On Galilean connections and the first jet bundle |
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Authors: | James D E Grant Bradley C Lackey |
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Institution: | 1.Gravitationsphysik, Fakult?t für Physik,Universit?t Wien,Wien,Austria;2.Trusted Systems Research Group,National Security Agency,Fort G.G. Meade,USA |
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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. |
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