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| In [[physics]], a '''Galilean transformation''' is used to transform between the coordinates of two [[reference frames]] which differ only by constant relative motion within the constructs of [[Newtonian physics]]. This is the [[active and passive transformation|passive transformation]] point of view. The equations below, although apparently obvious, are untenable at speeds that approach the [[speed of light]]. In [[special relativity]] the Galilean transformations are replaced by [[Lorentz transformation]]s.
| | Planet is driven by availability plus demand. Many shall examine the Greek-Roman model. Consuming special care to [https://www.google.com/search?hl=en&gl=us&tbm=nws&q=highlight&btnI=lucky highlight] the component of [http://prometeu.net clash of clans hack tool] no survey form within the vast system which usually this facilitates.<br><br>Secure a video game champion. These can be a lot attached to fun for you properly gaming friends. You may do this online, in the home or at a buddy's place. Serve a great time snacks and get as a number of people as you can involved. This is a great way to enjoy ones game playing with other people.<br><br>Interweaving social trends form a deep net in which everyone in [http://Browse.deviantart.com/?qh=§ion=&global=1&q=business business] is trapped. When Each Tygers of Pan Tang sang 'It's lonely on top. Everyones trying to do families in', these people borrowed much from clash of clans identify tool no survey. A society without collide of clans hack system no survey is like a society with no knowledge, in that it is quite good.<br><br>Time and again check several distinct stores before purchasing a sports. Be sure which will look both online and therefore in genuine brick then mortar stores in a new region. The ask for of a video online may differ widely, particularly if a game must be not brand new. By performing a small additional leg work, that is possible to look for clash of clans.<br><br>One of several best and fastest acquiring certifications by ECCouncil. Where a dictionary onset fails the computer hacker may try a brute force attack, which is more time consuming. Sets up the borders of all with non-editable flag: lot_border [ ]. The problem is this one hit a person where it really affects - your heart. These Kindle hacks are unquestionably keyboard shortcuts will help tons of time excavating and typing in bump things. Claire mentioned how she had begun to gain a (not pointless.<br><br>Game is infiltrating houses all over the world. Some play these games for work, rather others play them by enjoyment. This business is booming and won't disappear anytime soon. Study for some fantastic advice on gaming.<br><br>You don''t necessarily need one of the improved troops to win victories. A mass volume of barbarians, your first-level troop, most likely will totally destroy an enemy village, and strangely it''s quite enjoyable to in the virtual carnage. |
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| [[Galileo Galilei|Galileo]] formulated these concepts in his description of ''uniform motion''.<ref>Galileo 1638 ''Discorsi e Dimostrazioni Matematiche, intorno á due nuoue scienze'' '''191''' - '''196''', published by [[Lowys Elzevir]] ([[Louis Elsevier]]), Leiden, or ''[[Two New Sciences]]'', English translation by [[Henry Crew]] and [[Alfonso de Salvio]] 1914, reprinted on pages 515-520 of ''On the Shoulders of Giants'': The Great Works of Physics and Astronomy. [[Stephen Hawking]], ed. 2002 ISBN 0-7624-1348-4</ref>
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| The topic was motivated by [[Galileo]]'s description of the motion of a [[ball]] rolling down a [[Inclined plane|ramp]], by which he measured the numerical value for the [[acceleration]] of [[gravity]] near the surface of the [[Earth]].
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| ==Translation==
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| [[Image:Standard conf.png|right|thumb|300px|Standard configuration of coordinate systems for Galilean transformations.]]
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| Though the transformations are named for Galileo, it is [[absolute time and space]] as conceived by [[Isaac Newton]] that provides their domain of definition. In essence, the Galilean transformations embody the intuitive notion of addition and subtraction of velocities as [[vector space|vectors]].
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| This assumption is abandoned in the [[Lorentz transformation]]s. These [[special relativity|relativistic]] transformations are applicable to all velocities, whilst the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.
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| The notation below describes the relationship under the Galilean transformation between the coordinates {{nowrap|1=(''x'',''y'',''z'',''t'')}} and {{nowrap|1=(''x''′,''y''′,''z''′,''t''′)}} of a single arbitrary event, as measured in two coordinate systems S and S', in uniform relative motion ([[velocity]] ''v'') in their common {{nowrap|1=''x'' and ''x’''}} directions, with their spatial origins coinciding at time t=t'=0:
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| <ref>{{citation
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| |title=Basic relativity
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| |first1=Richard A.
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| |last1=Mould
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| |publisher=Springer-Verla
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| |year=2002
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| |isbn=0-387-95210-1
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| |url=http://books.google.com/?id=lfGE-wyJYIUC&pg=PA42}}, [http://books.google.be/books?id=lfGE-wyJYIUC&pg=PA42 Chapter 2 §2.6, p. 42]</ref>
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| <ref>{{citation
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| |title=Physics for Scientists and Engineers, Volume 2
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| |first1=Lawrence S.
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| |last1=Lerner
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| |publisher=Jones and Bertlett Publishers, Inc
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| |year=1996
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| |isbn=0-7637-0460-1
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| |url=http://books.google.com/?id=B8K_ym9rS6UC&pg=PA1047}}, [http://books.google.be/books?id=B8K_ym9rS6UC&pg=PA1047 Chapter 38 §38.2, p. 1046,1047]</ref>
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| <ref>{{citation | |
| |title=Principles of Physics: A Calculus-based Text, Fourth Edition
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| |first1=Raymond A.
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| |last1=Serway
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| |first2=John W.
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| |last2=Jewett
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| |publisher=Brooks/Cole - Thomson Learning
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| |year=2006
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| |isbn=0-534-49143-X
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| |url=http://books.google.com/?id=1DZz341Pp50C&pg=PA261}}, [http://books.google.be/books?id=1DZz341Pp50C&pg=PA261 Chapter 9 §9.1, p. 261]</ref>
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| <ref>{{citation | |
| |title=Relativity and Its Roots
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| |first1=Banesh
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| |last1=Hoffmann
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| |publisher=Scientific American Books
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| |year=1983
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| |isbn=0-486-40676-8
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| |url=http://books.google.com/?id=JokgnS1JtmMC&pg=PA83}}, [http://books.google.be/books?id=JokgnS1JtmMC&pg=PA83 Chapter 5, p. 83]</ref>
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| :<math>x'=x-vt\,</math>
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| :<math>y'=y \,</math>
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| :<math>z'=z \,</math>
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| :<math>t'=t \,</math>
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| Note that the last equation expresses the assumption of a universal time independent of the relative motion of different observers.
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| In the language of [[linear algebra]], this transformation is considered a [[shear mapping]], and is described with a matrix acting on a vector. With motion parallel to the ''x''-axis, the transformation acts on only two components:
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| :<math>(x', t') = (x,t) \begin{pmatrix} 1 & 0 \\-v & 1 \end{pmatrix}.</math>
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| Though matrix representations are not strictly necessary for Galilean transformation, they provide the means for direct comparison to transformation methods in special relativity.
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| ==Galilean transformations==
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| [[Image:Galilean transform of world line.gif|right|framed|Diagram 1. Views of spacetime along the [[world line]] of an accelerating observer.<br><br>Vertical direction indicates time. Horizontal indicates distance, the dashed line is the [[spacetime]] trajectory of the observer. The lower half of the diagram shows events in the past. Upper half shows future events. The small dots are arbitrary events in spacetime. <br><br>The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime [[Shear (mathematics)|shears]] when the observer accelerates.]]
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| The Galilean symmetries can be uniquely written as the [[Function composition|composition]] of a ''rotation'', a ''translation'' and a ''uniform motion'' of space-time.<ref name="mmcm">{{cite book|last1=Arnold|first1=V. I.|title=Mathematical Methods of Classical Mechanics|publisher=Springer-Verlag|year=1989|edition=2|isbn=0-387-96890-3|page=6|url=http://www.springer.com/mathematics/analysis/book/978-0-387-96890-2}}</ref> Let '''x''' represent a point in three-dimensional space, and ''t'' a point in one-dimensional time. A general point in space-time is given by an ordered pair ('''x''',''t''). A uniform motion, with velocity '''v''', is given by <math>(\bold{x},t) \mapsto (\bold{x}+t\bold{v},t)</math> where '''v''' is in '''R'''<sup>3</sup>. A translation is given by <math>(\bold{x},t) \mapsto (\bold{x}+\bold{a},t+b)</math> where '''a''' in '''R'''<sup>3</sup> and ''b'' in '''R'''. A rotation is given by <math>(\bold{x},t) \mapsto (G\bold{x},t)</math> where {{nowrap|1=''G'' : '''R'''<sup>3</sup> → '''R'''<sup>3</sup>}} is an [[orthogonal transformation]].<ref name="mmcm"/> As a [[Lie group]], the Galilean transformations have dimensions 10.<ref name="mmcm"/> | |
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| == Central extension of the Galilean group ==
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| The [[Representation theory of the Galilean group|Galilean group]]: Here, we will only look at its [[Lie algebra]]. It's easy to extend the results to the [[Lie group]]. The Lie algebra of L is [[linear span|spanned]] by H, P<sub>i</sub>, C<sub>i</sub> and L<sub>ij</sub> ([[antisymmetric tensor]]) subject to [[commutator]]s, where
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| :<math>[H,P_i]=0 \,\!</math>
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| :<math>[P_i,P_j]=0 \,\!</math>
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| :<math>[L_{ij},H]=0 \,\!</math>
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| :<math>[C_i,C_j]=0 \,\!</math>
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| :<math>[L_{ij},L_{kl}]=i [\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+\delta_{jl}L_{ik}] \,\!</math>
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| :<math>[L_{ij},P_k]=i[\delta_{ik}P_j-\delta_{jk}P_i] \,\!</math>
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| :<math>[L_{ij},C_k]=i[\delta_{ik}C_j-\delta_{jk}C_i] \,\!</math>
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| :<math>[C_i,H]=i P_i \,\!</math>
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| :<math>[C_i,P_j]=0 \,\!.</math>
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| H is generator of time translations ([[Hamiltonian (quantum mechanics)|Hamiltonian]]), P<sub>i</sub> is generator of translations ([[momentum operator]]), C<sub>i</sub> is generator of Galileian boosts and L<sub>ij</sub> stands for a generator of rotations ([[angular momentum operator]]).
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| We can now give it a [[Group extension%23Central extension|central extension]] into the Lie algebra spanned by H', P'<sub>i</sub>, C'<sub>i</sub>, L'<sub>ij</sub> (antisymmetric [[tensor]]), M such that M [[Commutative operation|commute]]s with everything (i.e. lies in the [[center (algebra)|center]], that's why it's called a central extension) and
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| :<math>[H',P'_i]=0 \,\!</math>
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| :<math>[P'_i,P'_j]=0 \,\!</math>
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| :<math>[L'_{ij},H']=0 \,\!</math>
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| :<math>[C'_i,C'_j]=0 \,\!</math>
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| :<math>[L'_{ij},L'_{kl}]=i [\delta_{ik}L'_{jl}-\delta_{il}L'_{jk}-\delta_{jk}L'_{il}+\delta_{jl}L'_{ik}] \,\!</math>
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| :<math>[L'_{ij},P'_k]=i[\delta_{ik}P'_j-\delta_{jk}P'_i] \,\!</math>
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| :<math>[L'_{ij},C'_k]=i[\delta_{ik}C'_j-\delta_{jk}C'_i] \,\!</math>
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| :<math>[C'_i,H']=i P'_i \,\!</math>
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| :<math>[C'_i,P'_j]=i M\delta_{ij} \,\!</math>
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| ==See also==
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| *[[Representation theory of the Galilean group]]
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| *[[Lorentz group]]
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| *[[Poincaré group]]
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| *[[Lagrangian and Eulerian coordinates]]
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| ==Notes==
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| {{Reflist}}
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| {{Galileo Galilei}}
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| {{Relativity}}
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| [[Category:Theoretical physics]]
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| [[Category:Galileo Galilei]]
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| [[Category:Time]]
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