified to include ]s, and are consistent with our observations of monopoles either existing or not existing; if they do not exist, the generalized equations reduce to the ones above, if they do, the equations become fully symmetric in electric and magnetic charges and currents. Indeed, there is a duality transformation where electric and magnetic charges can be "rotated into one another", and still satisfy Maxwell's equations.

;Pre-Maxwell laws

These laws were found before the formulation of Maxwell's equations. They 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.

*]
*]
*]

;Other laws

*]
*]
*]

==Photonics==

Classically, ] is based on a ]: light travels from one point in space to another in the shortest time.

*]

In ] laws are based on approximations in Euclidean geometry (such as the ]).

*]
*], ]

In ], laws are based on physical properties of materials.

*]
*]
*]

In actuality, optical properties of matter are significantly more complex and require quantum mechanics.

== Laws of quantum mechanics==

Quantum mechanics has its roots in ]. This leads to results which are not usually called "laws", but hold the same status, in that all of quantum mechanics follows from them.

One postulate that a particle (or a system of many particles) is described by a ], and this satisfies a quantum wave equation: namely the ] (which can be written as a non-relativistic wave equation, or a ]). Solving this wave equation predicts the time-evolution of the system's behaviour, analogous to solving Newton's laws in classical mechanics.

Other postulates change the idea of physical observables; using ]; some measurements can't be made at the same instant of time (]s), particles are fundamentally indistinguishable. Another postulate; the ] postulate, counters the usual idea of a measurement in science.

:{| class="wikitable" align="center"
|- valign="top"
| width="300px"| '''], ]'''

'''] (general form):''' Describes the time dependence of a quantum mechanical system.
:<math> i\hbar \frac{d}{dt} \left| \psi \right\rangle = \hat{H} \left| \psi \right\rangle </math>

The ] (in quantum mechanics) ''H'' is a ] acting on the state space, <math>| \psi \rangle </math> (see ]) is the instantaneous ] at time ''t'', position '''r''', ''i'' is the unit ], ''ħ'' = ''h''/2π is the reduced ].
|rowspan="2" scope="col" width="300px"|''']'''

''']:''' the ] of ]s is proportional to the ] of the light (the constant is ], ''h'').
:<math> E = h\nu = \hbar \omega </math>

''']length:''' this laid the foundations of wave–particle duality, and was the key concept in the ],
:<math> \mathbf{p} = \frac{h}{\lambda}\mathbf{\hat{k}} = \hbar \mathbf{k}</math>

''']:''' ] in position multiplied by uncertainty in ] is at least half of the ], similarly for time and ];
:<math>\Delta x \Delta p \ge \frac{\hbar}{2},\, \Delta E \Delta t \ge \frac{\hbar}{2} </math>

The uncertainty principle can be generalized to any pair of observables - see main article.
|-
| '''Wave mechanics'''

'''] (original form):'''
:<math> i\hbar \frac{\partial}{\partial t}\psi = -\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi </math>
|- style="border-top: 3px solid;"
|colspan="2" width="600px"| ''']:''' No two identical ]s can occupy the same quantum state (]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 ] of the particle. There is no way to keep track of particles physically, labels are only used mathematically to prevent confusion.
|-
|}

==Radiation laws==

Applying electromagnetism, thermodynamics, and quantum mechanics, to atoms and molecules, some laws of ] and light are as follows.

*]
*]
*]
*]

== Laws of chemistry ==
{{Main|Chemical law}}

'''Chemical laws''' are those ] relevant to ]. Historically, observations lead to many empirical laws, though now it is known that chemistry has its foundations in ].

;]

The most fundamental concept in chemistry is the ], which states that there is no detectable change in the quantity of matter during an ordinary ]. Modern physics shows that it is actually ] that is conserved, and that ]; a concept which becomes important in ]. ] leads to the important concepts of ], ], and ].

Additional laws of chemistry elaborate on the law of conservation of mass. ]'s ] says that pure chemicals are composed of elements in a definite formulation; we now know that the structural arrangement of these elements is also important.

]'s ] says that these chemicals will present themselves in proportions that are small whole numbers (i.e. 1:2 for ]:] ratio in water); although in many systems (notably ] and ]) the ratios tend to require large numbers, and are frequently represented as a fraction.

More modern laws of chemistry define the relationship between energy and its transformations.

;] and ]

* In equilibrium, molecules exist in mixture defined by the transformations possible on the timescale of the equilibrium, and are in a ratio defined by the intrinsic energy of the molecules—the lower the intrinsic energy, the more abundant the molecule. ] states that the system opposes changes in conditions from equilibrium states, i.e. there is an opposition to change the state of an equilibrium reaction.
* Transforming one structure to another requires the input of energy to cross an energy barrier; this can come from the intrinsic energy of the molecules themselves, or from an external source which will generally accelerate transformations. The higher the energy barrier, the slower the transformation occurs.
* There is a hypothetical intermediate, or ''transition structure'', that corresponds to the structure at the top of the energy barrier. The ] states that this structure looks most similar to the product or starting material which has intrinsic energy closest to that of the energy barrier. Stabilizing this hypothetical intermediate through chemical interaction is one way to achieve ].
* All chemical processes are reversible (law of ]) although some processes have such an energy bias, they are essentially irreversible.
* The reaction rate has the mathematical parameter known as the ]. The ] gives the temperature and ] dependence of the rate constant, an empirical law.

;]

*]
*]
*]

;Gas laws

*]
*]

;Chemical transport

*]
*]
*]

== Geophysical laws==

*]
*]
*]
*]

== See also ==
* ]
* ]
* ]
* ]

== Notes ==
{{Reflist}}

==External links==
* , a useful book in different formats containing many or the physical laws and formulae.
* , website containing most of the formulae in different disciplines.

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