Editing Galilean transformation (section)

==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>