TY - JOUR AU - Ghimenti, Marco PY - 2005/06/01 Y2 - 2024/03/28 TI - Geodesics in conical manifolds JF - Topological Methods in Nonlinear Analysis JA - TMNA VL - 25 IS - 2 SE - DO - UR - https://apcz.umk.pl/TMNA/article/view/TMNA.2005.012 SP - 235 - 261 AB - The aim of this paper is to extend the definition of geodesics toconical manifolds, defined as submanifolds of ${\mathbb R}^n$ with afinite number of singularities. We look for anapproach suitable both for the local geodesic problem and for thecalculus of variation in the large. We give a definitionwhich links the local solutions ofthe Cauchy problem (1.1) with variationalgeodesics, 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 toconical manifolds, and to estimate the number of geodesics (Theorem 3.4 and Corollary 3.5).In Section 4, we provide someapplications in which conical manifolds arise naturally: inparticular, we focus on the brachistochrone problem for africtionless particle moving in $S^n$ or in ${\mathbb R}^n$ in the presence ofa potential $U(x)$ unbounded from below. We conclude with anappendix in which the main results are presented in a general framework. ER -