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
h
L=0
. If the Lagrangian
L
is homogeneous of order
k≠1
then
L
is a conservation law and hence
d
h
L=0
. We give an example of nonhomogeneous Lagrangians for which
d
h
L≠0
.