action integral . In the associated coordinates on TM the first variation of the action integral reads as where X:[a,b]→R is the variation vector field associated with the variation γs:[a,b]→M around γ(t) = γ0(t). This first variation formula can be recast in a more informative form by introducing the following concepts: The covector with is the conjugate momentum of . The corresponding one-form with is the Hilbert-form associated with the Lagrangian. The bilinear form with is the fundamental tensor of the Lagrangian at . The Lagrangian satisfies the Legendre condition if the fundamental tensor is non-degenerate at every . Then the inverse matrix of is denoted by . The Energy associated with the Lagrangian is . If the Legendre condition is satisfied, then dα∈Ω2(TM) is a symplectic form, and there exists a unique Hamiltonian vector field H on TM corresponding to the Hamiltonian function E such that . Let (Xi,Yi) be the components of the Hamiltonian vector field H in the associated coordinates on TM. Then and so we see that the Hamiltonian vector field H is a semispray on the configuration space M with the spray coefficients Now the first variational formula can be rewritten as and we see γ[a,b]→M is stationary for the action integral with fixed end points if and only if its tangent curve γ':[a,b]→TM is an integral curve for the Hamiltonian vector field H. Hence the dynamics of mechanical systems are described by semisprays arising from action integrals. Geodesic spray The locally length minimizing curves of Riemannian and Finsler manifolds are called geodesics. Using the framework of Lagrangian mechanics one can describe these curves with spray structures. Define a Lagrangian function on TM by where F:TM→R is the Finsler function. In the Riemannian case one uses F2(x,ξ) = gij(x)ξiξj. Now introduce the concepts from the section above. In the Riemannian case it turns out that the fundamental tensor gij(x,ξ) is simply the Riemannian metric gij(x). In the general case the homogeneity condition of the Finsler-function implies the following formulae: In terms of classical mechanical the last equation states that all the energy in the system (M,L) is in the kinetic form. Furthermore, one obtains the homogeneity properties of which the last one says that the Hamiltonian vector field H for this mechanical system is a full spray. The constant speed geodesics of the underlying Finsler (or Riemannian) manifold are described by this spray for the