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].


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