## 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; $$\varphi : \mathbb{R}_+ \to \mathbb{R}$$, $$\varphi \in C^\infty$$. Then $$L = \varphi (F(x,y))$$ is a Lagrangian, called by the authors a $$\varphi$$-Lagrangian associated to $$F^n$$. It is shown: if $$\varphi'(t) \neq 0$$ and $$\varphi' (t) + 2t \varphi''(t) \neq 0$$ 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, $$\psi : \mathbb{R}\to \mathbb{R}_+$$, $$\psi \in C^\infty$$, $$\psi (L (x,y))$$ 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].

### MSC:

 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)