Description

• Title:
  Geodesics in conical manifolds
• Creator:
  Ghimenti, Marco
• Description:
  The aim of this paper is to extend the definition of geodesics to conical manifolds, defined as submanifolds of ${\mathbb R}^n$ with a finite number of singularities. We look for an approach suitable both for the local geodesic problem and for the calculus of variation in the large. We give a definition which links the local solutions of the Cauchy problem (1.1) with variational geodesics, i.e. critical points of the energy functional. We prove a deformation lemma (Theorem 2.2) which leads us to extend the Lusternik-Schnirelmann theory to conical manifolds, and to estimate the number of geodesics (Theorem 3.4 and Corollary 3.5). In Section 4, we provide some applications in which conical manifolds arise naturally: in particular, we focus on the brachistochrone problem for a frictionless particle moving in $S^n$ or in ${\mathbb R}^n$ in the presence of a potential $U(x)$ unbounded from below. We conclude with an appendix in which the main results are presented in a general framework.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2005.06.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 25, No 2 (June 2005); 235 - 261;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2005.012/9265
• Subject:
  Geodesics;
  nonsmooth critical point theory;
  nonsmooth manifolds;
  58E10;
  58E05;
  49J52
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2005.012
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other