Topological Methods in Nonlinear Analysis

Geodesics in conical manifolds

Marco Ghimenti

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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.

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Topol. Methods Nonlinear Anal., Volume 25, Number 2 (2005), 235-261.

First available in Project Euclid: 23 June 2016

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Ghimenti, Marco. Geodesics in conical manifolds. Topol. Methods Nonlinear Anal. 25 (2005), no. 2, 235--261.

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