Galilean transformation

From Academic Kids

The Galilean transformation is used to transform between the coordinates of two coordinate systems in constant relative motion in Newtonian physics. This is the passive transformation point of view. The equations below, although apparently obvious, break down at speeds that approach the speed of light.

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History

Galileo formulated these concepts in his description of uniform motion 1 The topic was motivated by Galileo's description of the motion of a ball rolling down a ramp, by which he discovered the numerical value for the acceleration of gravity, at the surface of the Earth. The descriptions below are another mathematical notation for this concept.

Translation (one dimension)

Unlike the Galilean transformation, the relativistic Lorentz transformation can be shown to apply at all velocities so far measured, and the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.

The notation below describes the relationship of two coordinate systems (x′ and x) in constant relative motion (velocity u) in the x-direction. All other parameters (t, y, z) are unchanged in the transformation from x′ to x coordinates.

<math>t^'=t<math>
<math>x^'=x-ut<math>
<math>y^'=y<math>
<math>z^'=z<math>

Galilean transformations

Under the Erlangen program, the space-time (no longer spacetime) of nonrelativistic physics is described by the symmetry group generated by Galilean transformations, spatial and time translations and rotations.

The Galilean symmetries (interpreted as active transformations)

Spatial translations:

<math>t\rightarrow t<math>
<math>\vec{x}\rightarrow \vec{x}+\vec{a}<math>

Time translations:

<math>t\rightarrow t+\tau<math>
<math>\vec{x}\rightarrow \vec{x}<math>

Boosts:

<math>t\rightarrow t<math>
<math>\vec{x}\rightarrow \vec{x}+\vec{v}t<math>

Rotations:

<math>t\rightarrow t<math>
<math>\vec{x}\rightarrow \mathbf{R}\vec{x}<math>

where R is an orthogonal matrix.

Central extension of the Galilean group

  1. The 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 spanned by E, Pi, Ci and Lij (antisymmetric tensor) subject to commutators (operators of the form [,]), where
<math>[E,P_i]=0<math>
<math>[P_i,P_j]=0<math>
<math>[L_{ij},E]=0<math>
<math>[C_i,C_j]=0<math>
<math>[L_{ij},L_{kl}]=i\hbar [\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+\delta_{jl}L_{ik}]<math>
<math>[L_{ij},P_k]=i\hbar[\delta_{ik}P_j-\delta_{jk}P_i]<math>
<math>[L_{ij},C_k]=i\hbar[\delta_{ik}C_j-\delta_{jk}C_i]<math>
<math>[C_i,E]=i\hbar P_i<math>
<math>[C_i,P_j]=0<math>

We can now give it a central extension into the Lie algebra spanned by E', P'i, C'i, L'ij (antisymmetric tensor), M such that M commutes with everything (i.e. lies in the center, that's why it's called a central extension) and

<math>[E',P'_i]=0<math>
<math>[P'_i,P'_j]=0<math>
<math>[L'_{ij},E']=0<math>
<math>[C'_i,C'_j]=0<math>
<math>[L'_{ij},L'_{kl}]=i\hbar [\delta_{ik}L'_{jl}-\delta_{il}L'_{jk}-\delta_{jk}L'_{il}+\delta_{jl}L'_{ik}]<math>
<math>[L'_{ij},P'_k]=i\hbar[\delta_{ik}P'_j-\delta_{jk}P'_i]<math>
<math>[L'_{ij},C'_k]=i\hbar[\delta_{ik}C'_j-\delta_{jk}C'_i]<math>
<math>[C'_i,E']=i\hbar P'_i<math>
<math>[C'_i,P'_j]=i\hbar M\delta_{ij}<math>

Notes

  • Note 1: Galileo 1638 Discorsi e Dimostrazioni Matematiche, intorno 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

See also

representation theory of the Galilean group, Poincaré group

de:Galilei-Transformation ja:ガリレイ変換 pl:Transformacja Galileusza pt:Transformao de Galileu sl:Galilejeva transformacija

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