The Variational Problem in Lagrange Spaces Endowed with a Special Type of (α, β)-Metrics

Abstract

In this paper, we will continue our investigation on the new recently introduced $(\alpha, \beta)$-metric $F=\beta+\frac{a\alpha^{2}+\beta^{2}}{\alpha}$ in \cite{Pis1}; where $\alpha$ is a Riemannian metric; $\beta$ is an 1-form and $a\in (0,1]$ is a real positive scalar. We will investigate the variational problem in Lagrange spaces endowed with this type of metrics. Also, we will study the dually local flatness for this type of metric and we will proof that this kind of metric can be reduced to a locally Minkowskian metric. Finally, we will introduce the 2-Killing equation in Finsler spaces.