Editing Galilean transformation

{{Short description|Concept in physics and mathematics}}
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]]. These transformations together with spatial rotations and translations in space and time form the '''inhomogeneous Galilean group''' (assumed throughout below). Without the translations in space and time the group is the '''homogeneous Galilean group'''. The Galilean group is the [[group of motions]] of [[Galilean relativity]] acting on the four dimensions of space and time, forming the '''Galilean geometry'''. This is the [[active and passive transformation|passive transformation]] point of view. In [[special relativity]] the homogeneous and inhomogeneous Galilean transformations are, respectively, replaced by the [[Lorentz transformations]] and [[Poincaré transformation]]s; conversely, the [[group contraction]] in the [[classical limit]] {{math|''c'' → ∞}} of Poincaré transformations yields Galilean transformations.

The equations below are only physically valid in a Newtonian framework, and not applicable to coordinate systems moving relative to each other at speeds approaching the [[speed of light]].

[[Galileo Galilei|Galileo]] formulated these concepts in his description of ''uniform motion''.<ref>{{harvnb|Galilei|1638i|loc=191&ndash;196 (in Italian)}}<br>{{harvnb|Galilei|1638e|loc=(in English)}}<br>{{harvnb|Copernicus|Kepler|Galilei|Newton|2002|pp=515&ndash;520}}</ref>
The topic was motivated by his 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]].

==Translation==
[[Image:Standard conf.png|right|thumb|300px|Standard configuration of coordinate systems for Galilean transformations]]
Although the transformations are named for Galileo, it is the [[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]].

The notation below describes the relationship under the Galilean transformation between the coordinates {{math|(''x'', ''y'', ''z'', ''t'')}} and {{math|(''x''′, ''y''′, ''z''′, ''t''′)}} of a single arbitrary event, as measured in two coordinate systems {{math|S}} and {{math|S′}}, in uniform relative motion ([[velocity]] {{math|''v''}}) in their common {{math|''x''}} and {{math|''x''′}} directions, with their spatial origins coinciding at time {{math|1=''t'' = ''t''′ = 0}}:<ref>{{harvnb|Mould|2002|loc=[https://books.google.com/books?id=lfGE-wyJYIUC&pg=PA42 Chapter 2 §2.6, p. 42]}}</ref><ref>{{harvnb|Lerner|1996|loc=[https://books.google.com/books?id=B8K_ym9rS6UC&pg=PA1047 Chapter 38 §38.2, p. 1046,1047]}}</ref><ref>{{harvnb|Serway|Jewett|2006|loc=[https://books.google.com/books?id=1DZz341Pp50C&pg=PA261 Chapter 9 §9.1, p. 261]}}</ref><ref>{{harvnb|Hoffmann|1983|loc=[https://books.google.com/books?id=JokgnS1JtmMC&pg=PA83 Chapter 5, p. 83]}}</ref>

:<math>x' = x - v t </math>
:<math>y' = y </math>
:<math>z' = z </math>
:<math>t' = t .</math>

Note that the last equation holds for all Galilean transformations up to addition of a constant, and expresses the assumption of a universal time independent of the relative motion of different observers.

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:
:<math>\begin{pmatrix} x' \\t' \end{pmatrix} = \begin{pmatrix} 1 & -v \\0 & 1 \end{pmatrix}\begin{pmatrix} x \\t \end{pmatrix} </math>
Though matrix representations are not strictly necessary for Galilean transformation, they provide the means for direct comparison to transformation methods in special relativity.

==Galilean transformations==
The Galilean symmetries can be uniquely written as the [[Function composition|composition]] of a ''rotation'', a ''translation'' and a ''uniform motion'' of spacetime.<ref name="mmcm">{{harvnb|Arnold|1989|p=6}}</ref> Let {{math|'''x'''}} represent a point in three-dimensional space, and {{math|''t''}} a point in one-dimensional time. A general point in spacetime is given by an ordered pair {{math|('''x''', ''t'')}}.

A uniform motion, with velocity {{math|'''v'''}}, is given by 
:<math>(\mathbf{x},t) \mapsto (\mathbf{x}+t\mathbf{v},t),</math>
where {{math|'''v''' ∈ '''R'''<sup>3</sup>}}. A translation is given by
:<math>(\mathbf{x},t) \mapsto (\mathbf{x}+\mathbf{a},t+s),</math>
where {{math|'''a''' ∈ '''R'''<sup>3</sup>}} and {{math|''s'' ∈ '''R'''}}. A rotation is given by
:<math>(\mathbf{x},t) \mapsto (R\mathbf{x},t),</math> 
where {{math|1=''R'' : '''R'''<sup>3</sup> → '''R'''<sup>3</sup>}} is an [[orthogonal transformation]].<ref name="mmcm"/>

As a [[Lie group]], the group of Galilean transformations has [[dimension]] 10.<ref name="mmcm"/>

==Galilean group==
Two Galilean transformations {{math| ''G''(''R'', '''v''', '''a''', ''s'')}} and {{math| ''G''(''R' '', '''v'''&prime;, '''a'''&prime;, ''s''&prime;)}} [[composition of functions|compose]] to form a third Galilean transformation, 
:{{math|1= ''G''(''R''&prime;, '''v'''&prime;, '''a'''&prime;, ''s''&prime;) ⋅ ''G''(''R'', '''v''', '''a''', ''s'') = ''G''(''R&prime; R'', ''R''&prime; '''v''' + '''v'''&prime;, ''R''&prime; '''a''' + '''a'''&prime; + '''v'''&prime; ''s'', ''s''&prime; + ''s'')}}. 
The set of all Galilean transformations {{math|Gal(3)}} forms a [[Group (mathematics)|group]] with composition as the group operation.

The group is sometimes represented as a matrix group with [[spacetime]] events {{math|('''x''', ''t'', 1)}} as vectors where {{math|''t''}} is real and {{math|'''x''' ∈ '''R'''<sup>3</sup>}} is a position in space. 
The [[Group action (mathematics)|action]] is given by<ref>[http://www.emis.de/journals/APPS/v11/A11-na.pdf]{{harvnb|Nadjafikhah|Forough|2009}}</ref>
:<math>\begin{pmatrix}R & v & a \\ 0 & 1 & s \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix}  x\\ t\\ 1\end{pmatrix}   = \begin{pmatrix}  R x+vt  +a\\ t+s\\   1\end{pmatrix},</math>
where {{math|''s''}} is real and {{math|''v'', ''x'', ''a'' ∈ '''R'''<sup>3</sup>}} and {{math|''R''}} is a [[rotation matrix]]. 
The composition of transformations is then accomplished through [[matrix multiplication]]. Care must be taken in the discussion whether one restricts oneself to the connected component group of the orthogonal transformations.

{{math|Gal(3)}} has named subgroups. The identity component is denoted {{math|SGal(3)}}.

Let {{math|''m''}} represent the transformation matrix with parameters {{math|''v'', ''R'', ''s'', ''a''}}:
* <math>\{ m : R = I_3 \} , </math> anisotropic transformations.
* <math>\{ m : s = 0 \} , </math> isochronous transformations.
* <math>\{ m : s = 0, v = 0 \} , </math> spatial Euclidean transformations.
* <math>G_1 = \{ m : s = 0, a = 0 \},</math> uniformly special transformations / homogeneous transformations, isomorphic to Euclidean transformations.
* <math>G_2 = \{ m :  v = 0,  R = I_3  \} \cong \left(\mathbf{R}^4, +\right),</math> shifts of origin / translation in Newtonian spacetime.
* <math>G_3 = \{ m : s = 0, a = 0, v = 0 \} \cong \mathrm{SO}(3),</math> rotations (of reference frame) (see [[SO(3)]]), a compact group. 
* <math>G_4 = \{ m : s = 0, a = 0, R = I_3 \} \cong \left(\mathbf{R}^3, +\right),</math> uniform frame motions / boosts.

The parameters {{math|''s'', ''v'', ''R'', ''a''}} span ten dimensions. Since the transformations depend continuously on {{math|''s'', ''v'', ''R'', ''a''}}, {{math|Gal(3)}} is a [[continuous group]], also called a topological group.

The structure of {{math|Gal(3)}} can be understood by reconstruction from subgroups. The [[semidirect product]] combination (<math>A \rtimes B </math>) of groups is required. 
#<math>G_2 \triangleleft \mathrm{SGal}(3)</math> ({{math|''G''<sub>2</sub>}} is a [[normal subgroup]])
#<math>\mathrm{SGal}(3) \cong G_2 \rtimes G_1</math>
#<math>G_4 \trianglelefteq G_1</math>
#<math>G_1 \cong G_4 \rtimes G_3</math>
#<math>\mathrm{SGal}(3) \cong \mathbf{R}^4 \rtimes (\mathbf{R}^3 \rtimes \mathrm{SO}(3)) .</math>

==Origin in group contraction==
The [[Lie algebra]] of the [[Representation theory of the Galilean group|Galilean group]] is [[linear span|spanned]] by {{math|''H'', ''P<sub>i</sub>'', ''C<sub>i</sub>''}} and {{math|''L<sub>ij</sub>''}} (an [[antisymmetric tensor]]), subject to [[commutator|commutation relations]], where
:<math>[H,P_i]=0 </math>
:<math>[P_i,P_j]=0 </math>
:<math>[L_{ij},H]=0 </math>
:<math>[C_i,C_j]=0 </math>
:<math>[L_{ij},L_{kl}]=i [\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+\delta_{jl}L_{ik}] </math>
:<math>[L_{ij},P_k]=i[\delta_{ik}P_j-\delta_{jk}P_i] </math>
:<math>[L_{ij},C_k]=i[\delta_{ik}C_j-\delta_{jk}C_i] </math>
:<math>[C_i,H]=i P_i \,\!</math>
:<math>[C_i,P_j]=0 ~.</math>
{{mvar|H}} is the generator of time translations ([[Hamiltonian (quantum mechanics)|Hamiltonian]]), {{math|''P<sub>i</sub>''}} is the generator of translations ([[momentum operator]]), {{math|''C<sub>i</sub>''}} is the generator of rotationless Galilean transformations (Galileian boosts),<ref>{{cite book |title=Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces |edition=illustrated |first1=A. A. |last1=Ungar |publisher=Springer Science & Business Media |year=2006 |isbn=978-0-306-47134-6 |page=336 |url=https://books.google.com/books?id=MTTaBwAAQBAJ}} [https://books.google.com/books?id=MTTaBwAAQBAJ&pg=PA336 Extract of page 336]</ref> and {{math|''L<sub>ij</sub>''}} stands for a generator of rotations ([[angular momentum operator]]).

This Lie Algebra is seen to be a special [[classical limit]] of the algebra of the [[Poincaré group#Technical explanation|Poincaré group]], in the limit {{math|''c'' → ∞}}. Technically, the Galilean group is a celebrated [[group contraction]] of the Poincaré group (which, in turn, is a [[group contraction]] of the de Sitter group {{math|SO(1,4)}}).<ref>{{harvnb|Gilmore|2006}}</ref>
Formally, renaming the generators of momentum and boost of the latter as in
:{{math|''P''<sub>0</sub> ↦ ''H'' / ''c''}} 
:{{math|''K<sub>i</sub>'' ↦ ''c'' ⋅ ''C<sub>i</sub>''}}, 
where {{math|''c''}} is the speed of light (or any unbounded function thereof), the commutation relations (structure constants) in the limit {{math|''c'' → ∞}} take on the relations of the former. 
Generators of time translations and rotations are identified. Also note the group invariants {{math|''L''<sub>''mn''</sub> ''L''<sup>''mn''</sup>}} and  {{math|''P''<sub>''i''</sub> ''P''{{i sup|''i''}}}}.

In matrix form, for {{math|1=''d'' = 3}}, one may consider the ''regular representation'' (embedded in {{math|GL(5; '''R''')}}, from which it could be derived by a single group contraction, bypassing the Poincaré group), 
:
<math>
 iH=   \left( {\begin{array}{ccccc}
   0 &  0 &  0 & 0 &  0\\
 0 &  0 &  0 &  0 &  0\\
 0 &  0 &  0 &  0 &  0\\
   0 & 0 & 0 & 0 & 1\\
   0 & 0 & 0 & 0 & 0\\
  \end{array} } \right)    , \qquad            
   </math>
<math>
  i\vec{a}\cdot\vec{P}= 
  \left( {\begin{array}{ccccc}
    0&0&0&0 & a_1\\
   0&0&0&0  & a_2\\
   0&0&0&0  & a_3\\
   0 & 0 & 0 & 0& 0\\
   0 & 0 & 0 & 0 & 0\\
  \end{array} } \right), \qquad
  </math>
<math>
  i\vec{v}\cdot\vec{C}= 
  \left( {\begin{array}{ccccc}
  0 & 0 & 0 & v_1 & 0\\
  0 & 0 & 0 & v_2 & 0\\
  0 & 0 & 0 & v_3 & 0\\
   0 & 0 & 0 & 0 & 0\\
   0 & 0 & 0 & 0 & 0\\
  \end{array} } \right), \qquad
  </math>
<math>
i \theta_i \epsilon^{ijk} L_{jk} = 
  \left( {\begin{array}{ccccc}
   0& \theta_3 & -\theta_2 & 0 & 0\\
   -\theta_3 & 0 &  \theta_1& 0 & 0\\
   \theta_2 & -\theta_1 & 0 & 0 & 0\\
   0 & 0 & 0 & 0 & 0\\
   0 & 0 & 0 & 0 & 0\\
  \end{array} } \right )  ~.  </math>

The infinitesimal group element is then 
::<math>
G(R,\vec{v},\vec{a},s)=1\!\!1_5   +       \left( {\begin{array}{ccccc}
   0& \theta_3 & -\theta_2 & v_1& a_1\\      -\theta_3 & 0 &  \theta_1& v_2 & a_2\\
   \theta_2 & -\theta_1 & 0 & v_3 & a_3\\      0 & 0 & 0 & 0 & s\\
   0 & 0 & 0 & 0 & 0\\     \end{array} } \right )  +\ ... ~.  </math>

== Central extension of the Galilean group ==
One may consider<ref>{{harvnb|Bargmann|1954}}</ref> a [[Lie algebra extension#Central|central extension]] of the Lie algebra of the Galilean group, spanned by {{math|''H''′, ''P''′<sub>''i''</sub>, ''C''′<sub>''i''</sub>, ''L''′<sub>''ij''</sub>}} and an operator ''M'': 
The so-called '''Bargmann algebra''' is obtained by imposing <math>[C'_i,P'_j]=i M\delta_{ij}</math>, such that {{math|''M''}} lies in the [[center (algebra)|center]], i.e. [[Commutative operation|commute]]s with all other operators.

In full, this algebra is given as
:<math>[H',P'_i]=0 \,\!</math>
:<math>[P'_i,P'_j]=0 \,\!</math>
:<math>[L'_{ij},H']=0 \,\!</math>
:<math>[C'_i,C'_j]=0 \,\!</math>
:<math>[L'_{ij},L'_{kl}]=i [\delta_{ik}L'_{jl}-\delta_{il}L'_{jk}-\delta_{jk}L'_{il}+\delta_{jl}L'_{ik}] \,\!</math>
:<math>[L'_{ij},P'_k]=i[\delta_{ik}P'_j-\delta_{jk}P'_i] \,\!</math>
:<math>[L'_{ij},C'_k]=i[\delta_{ik}C'_j-\delta_{jk}C'_i] \,\!</math>
:<math>[C'_i,H']=i P'_i \,\!</math>
and finally 
:<math>[C'_i,P'_j]=i M\delta_{ij} ~.</math>
where the new parameter <math>M</math> shows up. 
This extension and [[projective representation]]s that this enables is determined by its [[Group cohomology#Projective representations and group extensions|group cohomology]].

==See also==
*[[Galilean invariance]]
*[[Representation theory of the Galilean group]]
*[[Galilei-covariant tensor formulation]]
*[[Poincaré group]]
*[[Lorentz group]]
*[[Lagrangian and Eulerian coordinates]]

==Notes==
{{Reflist}}

==References==
*{{cite book|last1=Arnold|first1=V. I.|author-link=Vladimir Arnold|title=Mathematical Methods of Classical Mechanics|publisher=Springer-Verlag|date=1989|edition=2|isbn=0-387-96890-3|page=[https://archive.org/details/mathematicalmeth0000arno/page/6 6]|url=https://archive.org/details/mathematicalmeth0000arno/page/6}}
*{{cite journal|last=Bargmann|first=V.|author-link=Valentine Bargmann|year=1954|title=On Unitary Ray Representations of Continuous Groups|journal=Annals of Mathematics|series=2|volume=59|issue=1|pages=1&ndash;46|doi=10.2307/1969831|jstor=1969831}}
*{{cite book|year=2002|first1=Nicolaus|last1=Copernicus|author-link1=Nicolaus Copernicus|first2=Johannes|last2=Kepler|author-link2=Johannes Kepler|first3=Galileo|last3=Galilei|author-link3=Galileo Galilei|first4=Isaac|last4=Newton|author-link4=Isaac Newton|first5=Albert|last5=Einstein|author-link5=Albert Einstein|editor-last=Hawking|editor-first=Stephen|editor-link=Stephen Hawking|pages=[https://archive.org/details/isbn_9780762413485/page/515 515&ndash;520]|title=On the Shoulders of Giants: The Great Works of Physics and Astronomy|isbn=0-7624-1348-4|publisher=[[Running Press]]|location=Philadelphia, London|url-access=registration|url=https://archive.org/details/isbn_9780762413485/page/515}}
*{{cite book|last=Galilei|first=Galileo|author-link=Galileo Galilei|year=1638i|title=Discorsi e Dimostrazioni Matematiche, intorno á due nuoue scienze|pages=191&ndash;196|publisher=[[Elsevier]]|location=Leiden|language=it}}
*{{cite book|first=Galileo|last=Galilei|year=1638e|title=[[Discourses and Mathematical Demonstrations Relating to Two New Sciences]]|trans-title=Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze|others=Translated to English 1914 by [[Henry Crew]] and Alfonso de Salvio}}
*{{cite book|last=Gilmore|first=Robert|year=2006|title=Lie Groups, Lie Algebras, and Some of Their Applications|publisher=[[Dover Publications]]|series=Dover Books on Mathematics|isbn=0486445291}}
*{{citation|title=Relativity and Its Roots|first1=Banesh|last1=Hoffmann|publisher=Scientific American Books|year=1983|isbn=0-486-40676-8 |url=https://books.google.com/books?id=JokgnS1JtmMC&pg=PA83}}, [https://books.google.com/books?id=JokgnS1JtmMC&pg=PA83 Chapter 5, p. 83]
*{{citation|title=Physics for Scientists and Engineers|volume= 2|first1=Lawrence S.|last1=Lerner|publisher=Jones and Bertlett Publishers, Inc|year=1996|isbn=0-7637-0460-1|url=https://books.google.com/books?id=B8K_ym9rS6UC&pg=PA1047}}, [https://books.google.com/books?id=B8K_ym9rS6UC&pg=PA1047 Chapter 38 §38.2, p. 1046,1047]
*{{citation|title=Basic relativity|first1=Richard A.|last1=Mould|publisher=Springer-Verlag|year=2002|isbn=0-387-95210-1|url=https://books.google.com/books?id=lfGE-wyJYIUC&pg=PA42}}, [https://books.google.com/books?id=lfGE-wyJYIUC&pg=PA42 Chapter 2 §2.6, p. 42]
*{{cite journal |first=Mehdi|last1=Nadjafikhah|first2=Ahmad-Reza|last2=Forough|year=2009|title=Galilean Geometry of Motions|journal=Applied Sciences| volume=11|pages= 91–105|url=http://www.emis.de/journals/APPS/v11/A11-na.pdf}}
*{{citation|title=Principles of Physics: A Calculus-based Text|edition=4th|first1=Raymond A.|last1=Serway|first2=John W.|last2=Jewett|publisher=Brooks/Cole - Thomson Learning|year=2006|bibcode=2006ppcb.book.....J|isbn=0-534-49143-X|url=https://books.google.com/books?id=1DZz341Pp50C&pg=PA261}}, [https://books.google.com/books?id=1DZz341Pp50C&pg=PA261 Chapter 9 §9.1, p. 261]

{{Galileo Galilei}}
{{Relativity}}
{{Authority control}}

[[Category:Theoretical physics]]
[[Category:Time in physics]]