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A new class of spray-generating Lagrangians. (English) Zbl 0844.53046
Antonelli, P. L. (ed.) et al., Lagrange and Finsler geometry: applications to physics and biology. Proceedings of a conference. Dordrecht: Kluwer Academic Publishers. Fundam. Theor. Phys. 76, 81-92 (1996).
Let F n =(M,F(x,y)) be a Finsler manifold; ϕ: + , ϕC . Then L=ϕ(F(x,y)) is a Lagrangian, called by the authors a ϕ-Lagrangian associated to F n . It is shown: if ϕ ' (t)0 and ϕ ' (t)+2tϕ '' (t)0 for all t, then L is regular and L n =(M,L) is a ϕ-Lagrangian space. Conversely, if L n is a Lagrange space, ψ: + , ψC , ψ(L(x,y)) is homogeneous of degree 1 in y and neither of ψ(t), ψ ' (t), ψ '' (t) vanishes for all t, then (M,ψ(L(x,y))) is an F n . It is proved that any ϕ-Lagrange L n is projective to the associated F n . Then canonical, d-, Berwald-connections and sprays of ϕ-Lagrange spaces are studied and compared with those of the associated F n .
MSC:
53C60Finsler spaces and generalizations (areal metrics)