Spray (mathematics) - Geodesic Spray

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) = g_ij(x)ξiξj. Now introduce the concepts from the section above. In the Riemannian case it turns out that the fundamental tensor g_ij(x,ξ) is simply the Riemannian metric g_ij(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 following reasons:

• Since g_ξ 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 Φ_Ht(ξ) is called the geodesic flow.