arXiv subjects

L. Klapka

Publications and source records attributed to L. Klapka.

2 recordsLinked to original sources

arXiv2003

The Cauchy atlas on the manifold of all complete ODE solutions

In this paper the necessary and sufficient conditions for a mapping to be the dependence of the complete solution of some differentiable first-order ordinary differential equation on the initial Cauchy condition are deduced. The result is obtained by studying the Cauchy atlas on a manifold of complete solutions. The proof is constructive - the corresponding differential equation is obtained. The autonomous case and the linear case are discussed. The relation to the Sincov functional equation is clarified.

math.CA↗

arXiv1997

The functional definition of geodesics

In this paper a functional definition of geodesics is introduced which allows to generalize the notion of a geodesic from smooth to topological manifolds. It is shown that in the smooth case the new definition coincides with the classical definition of geodesics of a linear connection. If the smoothness is not required, it is shown by an example that the new definition includes geodesics of non-linear, homogeneous connections. Moreover, an example of generalized geodesics which does not arise from any connection is presented here.

dg-ga↗