following reasons: Since gξ is positive definite for Finsler spaces, every short enough stationary curve for the length functional is length minimizing. Every stationary curve for the action integral is of constant speed , since the energy is automatically a constant of motion. For any curve of constant speed the action integral and the length functional are related by Therefore, a curve is stationary to the action integral if and only if it is of constant speed and stationary to the length functional. The Hamiltonian vector field H is called the geodesic spray of the Finsler manifold (M,F) and the corresponding flow ΦHt(ξ) is called the geodesic flow. Correspondence with nonlinear connections A semispray on a smooth manifold defines an Ehresmann-connection on the slit tangent bundle through its horizontal and vertical projections This connection on TM\0 always has a vanishing torsion tensor, which is defined as the Frölicher-Nijenhuis bracket T=[J,v]. In more elementary terms the torsion can be defined as Introducing the canonical vector field V on TM\0 and the adjoint structure Θ of the induced connection the horizontal part of the semispray can be written as hH=ΘV. The vertical part ε=vH of the semispray is known as the first spray invariant, and the semispray H itself decomposes into The first spray invariant is related to the tension of the induced non-linear connection through the ordinary differential equation Therefore, the first spray invariant ε (and hence the whole semi-spray H) can be recovered from the non-linear connection by From this relation one also sees that the induced connection is homogeneous if and only if H is a full spray. Jacobi fields of sprays and semisprays A good source for Jacobi fields of semisprays is Section 4.4, Jacobi equations of a semispray of the publicly available book Finsler-Lagrange Geometry by Bucătaru and Miron. Of particular note is their concept of a dynamical covariant derivative. In another paper Bucătaru, Constantinescu and Dahl relate this concept to that of the Kosambi biderivative operator. For a good introduction to Kosambi's methods, see the article, What is Kosambi-Cartan-Chern theory?. References . . Differential geometry Finsler geometry