|Title||On Galilean connections and the first jet bundle|
|Publication Type||Journal Article|
|Year of Publication||2012|
|Authors||DE Grant, J, Lackey, B|
|Journal||Central European Journal of Mathematics|
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.