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On Galilean connections and the first jet bundle

James Grant; Bradley Lackey

Open Mathematics (2012)

  • Volume: 10, Issue: 5, page 1889-1895
  • ISSN: 2391-5455

Abstract

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We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations - sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse - the “fundamental theorem” - that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion.

How to cite

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James Grant, and Bradley Lackey. "On Galilean connections and the first jet bundle." Open Mathematics 10.5 (2012): 1889-1895. <http://eudml.org/doc/269376>.

@article{JamesGrant2012,
abstract = {We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations - sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse - the “fundamental theorem” - that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion.},
author = {James Grant, Bradley Lackey},
journal = {Open Mathematics},
keywords = {Galilean group; Cartan connections; Jet bundles; 2nd order ODE; jet bundles},
language = {eng},
number = {5},
pages = {1889-1895},
title = {On Galilean connections and the first jet bundle},
url = {http://eudml.org/doc/269376},
volume = {10},
year = {2012},
}

TY - JOUR
AU - James Grant
AU - Bradley Lackey
TI - On Galilean connections and the first jet bundle
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1889
EP - 1895
AB - We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations - sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse - the “fundamental theorem” - that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion.
LA - eng
KW - Galilean group; Cartan connections; Jet bundles; 2nd order ODE; jet bundles
UR - http://eudml.org/doc/269376
ER -

References

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  9. [9] Lackey B., Metric equivalence of path spaces, Nonlinear Studies, 2002, 7(2), 241–250 Zbl1001.53008
  10. [10] Lie S., Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Teubner, Leipzig, 1891 Zbl43.0373.01
  11. [11] Saunders D.J., The Geometry of Jet Bundles, London Math. Soc. Lecture Note Ser., 142, Cambridge University Press, Cambridge, 1989 http://dx.doi.org/10.1017/CBO9780511526411 Zbl0665.58002
  12. [12] Sharpe R.W., Differential Geometry, Grad. Texts in Math., 166, Springer, New York, 1997 
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