Editing Galilean transformation (section)

==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"/>