Galilean Transformations

HISTORY

Galileo formulated these concepts in his description of ''uniform motion''
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)

The Galilean transformation is nothing more than careful addition and subtraction of velocity vectors.

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.

:

t^'=t

x^'=x-ut

y^'=y

z^'=z

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 Transformation s):

Spatial translations:
:

t
ightarrow t

ec{x}
ightarrow ec{x}+ec{a}

Time translations:
:

t
ightarrow t+	au

ec{x}
ightarrow ec{x}

Boosts:
:

t
ightarrow t

ec{x}
ightarrow ec{x}+ec{v}t

Rotations:
:

t
ightarrow t

ec{x}
ightarrow \mathbf{R}ec{x}

where R is an Orthogonal Matrix .

CENTRAL EXTENSION OF THE GALILEAN GROUP

# The . It's easy to extend the results to the Lie Group . The Lie algebra of L is Spanned by E, P_i, C_i and L_ij ( Antisymmetric Tensor ) subject to Commutator s ( Operator s of the form {Link without Title} ), where
:

{Link without Title} =0
:

[\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+\delta_{jl}L_{ik}

{Link without Title} =i\hbar {Link without Title}
:

{Link without Title} =i\hbar P_i
:

{Link without Title} =0

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 Commute s with everything (i.e. lies in the Center , that's why it's called a central extension) and
:

{Link without Title} =0
:

[\delta_{ik}L'_{jl}-\delta_{il}L'_{jk}-\delta_{jk}L'_{il}+\delta_{jl}L'_{ik}

{Link without Title} =i\hbar {Link without Title}
:

{Link without Title} =i\hbar P'_i
:

{Link without Title} =i\hbar M\delta_{ij}

NOTES

Galileo , ed. 2002 ISBN 0-7624-1348-4

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Galilean Transformations