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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.
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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
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| |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
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| |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== | |
| [[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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