Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system

Erasmo Caponio, Miguel Angel Javaloyes, Antonio Masiello

TL;DR

This work analyzes geodesics of Randers-Kropina (singular Finsler) metrics by embedding them into a spacetime framework with a causal Killing vector field and linking geodesic existence to affine control theory. It constructs regular approximations $F_ abla$ via standard stationary spacetimes, proves convergence from lightlike pregeodesics in the perturbed spacetime to pregeodesics of the Randers-Kropina metric, and uses Lusternik-Schnirelman theory together with a corank-one, completely nonholonomic distribution to obtain infinitely many geodesics between two points on non-contractible manifolds. The main results include a convergence theorem for pregeodesics as $ abla o0$, a homotopy equivalence between admissible-control trajectories and Sobolev path spaces, and a multiplicity theorem ensuring an unbounded sequence of Randers-Kropina geodesics, with implications for both Zermelo navigation and spacetime lightlike geodesics. These findings advance understanding of nonholonomic geodesic multiplicity in singular Finsler settings and provide a robust framework for passing to the singular limit in geometric control problems.

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 nonholonomic constraints that the velocity vectors must satisfy, this is achieved thanks to a recent result about the homotopy type of the set of solutions of an affine control system with a controlled drift and related to a corank one, completely nonholonomic distribution of step 2.

Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system

TL;DR

This work analyzes geodesics of Randers-Kropina (singular Finsler) metrics by embedding them into a spacetime framework with a causal Killing vector field and linking geodesic existence to affine control theory. It constructs regular approximations via standard stationary spacetimes, proves convergence from lightlike pregeodesics in the perturbed spacetime to pregeodesics of the Randers-Kropina metric, and uses Lusternik-Schnirelman theory together with a corank-one, completely nonholonomic distribution to obtain infinitely many geodesics between two points on non-contractible manifolds. The main results include a convergence theorem for pregeodesics as , a homotopy equivalence between admissible-control trajectories and Sobolev path spaces, and a multiplicity theorem ensuring an unbounded sequence of Randers-Kropina geodesics, with implications for both Zermelo navigation and spacetime lightlike geodesics. These findings advance understanding of nonholonomic geodesic multiplicity in singular Finsler settings and provide a robust framework for passing to the singular limit in geometric control problems.

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 nonholonomic constraints that the velocity vectors must satisfy, this is achieved thanks to a recent result about the homotopy type of the set of solutions of an affine control system with a controlled drift and related to a corank one, completely nonholonomic distribution of step 2.
Paper Structure (5 sections, 9 theorems, 50 equations)

This paper contains 5 sections, 9 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Randers-Kropina metrics and spacetimes with a causal Killing vector field
  3. The perturbation with standard stationary spacetimes
  4. The control system and admissible curves
  5. Multiple geodesics between two points

Key Result

Theorem 2.1

Let $\sigma$ be a piecewise smooth admissible curve in $(S,F)$. Then $\sigma$ is a geodesic of the Randers-Kropina space $(S,F)$ parametrized with $F(\dot\sigma)=1$ if and only if the curve $\gamma$ defined by $\gamma(t)=(\sigma(t), t)$ is a future-pointing lightlike pregeodesic of $(S\times \mathbb

Theorems & Definitions (26)

  • Theorem 2.1: CaJaSa14, Theorem 5.5
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Remark 3.6
  • ...and 16 more