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| {{hatnote|This article discusses the [[Science|scientific]] and [[mathematics|mathematical]] details, for the [[philosophy]] of scientific laws see [[scientific law]].}}
| | == 'Xiao Yan little brother is not the Central Plains people == |
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| {{Refimprove|date=February 2008}}
| | And 'medicine' division of the benefits of it, beat to death a rival, is the most frightening.<br><br>'Yes, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ レディース 電波ソーラー腕時計] Han Chong brother, to ask, here is what the Central Plains?' Xiao Yan hesitated, suddenly asked, he did not want pains through space channels, the results had him give Hu 'chaos' lost in the Continental somewhere to go.<br><br>'ah, here is [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ gps 時計] the Northern Plains domain.' Han Chong nodded, immediately some surprised and said: 'Xiao Yan little brother is not the Central Plains people?'<br><br>Xiao Yan smile a cry, fell not conceal the space wormhole briefly say something again.<br><br>'space storm turned [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 説明書] out to be met, no wonder that the injury would be so heavy, but this, too, good luck, I met a storm in space wormhole in space, few people can escape [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 時計 プロトレック] Han year we have [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 時計 カシオ] once a guard wearing When the storm had all the space wormhole is swallowed. 'Heard, Han Chong had just suddenly, immediately breathtaking smack smack |
| The '''laws of science''' or '''scientific laws''' are statements that describe, predict, and perhaps explain ''why'', a range of [[phenomena]] behave as they appear to in [[nature]].<ref>{{OED|law of nature}}</ref> The term "law" has diverse usage in many cases: approximate, accurate, broad or narrow theories, in all [[natural science|natural scientific]] disciplines ([[physics]], [[chemistry]], [[biology]], [[geology]], [[astronomy]] etc.). An analogous term for a scientific law is a '''[[principle]]'''.
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://bbs.hnlmjd.com/home.php?mod=space&uid=312811 http://bbs.hnlmjd.com/home.php?mod=space&uid=312811]</li> |
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| | <li>[http://www.dmrrk.com/plus/feedback.php?aid=158 http://www.dmrrk.com/plus/feedback.php?aid=158]</li> |
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| | <li>[http://www.wdhnxy.com/plus/feedback.php?aid=135 http://www.wdhnxy.com/plus/feedback.php?aid=135]</li> |
| | |
| | </ul> |
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| Scientific laws:
| | == is gradually calmed down == |
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| #'''summarize''' a large collection of facts determined by [[experiment]] into a single statement,
| | Amount, and in this world, no one did not want to have this power ...<br><br>retreat with fire snake, riots [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html casio 時計] magma to 'cave', is gradually calmed down, but to be safe, 'Xiao Yan' did not immediately set [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオの時計] off to find different fire trail, but standing [http://alleganycountyfair.org/sitemap.xml http://alleganycountyfair.org/sitemap.xml] in the air waiting for a Nearly ten minutes, confirmed that the snake is true retreat after the fire, just a sigh of relief, the two front-channel arbitrary direction and shook [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 メンズ] his hand, and then pick up the wings, magma began slowly to 'cave' Search among them.<br><br>'Teacher, this [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ gps 時計] will have a different fire traces of it exist?' repulsed the fire snake, Xiao Yan is gradually re-mastered the body, the moment softly inquired.<br><br>'look at the environment and the extent of violent energy here, and then there is the other end of the previous fire snake and animals, there should be a lot of differences there may be fire ...' 'medicine' old smiled and replied. |
| #can usually be '''formulated [[mathematics|mathematically]]''' as one or several statements or [[equation]], or at least stated in a single sentence, so that it can be used to predict the outcome of an experiment, given the [[initial conditions|initial]], [[boundary conditions|boundary]], and other physical conditions of the processes which take place,
| | 相关的主题文章： |
| #are strongly supported by '''[[empirical]] [[evidence]]''' - they are 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 like mathematical [[theorems]] or [[Identity (mathematics)|identities]], and it is always possible for a law to be overturned by future observations.
| | <ul> |
| #are often quoted as a fundamental controlling influence rather than a description of observed facts, e.g. "the laws of motion require that..."
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| | <li>[http://015o.com/plus/feedback.php?aid=261 http://015o.com/plus/feedback.php?aid=261]</li> |
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| | <li>[http://www.ytzjj.com/bbs/home.php?mod=space&uid=26964 http://www.ytzjj.com/bbs/home.php?mod=space&uid=26964]</li> |
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| | <li>[http://jinxiang.cn/tieba/forum.php?mod=viewthread&tid=80787 http://jinxiang.cn/tieba/forum.php?mod=viewthread&tid=80787]</li> |
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| | </ul> |
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| Laws differ from [[hypotheses]] and [[postulates]], which are proposed during the [[Scientific method|scientific process]] 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.
| | == the former even stronger == |
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| 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 ontological commitments nor statements of logical absolution.
| | Different atmosphere, but also [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html カシオ腕時計 g-shock] slowly between this piece appeared in the world.<br><br>'Hum!'<br><br>With this channel black silhouette of inflammation occur, it filled the emptiness swallow inflammation, suddenly issued a singular voice whining, as if [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 腕時計 チタン] to welcome with their owners [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html casio 腕時計 メンズ] in general<br><br>'nothingness swallow inflammation'<br><br>Xiao Yan's eyes, keeping a close eye reminding [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html 腕時計 casio] us of scenes that a devil-like figure, the [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 電波ソーラー時計] heart is suddenly violently beating up his perception of different fire is naturally the first thing is to see that this figure, is the inflammation of the body swallowing nothingness!<br><br>true nothingness swallow inflammation!<br><br>'This nothingness swallow inflammation, turned out to form a cross-fire of the Spirit,' 'medicine' old looked at Xiao Yan, a dry mouth, inflammation of this nothingness swallow fire with fire demon demon Joren space is the same existence, even, compared to net lotus fire demon, the former even stronger!<br><br>'good atmosphere of terror |
| | | 相关的主题文章： |
| Fundamentally, ''all'' scientific laws follow from physics, laws which occur in other sciences ultimately follow from [[physical law]]s. Often, from mathematically fundamental viewpoints, [[universal constant]]s emerge from scientific laws.
| | <ul> |
| | | |
| == Conservation laws ==
| | <li>[http://www.bbs.read-walker.com/forum.php?mod=viewthread&tid=126254 http://www.bbs.read-walker.com/forum.php?mod=viewthread&tid=126254]</li> |
| ===Conservation and symmetry===
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| | | <li>[http://237.newfine.net/home.php?mod=spacecp&ac=blog&blogid= http://237.newfine.net/home.php?mod=spacecp&ac=blog&blogid=]</li> |
| Most significant laws in science are [[conservation laws]]. These fundamental laws follow from homogeneity of [[space]], [[time]] and [[phase (waves)|phase]], in other words ''symmetry''.
| | |
| | | <li>[http://www.jumeishop.cn/plus/view.php?aid=36848 http://www.jumeishop.cn/plus/view.php?aid=36848]</li> |
| * '''[[Noether's theorem]]:''' Any quantity which has a continuous differentiable symmetry in the action has an associated conservation law.
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| | | </ul> |
| * [[Conservation of mass]] 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.
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| * '''[[Conservation of energy]]''', '''[[Conservation of momentum|momentum]]''' and '''[[Conservation of angular momentum|angular momentum]]''' for isolated systems can be found to be symmetries in time, translation, and rotation.
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| * '''[[Conservation of charge]]''' was also realized since charge has never been observed to be created or destroyed, and only found to move from place to place.
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| ===Continuity and transfer===
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| Conservation laws can be expressed using the general [[continuity equation]] (for a conserved quantity) can be written in differential form as:
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| :<math>\frac{\partial \rho}{\partial t}=-\nabla \cdot \mathbf{J} </math>
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| where ρ is some quantity per unit volume, '''J''' is the [[flux]] of that quantity (change in quantity per unit time per unit area). Intuitively, the [[divergence]] (denoted ∇•) of a [[vector field]] 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.
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| :{| class="wikitable" align="center"
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| ! scope="col" width="150" | Physics, conserved quantity
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| ! scope="col" width="140" | Conserved quantity ''q''
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| ! scope="col" width="140" | Volume density ''ρ'' (of ''q'')
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| ! scope="col" width="140" | Flux '''J''' (of ''q'')
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| ! scope="col" width="10" | Equation
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| | [[Hydrodynamics]], [[fluid]]s <br />
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| | ''m'' = [[mass]] (kg)
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| | ''ρ'' = volume [[mass density]] (kg m<sup>−3</sup>)
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| | ''ρ'' '''u''', where<br/ >
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| '''u''' = [[velocity field]] of fluid (m s<sup>−1</sup>)
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| | <math> \frac{\partial \rho}{\partial t} = - \nabla \cdot (\rho \mathbf{u}) </math>
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| |-
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| | [[Electromagnetism]], [[electric charge]]
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| | ''q'' = electric charge (C)
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| | ''ρ'' = volume electric [[charge density]] (C m<sup>−3</sup>)
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| | '''J''' = electric [[current density]] (A m<sup>−2</sup>)
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| | <math> \frac{\partial \rho}{\partial t} = - \nabla \cdot \mathbf{J} </math>
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| |-
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| | [[Thermodynamics]], [[energy]]
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| | ''E'' = energy (J)
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| | ''u'' = volume [[energy density]] (J m<sup>−3</sup>)
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| | '''q''' = [[heat flux]] (W m<sup>−2</sup>)
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| | <math> \frac{\partial u}{\partial t}=- \nabla \cdot \mathbf{q} </math>
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| |-
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| | [[Quantum mechanics]], [[probability]]
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| || ''P'' = ('''r''', ''t'') = ∫|Ψ|<sup>2</sup>d<sup>3</sup>'''r''' = [[probability distribution]]
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| || ''ρ'' = ''ρ''('''r''', ''t'') = |Ψ|<sup>2</sup> = [[probability density function]] (m<sup>−3</sup>),<br />
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| Ψ = [[wavefunction]] of quantum system
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| || '''j''' = [[probability current]]/flux
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| | <math> \frac{\partial |\Psi|^2}{\partial t}=-\nabla \cdot \mathbf{j} </math>
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| |}
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| More general equations are the [[convection–diffusion equation]] and [[Boltzmann transport equation]], which have their roots in the continuity equation.
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| ==Laws of classical mechanics==
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| ===Principle of least action===
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| {{Main|Principle of least action}}
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| All of classical mechanics, including [[Newton's laws]], [[Lagrangian mechanics|Lagrange's equations]], [[Hamiltonian mechanics|Hamilton's equations]], etc., can be derived from this very simple principle:
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| :<math> \delta \mathcal{S} = \delta\int_{t_1}^{t_2} L(\mathbf{q}, \mathbf{\dot{q}}, t) dt = 0 </math>
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| where <math> \mathcal{S} </math> is the [[action (physics)|action]]; the integral of the [[Lagrangian]]
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| :<math> L(\mathbf{q}, \mathbf{\dot{q}}, t) = T(\mathbf{\dot{q}}, t)-V(\mathbf{q}, \mathbf{\dot{q}}, t)</math>
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| 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 [[configuration space|configuration]] of the system), and [[potential energy]] is ''V'' (a function of the configuration and its rate of change). The configuration of a system which has ''N'' [[Degrees of freedom (mechanics)|degrees of freedom]] is defined by [[generalized coordinates]] '''q''' = (''q''<sub>1</sub>, ''q''<sub>2</sub>, ... ''q<sub>N</sub>'').
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| There are [[Canonical coordinates|generalized momenta]] conjugate to these coordinates, '''p''' = (''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p<sub>N</sub>''), where:
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| :<math>p_i = \frac{\partial L}{\partial \dot{q}_i}</math>
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| 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 [[generalized coordinates]] in the [[configuration space]], i.e. the curve '''q'''(''t''), parameterized by time (see also [[parametric equation]] for this concept).
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| The action is a ''[[functional (mathematics)|functional]]'' rather than a ''[[function (mathematics)|function]]'', 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 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 its ''not'' as simple as "differentiating a function and setting it to zero, then solving the equations to find the points of [[maxima and minima]] etc", rather this idea is applied to the entire "shape" of the function, see [[calculus of variations]] 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>
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| Notice ''L'' is ''not'' the total energy ''E'' of the system due to the difference, rather than the sum:
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| :<math>E=T+V</math>
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| 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 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.
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| :{| class="wikitable" align="center"
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| !scope="col" width="600px" colspan="2"| '''Laws of motion'''
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| |-
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| |colspan="2" |'''[[Principle of least action]]:'''
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| <math> \mathcal{S} = \int_{t_1}^{t_2} L \mathrm{d}t \,\!</math>
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| |- valign="top"
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| |rowspan="2" scope="col" width="300px" |'''The [[Euler-Lagrange equation]]s are:'''
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| :<math> \frac{\mathrm{d}}{\mathrm{d} t} \left ( \frac{\partial L}{\partial \dot{q}_i } \right ) = -\frac{\partial L}{\partial q_i} </math>
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| Using the definition of generalized momentum, there is the symmetry:
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| :<math> p_i = \frac{\partial L}{\partial \dot{q}_i}\quad \dot{p}_i = \frac{\partial L}{\partial {q}_i} </math>
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| |width="300px"| '''Hamilton's equations'''
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| :<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>
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| | |
| The Hamiltonian as a function of generalized coordinates and momenta has the general form: <br />
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| :<math>H (\mathbf{q}, \mathbf{p}, t) = \mathbf{p}\cdot\mathbf{\dot{q}}-L</math>
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| |-
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| |[[Hamilton-Jacobi equation]]
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| :<math>H \left(\mathbf{q}, \frac{\partial S}{\partial\mathbf{q}}, t\right) = -\frac{\partial S}{\partial t}</math>
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| |- style="border-top: 3px solid;"
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| |colspan="2" scope="col" width="600px" | '''Newton's laws'''
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| '''[[Newton's laws of motion]]'''
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| They are low-limit solutions to [[theory of relativity|relativity]]. Alternative formulations of Newtonian mechanics are [[Lagrangian mechanics|Lagrangian]] and [[Hamiltonian mechanics|Hamiltonian]] mechanics.
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| The laws can be summarized by two equations (since the 1st is a special case of the 2nd, zero resultant acceleration):
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| :<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>''ij''</sub> = force ''on'' body ''j'' ''by'' body ''i''.
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| For a dynamical system the two equations (effectively) combine into one:
| |
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| :<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>
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| | |
| 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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| |-
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| |}
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| From the above, any equation of motion in classical mechanics can be derived.
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| ;Corollaries in mechanics
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| *[[Euler's laws of motion]]
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| *[[Euler's equations (rigid body dynamics)]]
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| ;Corollaries in [[fluid mechanics]]
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| 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.
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| *[[Archimedes' principle]]
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| *[[Bernoulli's principle]]
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| *[[Poiseuille's law]]
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| *[[Stoke's law]]
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| *[[Navier–Stokes equations]]
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| *[[Faxén's law]]
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| ==Laws of gravitation and relativity==
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| ===Modern laws===
| |
| | |
| ;[[Special relativity]]
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| Postulates of special relativity are not "laws" in themselves, but assumptions of their nature in terms of ''relative motion''.
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| Often two are stated as "the laws of physics are the same in all [[inertial frames]]" and "the [[speed of light]] is constant". However the second is redundant, since the speed of light is predicted by [[Maxwell's equations]]. Essentially there is only one.
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| The said posulate leads to the [[Lorentz transformations]] – the transformation law between two [[frame of reference]]s moving relative to each other. For any [[4-vector]]
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| :<math>A' =\Lambda A </math>
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| this replaces the [[Galilean transformation]] law from classical mechanics. The Lorentz transformations reduce to the Galilean transformations for low velocities much less than the speed of light ''c''.
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| 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 [[four-momentum]], the magnitude can derive the famous invariant equation for mass-energy and momentum conservation (see [[invariant mass]]):
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| | |
| :<math> E^2 = (pc)^2 + (mc^2)^2 </math>
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| | |
| in which the (more famous) [[mass-energy equivalence]] ''E'' = ''mc''<sup>2</sup> is a special case.
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| | |
| ;[[General relativity]]
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| General relativity is governed by the [[Einstein field equation]]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 [[metric tensor]]. Using the geodesic equation, the motion of masses falling along the geodesics can be calculated.
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| ;[[Gravitomagnetism]]
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| 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 ''[[Gravitomagnetism|gravitomagnetic field]]''. 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>
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| :{| class="wikitable" align="center"
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| |- valign="top"
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| |scope="col" width="300px"|'''[[Einstein field equations]] (EFE):'''
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| :<math>R_{\mu \nu} + \left ( \Lambda - \frac{R}{2} \right ) g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}\,\!</math>
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| | |
| where Λ = [[cosmological constant]], ''R<sub>μν</sub>'' = [[Ricci curvature tensor]], ''T<sub>μν</sub>'' = [[Stress-energy tensor]], ''g<sub>μν</sub>'' = [[metric tensor]]
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| |scope="col" width="300px"|'''[[Geodesic equation]]:'''
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| :<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>
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| where Γ is a [[Christoffel symbol]] of the [[Christoffel symbols#Christoffel symbols of the second kind (symmetric definition)|second kind]], containing the metric.
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| |- style="border-top: 3px solid;"
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| |colspan="2"| '''GEM Equations'''
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| If '''g''' the gravitational field and '''H''' the gravitomagnetic field, the solutions in these limits are:
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| :<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>
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| :<math> \nabla \times \mathbf{H} = \frac{4}{c^2}\left( - 4 \pi G\mathbf{J} + \frac{\partial \mathbf{g}} {\partial t} \right) \,\!</math>
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| | |
| where ρ is the [[density|mass density]] and '''J''' is the mass current density or [[mass flux]].
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| |-
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| |colspan="2"| In addition there is the '''gravitomagnetic Lorentz force''':
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| :<math>\mathbf{F} = \gamma(\mathbf{v}) m \left( \mathbf{g} + \mathbf{v} \times \mathbf{H} \right) </math>
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| | |
| where ''m'' is the [[rest mass]] of the particlce and γ is the [[Lorentz factor]].
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| |-
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| |}
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| ===Classical laws===
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| {{Main|Kepler's laws of planetary motion|Newton's law of gravitation}}
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| Kepler's Laws, though originally discovered from planetary observations (also due to [[Tycho Brahe]]), are true for any ''[[central force]]s''.<ref>2.^ Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, ISBN 07-084018-0</ref>
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| :{| class="wikitable" align="center"
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| |- valign="top"
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| |scope="col" width="300px"|'''[[Newton's law of universal gravitation]]:'''
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| For two point masses:
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| :<math>\mathbf{F} = \frac{G m_1 m_2}{\left | \mathbf{r} \right |^2} \mathbf{\hat{r}} \,\!</math>
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| For a non uniform mass distribution of local mass density ''ρ'' ('''r''') of body of Volume ''V'', this becomes:
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| :<math> \mathbf{g} = G \int_{V} \frac{\mathbf{r} \rho \mathrm{d}{V}}{\left | \mathbf{r} \right |^3}\,\!</math>
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| |scope="col" width="300px"| '''[[Gauss' law for gravity]]:'''
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| An equivalent statement to Newton's law is:
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| :<math>\nabla\cdot\mathbf{g} = 4\pi G\rho \,\!</math>
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| |- style="border-top: 3px solid;"
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| |colspan="2" scope="col" width="600px"|'''Kepler's 1st Law:''' Planets move in an ellipse, with the star at a focus
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| :<math>\mathbf{r} = \frac{a}{1+e \cos\theta} \mathbf{\hat{r}} \,\!</math>
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| | |
| where
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| :<math> e = \sqrt{1- (b/a)^2} </math>
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| is the [[Eccentricity (mathematics)|eccentricity]] of the elliptic orbit, of semi-major axes ''a'' (planet [[Apsis|aphelion]]) and semi-minor axes ''b'' (planet [[Apsis|perihelion]]). This equation in itself is nothing physically fundamental; simply the [[Polar coordinate system|polar equation]] of an [[ellipse]] in vector form, in which the pole (origin of polar coordinate system) is positioned at a focus of the ellipse, centred on the central star.
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| |-
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| |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):
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| :<math>\frac{\mathrm{d}A}{\mathrm{d}t} = \frac{\left | \mathbf{L} \right |}{2m} \,\!</math>
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| where '''L''' is the orbital angular momentum of the particle (i.e. planet) of mass ''m'' about the focus of orbit,
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| |-
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| |colspan="2"|'''Kepler's 3rd Law:''' The square of the orbital time period ''T'' is proportional to the mean radius ''a'':
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| :<math>T^2 = \frac{4\pi^2}{G \left ( m + M \right ) }r^3\,\!</math>
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| where ''M'' is the mass of the central body (i.e. star).
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| |-
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| |}
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| ==Thermodynamics==
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| :{| class="wikitable" align="center"
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| |-
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| !colspan="2"|'''[[Laws of thermodynamics]]'''
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| |-valign="top"
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| |scope="col" width="150px"|'''[[First law of thermodynamics]]:''' The change in internal energy d''U'' in a closed system is accounted for entirely by the heat δ''Q'' absorbed by the system and the work δ''W'' done by the system:
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| :<math>\mathrm{d}U=\delta Q-\delta W\,</math>
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| '''[[Second law of thermodynamics]]:''' There are many statements of this law, perhaps the simplest is "the entropy of isolated systems never decreases",
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| :<math>\Delta S \ge 0</math>
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| meaning reversible changes have zero entropy change, irreversible process are positive, and impossible process are negative.
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| |rowspan="2" 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.
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| :<math>T_A = T_B \,, T_B=T_C \Rightarrow T_A=T_C\,\!</math>
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| | |
| '''[[Third law of thermodynamics]]:'''
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| :As the temperature ''T'' of a system approaches absolute zero, the entropy ''S'' approaches a minimum value ''C'': as ''T'' → 0, ''S'' → ''C''.
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| |-
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| | For homogeneous systems the first and second law can be combined into the '''[[Fundamental thermodynamic relation]]''':
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| :<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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| |- style="border-top: 3px solid;"
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| |colspan="2" width="500px"|'''[[Onsager reciprocal relations]]:''' sometimes called the ''Fourth Law of Thermodynamics''
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| :<math> \mathbf{J}_{u} = L_{uu}\, \nabla(1/T) - L_{ur}\, \nabla(m/T) \!</math>;
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| :<math> \mathbf{J}_{r} = L_{ru}\, \nabla(1/T) - L_{rr}\, \nabla(m/T) \!</math>.
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| |-
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| |}
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| *[[Newton's law of cooling]]
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| *[[Conduction (heat)|Fourier's law]]
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| *[[Ideal gas law]], combines a number of separately developed gas laws;
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| **[[Boyle's law]]
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| **[[Charles's law]]
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| **[[Gay-Lussac's law]]
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| **[[Avogadro's law]], into one)
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| :now improved by other [[equations of state]]
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| *[[Dalton's law]] (of partial pressures)
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| *[[Boltzmann equation]]
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| *[[Carnot's theorem]]
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| *[[Kopp's law]]
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| ==Electromagnetism==
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| | |
| [[Maxwell's equations]] give the time-evolution of the [[electric field|electric]] and [[magnetic field|magnetic]] fields due to [[electric charge]] and [[Electric current|current]] distributions. Given the fields, the [[Lorentz force]] law is the [[equation of motion]] for charges in the fields.
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| :{| class="wikitable" align="center"
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| |- valign="top"
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| |scope="col" width="300px"|'''[[Maxwell's equations]]'''
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| '''[[Gauss's law]] for electricity'''
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| :<math> \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} </math>
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| '''[[Gauss's law for magnetism]]'''
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| :<math>\nabla \cdot \mathbf{B} = 0 </math>
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| '''[[Faraday's law of induction|Faraday's law]]'''
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| :<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math> | |
| '''[[Ampère's circuital law]] (with Maxwell's correction)'''
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| :<math>\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t} \ </math>
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| |scope="col" width="300px"| '''[[Lorentz force]] law:'''
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| : <math>\mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)</math>
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| |- style="border-top: 3px solid;"
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| |colspan="2" scope="col" 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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| |-
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| |}
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| These equations can be modified to include [[magnetic monopole]]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
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| 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.
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| *[[Lenz's law]]
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| *[[Coulomb's Law]]
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| *[[Biot-Savart law]]
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| ;Other laws
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| *[[Ohm's law]]
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| *[[Kirchhoff's circuit laws|Kirchhoff's laws]]
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| *[[Joule's first law|Joule's law]]
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| ==Photonics== | |
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| Classically, [[optics]] is based on a [[variational principle]]: light travels from one point in space to another in the shortest time.
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| *[[Fermat's principle]]
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| In [[geometric optics]] laws are based on approximations in Euclidean geometry (such as the [[paraxial approximation]]).
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| *[[Law of reflection]]
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| *[[Law of refraction]], [[Snell's law]]
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| In [[physical optics]], laws are based on physical properties of materials.
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| *[[Brewster's law|Brewster's angle]]
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| *[[Malus's law]]
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| *[[Beer–Lambert law]]
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| In actuality, optical properties of matter are significantly more complex and require quantum mechanics.
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| == Laws of quantum mechanics==
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| Quantum mechanics has its roots in [[postulates of quantum mechanics|postulates]]. This leads to results which are not usually called "laws", but hold the same status, in that all of quantum mechanics follows from them.
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| One postulate that a particle (or a system of many particles) is described by a [[wavefunction]], and this satisfies a quantum wave equation: namely the [[Schrödinger equation]] (which can be written as a non-relativistic wave equation, or a [[relativistic wave equation]]). Solving this wave equation predicts the time-evolution of the system's behaviour, analogous to solving Newton's laws in classical mechanics.
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| Other postulates change the idea of physical observables; using [[operators (physics)|quantum operators]]; some measurements can't be made at the same instant of time ([[Uncertainty principle]]s), particles are fundamentally indistinguishable. Another postulate; the [[wavefunction collapse]] postulate, counters the usual idea of a measurement in science.
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| :{| class="wikitable" align="center"
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| |- valign="top"
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| | width="300px"| '''[[Quantum mechanics]], [[Quantum field theory]]'''
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| '''[[Schrödinger equation]] (general form):''' Describes the time dependence of a quantum mechanical system.
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| :<math> i\hbar \frac{d}{dt} \left| \psi \right\rangle = \hat{H} \left| \psi \right\rangle </math>
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| 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]].
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| |rowspan="2" scope="col" width="300px"|'''[[Wave-particle duality]]'''
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| | |
| '''[[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'').
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| :<math> E = h\nu = \hbar \omega </math>
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| | |
| '''[[Matter wave|De Broglie wave]]length:''' this laid the foundations of wave-particle duality, and was the key concept in the [[Schrödinger equation]],
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| :<math> \mathbf{p} = \frac{h}{\lambda}\mathbf{\hat{k}} = \hbar \mathbf{k}</math>
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| | |
| '''[[Heisenberg uncertainty principle]]:''' [[Uncertainty]] in position multiplied by uncertainty in [[momentum]] is at least half of the [[reduced Planck constant]], similarly for [[time]] and [[energy]];
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| :<math>\Delta x \Delta p \ge \frac{\hbar}{2},\, \Delta E \Delta t \ge \frac{\hbar}{2} </math>
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| | |
| The uncertainty principle can be generalized to any pair of observables - see main article.
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| |-
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| | '''Wave mechanics'''
| |
| | |
| '''[[Schrödinger equation]] (original form):'''
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| :<math> i\hbar \frac{\partial}{\partial t}\psi = -\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi </math>
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| |- style="border-top: 3px solid;"
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| |colspan="2" 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:
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| | |
| <math>\psi(\cdots\mathbf{r}_i\cdots\mathbf{r}_j\cdots) = (-1)^{2s}\psi(\cdots\mathbf{r}_j\cdots\mathbf{r}_i\cdots)</math>
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| | |
| 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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| |-
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| |}
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| | |
| ==Radiation laws== | |
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| Applying electromagnetism, thermodynamics, and quantum mechanics, to atoms and molecules, some laws of [[electromagnetic radiation]] and light are as follows.
| |
| | |
| *[[Stefan-Boltzmann law]]
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| *[[Planck's law of black body radiation]]
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| *[[Wien's displacement law]]
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| *[[Radioactive decay law]]
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| | |
| == Laws of chemistry ==
| |
| {{Main|Chemical law}}
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| | |
| '''Chemical laws''' are those [[natural law|laws of nature]] relevant to [[chemistry]]. Historically, observations lead to many empirical laws, though now it is known that chemistry has its foundations in [[quantum mechanics]].
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| | |
| ;[[Quantitative analysis (chemistry)|Quantitative analysis]]
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| | |
| The most fundamental concept in chemistry is the [[law of conservation of mass]], which states that there is no detectable change in the quantity of matter during an ordinary [[chemical reaction]]. Modern physics shows that it is actually [[energy]] that is conserved, and that [[Mass–energy equivalence|energy and mass are related]]; a concept which becomes important in [[nuclear chemistry]]. [[Conservation of energy]] leads to the important concepts of [[Chemical equilibrium|equilibrium]], [[thermodynamics]], and [[chemical kinetics|kinetics]].
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| | |
| Additional laws of chemistry elaborate on the law of conservation of mass. [[Joseph Proust]]'s [[law of definite composition]] 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.
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| | |
| [[John Dalton|Dalton]]'s [[law of multiple proportions]] says that these chemicals will present themselves in proportions that are small whole numbers (i.e. 1:2 for [[Oxygen]]:[[Hydrogen]] ratio in [[water]]); although in many systems (notably [[Biomolecule|biomacromolecules]] and [[minerals]]) the ratios tend to require large numbers, and are frequently represented as a fraction.
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| | |
| More modern laws of chemistry define the relationship between energy and its transformations.
| |
| | |
| ;[[Reaction kinetics]] and [[Chemical equilibrium|Equilibria]]
| |
| | |
| * 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. [[Le Chatelier's principle]] 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.
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| * There is a hypothetical intermediate, or ''transition structure'', that corresponds to the structure at the top of the energy barrier. The [[Hammond's postulate|Hammond–Leffler postulate]] 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 [[catalysis]].
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| * All chemical processes are reversible (law of [[microscopic reversibility]]) although some processes have such an energy bias, they are essentially irreversible.
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| * The reaction rate has the mathematical parameter known as the [[rate constant]]. The [[Arrhenius equation]] gives the temperature and [[activation energy]] dependence of the rate constant, an empirical law.
| |
| | |
| ;[[Thermochemistry]]
| |
| | |
| *[[Dulong–Petit law]]
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| *[[Gibbs–Helmholtz equation]]
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| *[[Hess's law]]
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| | |
| ;Gas laws
| |
| | |
| *[[Raoult's law]]
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| *[[Henry's law]]
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| | |
| ;Chemical transport
| |
| | |
| *[[Fick's laws of diffusion]]
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| *[[Graham's law]]
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| *[[Lamm equation]]
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| | |
| == Geophysical laws==
| |
| | |
| *[[Archie's law]]
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| *[[Buys-Ballot's law]]
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| *[[Birch's law]]
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| *[[Byerlee's law]]
| |
| | |
| ==Biological laws==
| |
| | |
| *Life is based on [[Cell (biology)|cells]].<ref name=nurse>{{cite video|url=http://www.guardian.co.uk/science/video/2010/nov/05/paul-nurse-life-information-networks|title=Sir Paul Nurse: Organisms are information networks|author=Paul Nurse|time=15:20|publisher=Newton via ''The Guardian''|date=2010-10-12|accessdate=2012-06-15}}</ref>
| |
| *All life has [[gene]]s.<ref name=nurse/>
| |
| *All life occurs through [[biochemistry]].<ref name=nurse/>
| |
| *[[Mendelian inheritance]]
| |
| | |
| == See also ==
| |
| * [[List of laws]]
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| * [[List of scientific laws named after people]]
| |
| | |
| == Notes ==
| |
| {{Reflist}}
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| | |
| ==External links==
| |
| * [http://www.xs4all.nl/~johanw/contents.html Physics Formulary], a useful book in different formats containing many or the physical laws and formulae.
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| * [http://www.eformulae.com/ Eformulae.com], website containing most of the formulae in different disciplines.
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| [[Category:Science-related lists|Laws in science]]
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| [[Category:Units of information (cognitive processes)]]
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| [[sl:Seznam fizikalnih zakonov]]
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| [[uk:Закон природи]]
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