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. If this requirement is dropped, H is called a semi-spray. Sprays arise naturally in Riemannian and Finsler geometry as the geodesic sprays whose integral curves are precisely the tangent curves of locally length minimizing curves. Semisprays arise naturally as the extremal curves of action integrals in Lagrangian mechanics. Generalizing all these examples, any (possibly nonlinear) connection on M induces a semispray H, and conversely, any semispray H induces a torsion-free nonlinear connection on M. If the original connection is torsion-free it coincides with the connection induced by H, and homogeneous torsion-free connections are in one-to-one correspondence with full sprays.
Formal definitions Let M be a differentiable manifold and (TM,πTM,M) its tangent bundle. 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:
(πTM)*Hξ = ξ. JH=V, where J is the tangent structure on TM and V is the canonical vector field on TM\0. j∘H=H, where j:TTM→TTM is the canonical flip and H is seen as a mapping TM→TTM. A semispray H on M is a (full) spray if any of the following equivalent conditions hold:
Hλξ = λ*(λHξ), where λ*:TTM→TTM is the push-forward of the multiplication λ:TM→TM by a positive scalar λ>0. The Lie-derivative of H along the canonical vector field V satisfies [V,H]=H. The integral curves t→ΦHt(ξ)∈TM\0 of H satisfy ΦHt(λξ)=λΦHλt(ξ) for any λ>0. Let ( x i , ξ i ) {\displaystyle (x^{i},\xi ^{i})} be the local coordinates on T M {\displaystyle TM} associated with the local coordinates ( x i {\displaystyle (x^{i}} ) on M {\displaystyle M} using the coordinate basis on each tangent space. Then H {\displaystyle H} is a semi-spray on M {\displaystyle M} if it has a local representation of the form
H ξ = ξ i ∂ ∂ x i | ( x , ξ ) − 2 G i ( x , ξ ) ∂ ∂ ξ i | ( x , ξ ) . {\displaystyle H_{\xi }=\xi ^{i}{\frac {\partial }{\partial x^{i}}}{\Big |}_{(x,\xi )}-2G^{i}(x,\xi ){\frac {\partial }{\partial \xi ^{i}}}{\Big |}_{(x,\xi )}.}
on each associated coordinate system on TM. The semispray H is a (full) spray, if and only if the spray coefficients Gi satisfy
G i ( x , λ ξ ) = λ 2 G i ( x , ξ ) , λ > 0. {\displaystyle G^{i}(x,\lambda \xi )=\lambda ^{2}G^{i}(x,\xi ),\quad \lambda >0.\,}
Semi-sprays in Lagrangian mechanics 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
S ( γ ) := ∫ a b L ( γ ( t ) , γ ˙ ( t ) ) d t {\displaystyle {\mathcal {S}}(\gamma ):=\int _{a}^{b}L(\gamma (t),{\dot {\gamma }}(t))dt} . In the associated coordinates on TM the first variation of the action integral reads as
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