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. This is the passive transformation point of view. The equations below, although apparently obvious, are untenable at speeds that approach the speed of light. In special relativity the Galilean transformations are replaced by Lorentz transformations.

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 measured the numerical value for the acceleration of gravity near the surface of the Earth.

Translation[edit]

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.

This assumption is abandoned in the Lorentz transformations. These relativistic transformations are applicable to all velocities, whilst the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.

The notation below describes the relationship under the Galilean transformation between the coordinates (x,y,z,t) and (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 t=t'=0: [2] [3] [4] [5]

x'=x-vt\,
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:

(x', t') = (x,t) \begin{pmatrix} 1 & 0 \\-v & 1 \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[edit]

The Galilean symmetries can be uniquely written as the composition of a rotation, a translation and a uniform motion of space-time.[6] Let x represent a point in three-dimensional space, and t a point in one-dimensional time. A general point in space-time is given by an ordered pair (x,t). A uniform motion, with velocity v, is given by (\bold{x},t) \mapsto (\bold{x}+t\bold{v},t) where v is in R3. A translation is given by (\bold{x},t) \mapsto (\bold{x}+\bold{a},t+b) where a in R3 and b in R. A rotation is given by (\bold{x},t) \mapsto (G\bold{x},t) where G : R3R3 is an orthogonal transformation.[6] As a Lie group, the Galilean transformations have dimensions 10.[6]

Central extension of the Galilean group[edit]

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 H, Pi, Ci and Lij (antisymmetric tensor) subject to commutators, 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 \,\!.

H is generator of time translations (Hamiltonian), Pi is generator of translations (momentum operator), Ci is generator of Galileian boosts and Lij stands for a generator of rotations (angular momentum operator).

We can now give it a central extension into the Lie algebra spanned by H', 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

[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 M\delta_{ij} \,\!

See also[edit]

Notes[edit]

  1. ^ Galileo 1638 Discorsi e Dimostrazioni Matematiche, intorno á 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
  2. ^ Mould, Richard A. (2002), Basic relativity, Springer-Verla, ISBN 0-387-95210-1 , Chapter 2 §2.6, p. 42
  3. ^ Lerner, Lawrence S. (1996), Physics for Scientists and Engineers, Volume 2, Jones and Bertlett Publishers, Inc, ISBN 0-7637-0460-1 , Chapter 38 §38.2, p. 1046,1047
  4. ^ Serway, Raymond A.; Jewett, John W. (2006), Principles of Physics: A Calculus-based Text, Fourth Edition, Brooks/Cole - Thomson Learning, ISBN 0-534-49143-X , Chapter 9 §9.1, p. 261
  5. ^ Hoffmann, Banesh (1983), Relativity and Its Roots, Scientific American Books, ISBN 0-486-40676-8 , Chapter 5, p. 83
  6. ^ a b c Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics (2 ed.). Springer-Verlag. p. 6. ISBN 0-387-96890-3.