The aim of this paper is to extend the definition of geodesics to conical manifolds, defined as submanifolds of Rn with a finite number of singularities. We look for an approach suitable both for the local ge- odesic 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 par- ticle moving in Sn or in Rn 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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