On Galilean connections and the first jet bundle

Author(s)
James Grant, Bradley C. Lackey
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.

Organisation(s)
Gravitational Physics
External organisation(s)
National Security Agency
Journal
Central European Journal of Mathematics
Volume
10
Pages
1889-1895
No. of pages
7
ISSN
1895-1074
DOI
https://doi.org/10.2478/s11533-012-0089-4
Publication date
2012
Peer reviewed
Yes
Austrian Fields of Science 2012
103019 Mathematical physics
Portal url
https://ucrisportal.univie.ac.at/en/publications/on-galilean-connections-and-the-first-jet-bundle(772d7a34-a6ea-43e1-a687-de6e1591ad93).html