In differential geometry, a spray is a vector field H on the tangent bundle TM that encodes a quasilinear second order system of ordinary differential equations on the base manifold M. Usually a spray is required to be homogeneous in the sense that its integral curves t→ΦHt(ξ)∈TM obey the rule ΦHt(λξ)=ΦHλt(ξ) in positive re-parameterizations.
Semisprays arise naturally as the extremal curves of action integrals in Lagrangian mechanics.
Then a vector field H on TM (that is, a section of the double tangent bundle TTM) is a semi-spray on M, if any of the three following equivalent conditions holds: A semispray H on M is a (full) spray if any of the following equivalent conditions hold: Let
The semispray H is a (full) spray, if and only if the spray coefficients Gi satisfy A physical system is modeled in Lagrangian mechanics by a Lagrangian function L:TM→R on the tangent bundle of some configuration space M. The dynamical law is obtained from the Hamiltonian principle, which states that the time evolution γ:[a,b]→M of the state of the system is stationary for the 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: 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 semi-spray 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.
The locally length minimizing curves of Riemannian and Finsler manifolds are called geodesics.
In the general case the homogeneity condition of the Finsler-function implies the following formulae: In terms of classical mechanics, 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: Therefore, a curve
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 semi-spray can be written as hH=ΘV.