Page 59 - programme book

GE-001
Metric Foliated Cocycles Adapted to Isotropic Systems of Second Order
Ordinary Differential Equations

Fatemeh Ahangari 1,a)

1 Department of Mathematics, Faculty of Mathematical Sciences,
Alzahra University, Tehran, P.O.Box 1993893973, Iran.

a) Corresponding author: fa.ahangari@gmail.com
a) f.ahangari@alzahra.ac.ir

Abstract. As it is well known, the inverse problem of Lagrangian mechanics is structurally based on the
fact that whether or not a given system of second order ordinary differential equations (SODEs) can
be characterized as the Euler-Lagrange equations of some Lagrangian. In particular, the essential
condition for a system of homogeneous SODEs to be metrizable via a Finsler function of scalar flag
curvature is that the given system of SODEs must be isotropic. In addition, projectively flat Finsler
functions possess isotropic geodesic sprays and accordingly have constant or scalar flag curvature. In
this paper, a comprehensive investigation of metric foliated cocycles which are geometrically
compatible with a given system of isotropic SODEs is presented. For this purpose, the constant flag
curvature (CFC) test, as well as the scalar flag curvature (SFC) test, presented by I. Bucataru and Z.
Muzsnay are applied in order to induce a metric structure which leads to construction of a foliated
cocycle equipped with a bundle-like metric on the tangent bundle. Consequently, by exhaustive
analysis of the main circumstances under which the projective deformations of a flat isotropic spray
are metrizable, we create a class of transverse foliated cocycles on the tangent space which are totally
adapted to the given system of isotropic SODEs.

Keywords: Isotropic Sprays, Bundle-like Metric, Flag Curvature, Foliated Cocycle, Metrizability.

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