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in kinematics and dynamics of relativity theory. Since rapidity is unbounded, the one-parameter group it stands upon is non-compact. The rapidity concept was introduced by
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is an infinitely small transformation of the one-parameter group that it generates. It is these infinitesimal transformations that generate a
1418:{\displaystyle \left.{\frac {d\varphi (t)}{dt}}\right|_{t=0}=\left.{\frac {d}{dt}}\right|_{t=0}e^{tX}=\left.(Xe^{tX})\right|_{t=0}=Xe^{0}=X}
42:
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This result can be used, for example, to show that any continuous homomorphism between matrix Lie groups is smooth.
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35:
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Any connected 1-dimensional Lie group is analytically isomorphic either to the additive group of real numbers
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was reduced to arbitrariness of which diameter of the unit hyperbola was used to determine a
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had all employed in their writings an equivalent mapping of the
Cartesian plane by operator
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is differentiable, even though this was not an assumption of the theorem. The matrix
558:
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693:. Furthermore, whenever a system of physical laws admits a one-parameter group of
514:, one-parameter groups correspond to one-dimensional subspaces of the associated
1624:
In that case the induced topology may not be the standard one of the real line.
756:
515:
344:
122:
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provided a calculus of relative motion with the one-parameter group indexed by
1683:
1040:{\displaystyle \varphi :\mathbb {R} \rightarrow \mathrm {GL} (n;\mathbb {C} )}
984:
matrices with complex entries. In that case, a basic result is the following:
728:
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523:
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332:
1690:, English translation by D.H. Delphenich, §8, link from Neo-classical Physics
1724:
712:
370:
of the vector field. The local flow of a vector field is used to define the
348:
252:
180:
1660:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
1658:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
740:
654:. In particular, every 1-dimensional Lie group is locally isomorphic to
1731:, page 58, Cambridge Tracts in Mathematics and Mathematical Physics #46
686:
366:- a one parameter group of local diffeomorphisms, sending points along
760:
1742:
Applied
Functional Analysis: Main Principles and Their Applications
1559:
719:
to calibrate spatio-temporal measurements has become common since
759:
the next year. The rapidity parameter amounts to the length of a
763:, a concept of the nineteenth century. Mathematical physicists
18:
896:
An important example in the theory of Lie groups arises when
362:. A smooth vector field on a manifold, at a point, induces a
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is a one-parameter group. Then there exists a unique
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49:. Unsourced material may be challenged and removed.
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1538:is injective. Think for example of the case where
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731:. Using the parametrization of the hyperbola with
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169:{\displaystyle \varphi :\mathbb {R} \rightarrow G}
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1601:is constructed by winding a straight line round
358:of a one-parameter group on a set is known as a
892:Stone's theorem on one-parameter unitary groups
563:Stone's theorem on one-parameter unitary groups
409:{\displaystyle \phi :\mathbb {R} \rightarrow G}
951:{\displaystyle \mathrm {GL} (n;\mathbb {C} )}
8:
16:Lie group homomorphism from the real numbers
494:{\displaystyle \phi (t)\phi (s)=\phi (s+t)}
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109:Learn how and when to remove this message
1688:Vorlesungen ĂĽber Continuierliche Gruppen
331:One-parameter groups were introduced by
1676:
1461:{\displaystyle \varphi (\mathbb {R} )}
819:{\displaystyle (\cosh {a}+r\sinh {a})}
279:{\displaystyle \varphi (\mathbb {R} )}
628:, the additive group of real numbers
7:
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520:Lie group–Lie algebra correspondence
416:is called one-parameter subgroup of
47:adding citations to reliable sources
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522:is the basis of a science begun by
286:, the image, will be a subgroup of
1133:{\displaystyle \varphi (t)=e^{tX}}
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529:Another important case is seen in
14:
1518:; this may happen in cases where
1437:A technical complication is that
1176:It follows from this result that
1165:{\displaystyle t\in \mathbb {R} }
689:, one-parameter groups describe
23:
1702:Geometry, topology, and physics
635:
621:{\displaystyle {\mathfrak {T}}}
597:{\displaystyle {\mathfrak {R}}}
436:if it satisfies the condition
34:needs additional citations for
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1492:may carry a topology that is
337:infinitesimal transformations
1511:{\displaystyle \mathbb {R} }
846:is the hyperbolic angle and
669:{\displaystyle \mathbb {R} }
576:gave the following theorem:
341:infinitesimal transformation
321:{\displaystyle \mathbb {R} }
197:{\displaystyle \mathbb {R} }
1216:can then be recovered from
347:that is used to describe a
1811:
1704:. CRC Press. p. 232.
958:, the group of invertible
889:
723:discussed it in 1908. The
1066:{\displaystyle n\times n}
977:{\displaystyle n\times n}
1621:at an irrational slope.
1594:{\displaystyle \varphi }
1531:{\displaystyle \varphi }
1229:{\displaystyle \varphi }
1189:{\displaystyle \varphi }
875:{\displaystyle r^{2}=+1}
769:William Kingdon Clifford
244:{\displaystyle \varphi }
1656:Hall, Brian C. (2015),
1639:One-parameter semigroup
725:principle of relativity
339:. According to Lie, an
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647:{\displaystyle \mod 1}
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328:as an additive group.
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58:"One-parameter group"
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773:Alexander Macfarlane
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43:improve this article
1740:Zeidler, E. (1995)
553:being the group of
531:functional analysis
127:one-parameter group
1795:Topological groups
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737:special relativity
702:conserved quantity
700:, then there is a
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351:of any dimension.
335:in 1893 to define
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138:group homomorphism
1644:Noether's theorem
1614:{\displaystyle T}
1574:{\displaystyle T}
1551:{\displaystyle G}
1485:{\displaystyle G}
1315:
1277:
1209:{\displaystyle X}
1086:{\displaystyle X}
909:{\displaystyle G}
839:{\displaystyle a}
761:hyperbolic versor
721:Hermann Minkowski
706:Noether's theorem
691:dynamical systems
568:In his monograph
555:unitary operators
546:{\displaystyle G}
429:{\displaystyle G}
299:{\displaystyle G}
224:{\displaystyle G}
210:topological group
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711:In the study of
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847:
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776:
715:the use of the
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368:integral curves
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1768:Corollary 3.50
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1667:978-3319134666
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1634:Integral curve
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753:E.T. Whittaker
717:unit hyperbola
695:differentiable
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679:
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677:
664:
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526:in the 1890s.
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372:Lie derivative
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206:additive group
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13:
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1711:9780750306065
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1496:than that on
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802:
799:
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770:
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758:
754:
750:
747:replaces the
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730:
726:
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718:
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636:
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566:
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560:
559:Hilbert space
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63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1761:
1756:Theorem 2.14
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765:James Cockle
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99:January 2015
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41:Please help
36:verification
33:
757:Alfred Robb
516:Lie algebra
345:Lie algebra
123:mathematics
1790:1 (number)
1785:Lie groups
1779:Categories
1729:Lie Groups
1700:Nakahara.
1684:Sophus Lie
1650:References
1093:such that
991:: Suppose
890:See also:
886:In GL(n,C)
729:world-line
698:symmetries
574:P. M. Cohn
570:Lie Groups
524:Sophus Lie
512:Lie theory
378:Definition
364:local flow
333:Sophus Lie
135:continuous
69:newspapers
1766:Hall 2015
1754:Hall 2015
1725:Paul Cohn
1589:φ
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181:real line
179:from the
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1470:subspace
1433:Topology
1144:for all
826:, where
749:velocity
745:rapidity
741:rapidity
604:, or to
506:Examples
382:A curve
1727:(1957)
1686:(1893)
1494:coarser
1073:matrix
989:Theorem
687:physics
681:Physics
533:, with
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