We consider a geodesic problem in a manifold endowed with a Randers-Kropina metric. This is a type of singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo's navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik-Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of non-holonomic constraints that the velocity vectors must satisfy, this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with a totally non-integrable distribution.
Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system / Caponio, Erasmo; Masiello, Antonio; Javaloyes, Miguel Angel. - In: TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS. - ISSN 1230-3429. - 61:1(2023), pp. 527-547. [10.12775/TMNA.2022.066]
Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system
Caponio, Erasmo;Masiello, Antonio
;
2023-01-01
Abstract
We consider a geodesic problem in a manifold endowed with a Randers-Kropina metric. This is a type of singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo's navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik-Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of non-holonomic constraints that the velocity vectors must satisfy, this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with a totally non-integrable distribution.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.