### Geodesics in conical manifolds

DOI: http://dx.doi.org/10.12775/TMNA.2005.012

#### Abstract

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.

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.

#### Keywords

Geodesics; nonsmooth critical point theory; nonsmooth manifolds

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.