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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; $\varphi : \bbfR_+ \to \bbfR$, $\varphi \in C^\infty$. Then $L = \varphi (F(x,y))$ is a Lagrangian, called by the authors a $\varphi$-Lagrangian associated to $F^n$. It is shown: if $\varphi'(t) \ne 0$ and $\varphi' (t) + 2t \varphi''(t) \ne 0$ for all $t$, then $L$ is regular and $L^n = (M,L)$ is a $\varphi$-Lagrangian space. Conversely, if $L^n$ is a Lagrange space, $\psi : \bbfR\to \bbfR_+$, $\psi \in C^\infty$, $\psi (L (x,y))$ is homogeneous of degree 1 in $y$ and neither of $\psi(t)$, $\psi'(t)$, $\psi''(t)$ vanishes for all $t$, then $(M, \psi (L (x,y)))$ is an $F^n$. It is proved that any $\varphi$-Lagrange $L^n$ is projective to the associated $F^n$. Then canonical, $d$-, Berwald-connections and sprays of $\varphi$-Lagrange spaces are studied and compared with those of the associated $F^n$. For the entire collection see [Zbl 0833.00033].

53C60Finsler spaces and generalizations (areal metrics)