Some basic properties of Lagrange spaces

Details Some basic properties of Lagrange spaces Some basic properties of Lagrange spaces

2005-07-27

arxiv.org/abs/math/0507560v1

Sci. Ann. Univ. Agric. Sci. Vet. Med. 48 (2005), no. 2, Proceedings of the 4th Annual Symposium on Mathematics Applied in Biol

Mathematics

Differential Geometry

Scientific paper

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 $d_hL\neq 0$.

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