Examples of Conics Arising in Two - Dimensional Finsler and Lagrange Geometries ∗

The well-known invariants of conics are computed for classes of Finsler and Lagrange spaces. For the Finsler case, some (α, β)-metrics namely Randers, Kropina and ”Riemann”-type metrics provides conics as indicatrices and a Randers-Funk metric on the unit disk is treated as example. The relations between algebraic and differential invariants of (α, β)-metrics are pointed out as a method to use the formers in terms of the Finsler metric. In the Lagrange framework, a polynomial of third order Lagrangian inspired by Tzitzeica is studied and examples for all three cases (elliptic, hyperbolic, parabolic) are given. Introduction This paper is devoted to a study of conics which are naturally associated in two-dimensional Lagrange, particularly Finsler geometry. We are interested in this dimension since the 2D Lagrange geometry may yields in a somehow intrinsic manner a conic and the particular case of 2D Finsler geometry is a subject of continuous research: see [1], [2], [5], [9]. Moreover, conics in two dimensional Finsler geometry were already studied by Matsumoto in [7] from the point of view of geodesics. The great importance of indicatrices in the Finslerian setting is pointed out by Okubo’s technique ([2, p. 13]) which shows that, in a certain sense, not