Some basic properties of Lagrange spaces

Abstract

Consider $L$ a regular Lagrangian, $S$ the canonical semispray, and $h$ the horizontal projector of the canonical nonlinear connection. We prove that if the Lagrangian is constant along the integral curves of the Euler-Lagrange equations then it is constant along the horizontal curves of the canonical nonlinear connection. In other words $S(L)=0$ implies $d_hL=0$. If the Lagrangian $L$ is homogeneous of order $k\neq 1$ then $L$ is a conservation law and hence $d_hL=0$. We give an example of nonhomogeneous Lagrangians for which $d_hL\neq 0$.