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Difference between revisions of "Galilean space"

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The space-time of classical mechanics according to Galilei–Newton, in which the distance between two events taking place at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043070/g0430701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043070/g0430702.png" /> at the moments of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043070/g0430703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043070/g0430704.png" /> is taken to be the time interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043070/g0430705.png" />, while if these events take place at the same time, it is considered to be the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043070/g0430706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043070/g0430707.png" />. For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043070/g0430708.png" />-dimensional Galilean space, the distance is defined as follows:
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The space-time of classical mechanics according to Galilei–Newton, in which the distance between two events taking place at the points $ M_{1} $ and $ M_{2} $ at different moments of time $ t_{1} $ and $ t_{2} $ respectively is taken to be the time interval $ |t_{1} - t_{2}| $, while if these events take place at the same time, it is then considered to be the Euclidean distance between the points $ M_{1} $ and $ M_{2} $. For an $ n $-dimensional Galilean space, the distance is defined as follows:
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$$
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d(\mathbf{x},\mathbf{y}) \stackrel{\text{df}}{=}
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\begin{cases}
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|x^{1} - y^{1}| & \text{if $ x^{1} \neq y^{1} $}; \\\\
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\displaystyle \sqrt{\sum_{i = 2}^{n} (x^{i} - y^{i})^{2}} & \text{if $ x^{1} = y^{1} $}.
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\end{cases}
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043070/g0430709.png" /></td> </tr></table>
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A Galilean space is a [[Semi-pseudo-Euclidean space|semi-pseudo-Euclidean space]] of nullity $ 1 $; it may be considered as the limit case of a pseudo-Euclidean space in which the isotropic cone degenerates to a plane. This limit transition corresponds to the limit transition from the special theory of relativity to classical mechanics.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043070/g04307010.png" /></td> </tr></table>
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====References====
  
A Galilean space is a [[Semi-pseudo-Euclidean space|semi-pseudo-Euclidean space]] of nullity one; it may be considered as the limit case of a pseudo-Euclidean space in which the isotropic cone degenerates to a plane. This limit transition corresponds to the limit transition from the special theory of relativity to classical mechanics.
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<table>
 
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<TR><TD valign="top">[1]</TD><TD valign="top">
====References====
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B.A. Rozenfel’d, “Non-Euclidean spaces”, Moscow (1969). (In Russian)</TD></TR>
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.A. Rozenfel'd,   "Non-Euclidean spaces" , Moscow (1969) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Penrose,   "Structure of space-time"  C.M. DeWitt (ed.) J.A. Wheeler (ed.) , ''Batelle Rencontres 1967 Lectures in Math. Physics'' , Benjamin (1968) pp. 121–235 (Chapt. VII)</TD></TR></table>
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<TR><TD valign="top">[2]</TD><TD valign="top">
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R. Penrose, “Structure of space-time”, C.M. DeWitt (ed.), J.A. Wheeler (ed.), ''Batelle Rencontres 1967 Lectures in Math. Physics'', Benjamin (1968), pp. 121–235 (Chapt. VII).</TD></TR>
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</table>

Revision as of 01:03, 15 December 2016

The space-time of classical mechanics according to Galilei–Newton, in which the distance between two events taking place at the points $ M_{1} $ and $ M_{2} $ at different moments of time $ t_{1} $ and $ t_{2} $ respectively is taken to be the time interval $ |t_{1} - t_{2}| $, while if these events take place at the same time, it is then considered to be the Euclidean distance between the points $ M_{1} $ and $ M_{2} $. For an $ n $-dimensional Galilean space, the distance is defined as follows: $$ d(\mathbf{x},\mathbf{y}) \stackrel{\text{df}}{=} \begin{cases} |x^{1} - y^{1}| & \text{if $ x^{1} \neq y^{1} $}; \\\\ \displaystyle \sqrt{\sum_{i = 2}^{n} (x^{i} - y^{i})^{2}} & \text{if $ x^{1} = y^{1} $}. \end{cases} $$

A Galilean space is a semi-pseudo-Euclidean space of nullity $ 1 $; it may be considered as the limit case of a pseudo-Euclidean space in which the isotropic cone degenerates to a plane. This limit transition corresponds to the limit transition from the special theory of relativity to classical mechanics.

References

[1] B.A. Rozenfel’d, “Non-Euclidean spaces”, Moscow (1969). (In Russian)
[2] R. Penrose, “Structure of space-time”, C.M. DeWitt (ed.), J.A. Wheeler (ed.), Batelle Rencontres 1967 Lectures in Math. Physics, Benjamin (1968), pp. 121–235 (Chapt. VII).
How to Cite This Entry:
Galilean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galilean_space&oldid=15236
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article