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On Galilean connections and the first jet bundle

100%

Grant J., Lackey B.

Open Mathematics

2012

tom 10

nr 5

1889-1895

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.

The geometry of the space of Cauchy data of nonlinear PDEs

68%

Moreno G.

Open Mathematics

2013

tom 11

nr 11

1960-1981

First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.

Ograniczanie wyników

2 Open Mathematics

1 Grant J.

1 Lackey B.

1 Moreno G.

1 2013

1 2012

Wyniki wyszukiwania

On Galilean connections and the first jet bundle

The geometry of the space of Cauchy data of nonlinear PDEs