Spray (mathematics)

In the associated coordinates on TM the first variation of the action integral reads as

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:

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.