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Galilean transformation
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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. 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 passive transformation point of view. The equations below, although apparently obvious, are valid only at speeds much less than the speed of light. In special relativity the Galilean transformations are replaced by Poincaré transformations; conversely, the group contraction in the classical limit {{nowrap|c → ∞}} of Poincaré transformations yields Galilean transformations.Galileo formulated these concepts in his description of uniform motion.{{harvnb|Galilei|1638I|loc=191–196 (in Italian)}}{{harvnb|Galilei|1638E|loc=(in English)}}{{harvnb|Copernicus|Kepler|Galilei|Newton|2002|pp=515–520}}The topic was motivated by his description of the motion of a ball rolling down a ramp, by which he measured the numerical value for the acceleration of gravity near the surface of the Earth.

Translation

missing image!
- Standard conf.png -
Standard configuration of coordinate systems for Galilean transformations.
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 vectors.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 x and x′ directions, with their spatial origins coinciding at time {{nowrap|1=t = t′ = 0}}:{{harvnb|Mould|2002|loc=Chapter 2 §2.6, p. 42}}{{harvnb|Lerner|2996|loc=Chapter 38 §38.2, p. 1046,1047}}{{harvnb|Serway|2006|loc=Chapter 9 §9.1, p. 261}}{{harvnb|Hoffmann|1983|loc=Chapter 5, p. 83}}
x' = x - v t y' = y z' = z t' = t .
Note that the last equation 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:
begin{pmatrix} x' t' end{pmatrix} = begin{pmatrix} 1 & -v & 1 end{pmatrix}begin{pmatrix} x t end{pmatrix}
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 composition of a rotation, a translation and a uniform motion of spacetime.{{harvnb|Arnold|1989|p=6}} 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
(mathbf{x},t) mapsto (mathbf{x}+tmathbf{v},t),
where {{math|v ∈ ℝ3}}. A translation is given by
(mathbf{x},t) mapsto (mathbf{x}+mathbf{a},t+s),
where {{math|a ∈ ℝ3}} and {{math|s ∈ ℝ}}. A rotation is given by
(mathbf{x},t) mapsto (Gmathbf{x},t),
where {{math|1=G : ℝ3 → ℝ3}} is an orthogonal transformation.As a Lie group, the group of Galilean transformations has dimension 10.

Galilean group

Two Galilean transformations {{math| G(R, v, a, s)}} compose to form a third Galilean transformation, {{math| G(R' , v' , a' , s' ) G(R, v, a, s) {{=}} G(R' R, R' v+v' , R' a+a' +v' s, s' +s)}}. The set of all Galilean transformations {{math|Gal(3)}} on space forms a 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 ∈ ℝ3}} is a position in space. The action is given byweblink{{harvnb|Nadjafikhah|Forough|2009}}
begin{pmatrix}R & v & a 0 & 1 & s 0 & 0 & 1 end{pmatrix} begin{pmatrix} x t 1end{pmatrix} = begin{pmatrix} R x+vt +a t+s 1end{pmatrix},
where {{math|s}} is real and {{math|v, x, a ∈ ℝ3}} and {{math|R}} is a rotation matrix.The composition of transformations is then accomplished through matrix multiplication. {{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}}:
G_1 = { m : s = 0, a = 0 } , uniformly special transformations. G_2= { m : v = 0, R = I_3 } cong (mathbb{R}^4 , +) , shifts of origin. G_3 = { m : s = 0, a = 0, v = 0 } cong mathrm{SO}(3) , rotations of reference frame (see SO(3)). G_4= { m : s = 0, a = 0, R = I_3 } cong (mathbb{R}^3, +) , uniform frame motions.
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 (A rtimes B ) of groups is required.
  1. G_2 triangleleft mathrm{SGal}(3) (G2 is a normal subgroup)
  2. mathrm{SGal}(3) cong G_2 rtimes G_1
  3. G_4 trianglelefteq G_1
  4. G_1 cong G_4 rtimes G_3
  5. mathrm{SGal}(3) cong mathbb{R}^4 rtimes (mathbb{R}^3 rtimes mathrm{SO}(3)) .

Origin in group contraction

Here, we only look at the Lie algebra of the Galilean group; it is then easy to extend the results to the Lie group.The relevant Lie algebra is spanned by {{math|H, Pi, Ci}} and {{math|Lij}} (an antisymmetric tensor), subject to commutation relations, where
[H,P_i]=0 [P_i,P_j]=0 [L_{ij},H]=0 [C_i,C_j]=0 [L_{ij},L_{kl}]=i [delta_{ik}L_{jl}-delta_{il}L_{jk}-delta_{jk}L_{il}+delta_{jl}L_{ik}] [L_{ij},P_k]=i[delta_{ik}P_j-delta_{jk}P_i] [L_{ij},C_k]=i[delta_{ik}C_j-delta_{jk}C_i] [C_i,H]=i P_i ,! [C_i,P_j]=0 ~.
{{mvar|H}} is the generator of time translations (Hamiltonian), Pi is the generator of translations (momentum operator), Ci is the generator of rotationless Galilean transformations (Galileian boosts),BOOK, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces, illustrated, A. A., Ungar, Springer Science & Business Media, 2006, 978-0-306-47134-6, 336,weblink Extract of page 336 and Lij 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, 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 SO(1,4)).{{harvnb|Gilmore|2006}}Renaming the generators of the latter as {{math| ϵimn Ji ↦ Lmn ; Pi ↦ Pi ; P0 ↦ H/c ; Ki ↦ cCi}}, where {{math|c}} is the speed of light, or any function thereof diverging as {{math|c → ∞}}, the commutation relations (structure constants) of the latter limit to that of the former.Note the group invariants {{math|LmnLmn}}, {{math|PiPi}}.In matrix form, for d=3, one may consider the regular representation (embedded in GL(5;ℝ), from which it could be derived by a single group contraction, bypassing the Poincaré group),
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


ivec{a}cdotvec{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


ivec{v}cdotvec{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
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 ) ~.
The infinitesimal group element is then
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_1 & a_2
theta_2 & -theta_1 & 0 & v_1 & a_3 0 & 0 & 0 & 0 & s
0 & 0 & 0 & 0 & 0 end{array} } right ) + ... ~.

Central extension of the Galilean group

One could, instead,{{harvnb|Bargmann|1954}} augment the Galilean group by a central extension of the Lie algebra spanned by {{math|H′, P′i, C′i, L′ij, M}}, such that {{math|M}} commutes with everything (i.e. lies in the center), and
[H',P'_i]=0 ,! [P'_i,P'_j]=0 ,! [L'_{ij},H']=0 ,! [C'_i,C'_j]=0 ,! [L'_{ij},L'_{kl}]=i [delta_{ik}L'_{jl}-delta_{il}L'_{jk}-delta_{jk}L'_{il}+delta_{jl}L'_{ik}] ,! [L'_{ij},P'_k]=i[delta_{ik}P'_j-delta_{jk}P'_i] ,! [L'_{ij},C'_k]=i[delta_{ik}C'_j-delta_{jk}C'_i] ,! [C'_i,H']=i P'_i ,! [C'_i,P'_j]=i Mdelta_{ij} ~.
This algebra is often referred to as the Bargmann algebra.

See also

Notes

{{Reflist}}

References

  • BOOK, harv, Arnold, V. I., Vladimir Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1989, 2, 0-387-96890-3, 6,weblink
  • JOURNAL, harv, Bargmann, V., Valentine Bargmann, 1954, On Unitary Ray Representations of Continuous Groups, Annals of Mathematics, 2, 59, 1, 1–46, 10.2307/1969831,
  • BOOK, 2002, Nicolaus, Copernicus, Nicolaus Copernicus, Johannes, Kepler, Johannes Kepler, Galileo, Galalei, Galileo Galilei, Isaac, Newton, Isaac Newton, Albert, Einstein, Albert Einstein, Hawking, Stephen, Stephen Hawking, 515–520, On the Shoulders of Giants: The Great Works of Physics and Astronomy, 0-7624-1348-4, Running Press, Philadelphia, London,
  • BOOK, harv, Galilei, Galileo, Galileo Galilei, 1638I, Discorsi e Dimostrazioni Matematiche, intorno á due nuoue scienze, 191–196, Elsevier, Leiden, Italian,
  • BOOK, harv, Galileo, Galilei, 1638E, Discourses and Mathematical Demonstrations Relating to Two New Sciences, Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze, Translated to English 1914 by Henry Crew and Alfonso de Salvio,
  • BOOK, harv, Gilmore, Robert, 2006, Lie Groups, Lie Algebras, and Some of Their Applications, Dover Publications, Dover Books on Mathematics, 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/?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/?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/?id=lfGE-wyJYIUC&pg=PA42}}, Chapter 2 §2.6, p. 42
  • WEB, 97-105, Mehdi, Nadjafikhah, Ahmad-Reza, Forough, 2009, Galilean Geometry of Motions,
Applied Sciences, 11, 91–105,weblink
  • {{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|isbn=0-534-49143-X|url=https://books.google.com/?id=1DZz341Pp50C&pg=PA261}}, Chapter 9 §9.1, p. 261
{{Galileo Galilei}}{{Relativity}}

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