Let M be a differentiable manifold and (TM,π_TM,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:

• (π_TM)_*H_ξ = 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:

• H_λξ = λ_*(λH_ξ), 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→Φ_H^t(ξ)∈TM\0 of H satisfy Φ_H^t(λξ)=λΦ_H^λt(ξ) for any λ>0.

Let ${\displaystyle (x^{i},\xi ^{i})}$  be the local coordinates on ${\displaystyle TM}$  associated with the local coordinates ${\displaystyle (x^{i}}$ ) on ${\displaystyle M}$  using the coordinate basis on each tangent space. Then ${\displaystyle H}$  is a semi-spray on ${\displaystyle M}$  if it has a local representation of the form

${\displaystyle H_{\xi }=\xi ^{i}{\frac {\partial }{\partial x^{i}}}{\Big |}_{(x,\xi )}-2G^{i}(x,\xi ){\frac {\partial }{\partial \xi ^{i}}}{\Big |}_{(x,\xi )}.}$ 

on each associated coordinate system on TM. The semispray H is a (full) spray, if and only if the spray coefficients Gⁱ satisfy

${\displaystyle G^{i}(x,\lambda \xi )=\lambda ^{2}G^{i}(x,\xi ),\quad \lambda >0.\,}$ 

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

${\displaystyle {\mathcal {S}}(\gamma ):=\int _{a}^{b}L(\gamma (t),{\dot {\gamma }}(t))dt}$ .

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

${\displaystyle {\frac {d}{ds}}{\Big |}_{s=0}{\mathcal {S}}(\gamma _{s})={\Big |}_{a}^{b}{\frac {\partial L}{\partial \xi ^{i}}}X^{i}-\int _{a}^{b}{\Big (}{\frac {\partial ^{2}L}{\partial \xi ^{j}\partial \xi ^{i}}}{\ddot {\gamma }}^{j}+{\frac {\partial ^{2}L}{\partial x^{j}\partial \xi ^{i}}}{\dot {\gamma }}^{j}-{\frac {\partial L}{\partial x^{i}}}{\Big )}X^{i}dt,}$ 

where X:[a,b]→R is the variation vector field associated with the variation γ_s:[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:

• The covector ${\displaystyle \alpha _{\xi }=\alpha _{i}(x,\xi )dx^{i}|_{x}\in T_{x}^{*}M}$  with ${\displaystyle \alpha _{i}(x,\xi )={\tfrac {\partial L}{\partial \xi ^{i}}}(x,\xi )}$  is the conjugate momentum of ${\displaystyle \xi \in T_{x}M}$ .
• The corresponding one-form ${\displaystyle \alpha \in \Omega ^{1}(TM)}$  with ${\displaystyle \alpha _{\xi }=\alpha _{i}(x,\xi )dx^{i}|_{(x,\xi )}\in T_{\xi }^{*}TM}$  is the Hilbert-form associated with the Lagrangian.
• The bilinear form ${\displaystyle g_{\xi }=g_{ij}(x,\xi )(dx^{i}\otimes dx^{j})|_{x}}$  with ${\displaystyle g_{ij}(x,\xi )={\tfrac {\partial ^{2}L}{\partial \xi ^{i}\partial \xi ^{j}}}(x,\xi )}$  is the fundamental tensor of the Lagrangian at ${\displaystyle \xi \in T_{x}M}$ .
• The Lagrangian satisfies the Legendre condition if the fundamental tensor ${\displaystyle \displaystyle g_{\xi }}$  is non-degenerate at every ${\displaystyle \xi \in T_{x}M}$ . Then the inverse matrix of ${\displaystyle \displaystyle g_{ij}(x,\xi )}$  is denoted by ${\displaystyle \displaystyle g^{ij}(x,\xi )}$ .
• The Energy associated with the Lagrangian is ${\displaystyle \displaystyle E(\xi )=\alpha _{\xi }(\xi )-L(\xi )}$ .

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

${\displaystyle \displaystyle dE=-\iota _{H}d\alpha }$ .

Let (Xⁱ,Yⁱ) be the components of the Hamiltonian vector field H in the associated coordinates on TM. Then

${\displaystyle \iota _{H}d\alpha =Y^{i}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}dx^{j}-X^{i}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}d\xi ^{j}}$ 

and

${\displaystyle dE={\Big (}{\frac {\partial ^{2}L}{\partial x^{i}\partial \xi ^{j}}}\xi ^{j}-{\frac {\partial L}{\partial x^{i}}}{\Big )}dx^{i}+\xi ^{j}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}d\xi ^{i}}$ 

so we see that the Hamiltonian vector field H is a semi-spray on the configuration space M with the spray coefficients

${\displaystyle G^{k}(x,\xi )={\frac {g^{ki}}{2}}{\Big (}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}\xi ^{j}-{\frac {\partial L}{\partial x^{i}}}{\Big )}.}$ 

Now the first variational formula can be rewritten as

${\displaystyle {\frac {d}{ds}}{\Big |}_{s=0}{\mathcal {S}}(\gamma _{s})={\Big |}_{a}^{b}\alpha _{i}X^{i}-\int _{a}^{b}g_{ik}({\ddot {\gamma }}^{k}+2G^{k})X^{i}dt,}$ 

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. Using the framework of Lagrangian mechanics one can describe these curves with spray structures. Define a Lagrangian function on TM by

${\displaystyle L(x,\xi )={\tfrac {1}{2}}F^{2}(x,\xi ),}$ 

where F:TM→R is the Finsler function. In the Riemannian case one uses F²(x,ξ) = g_ij(x)ξⁱξ^j. 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

${\displaystyle F(x,\lambda \xi )=\lambda F(x,\xi ),\quad \lambda >0}$ 

of the Finsler-function implies the following formulae:

${\displaystyle \alpha _{i}=g_{ij}\xi ^{i},\quad F^{2}=g_{ij}\xi ^{i}\xi ^{j},\quad E=\alpha _{i}\xi ^{i}-L={\tfrac {1}{2}}F^{2}.}$ 

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

${\displaystyle g_{ij}(\lambda \xi )=g_{ij}(\xi ),\quad \alpha _{i}(x,\lambda \xi )=\lambda \alpha _{i}(x,\xi ),\quad G^{i}(x,\lambda \xi )=\lambda ^{2}G^{i}(x,\xi ),}$ 

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 ${\displaystyle F(\gamma (t),{\dot {\gamma }}(t))=\lambda }$ , since the energy is automatically a constant of motion.
• For any curve ${\displaystyle \gamma :[a,b]\to M}$  of constant speed the action integral and the length functional are related by

${\displaystyle {\mathcal {S}}(\gamma )={\frac {(b-a)\lambda ^{2}}{2}}={\frac {\ell (\gamma )^{2}}{2(b-a)}}.}$ 

Therefore, a curve ${\displaystyle \gamma :[a,b]\to M}$  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_H^t(ξ) is called the geodesic flow.

A semi-spray ${\displaystyle H}$  on a smooth manifold ${\displaystyle M}$  defines an Ehresmann-connection ${\displaystyle T(TM\setminus 0)=H(TM\setminus 0)\oplus V(TM\setminus 0)}$  on the slit tangent bundle through its horizontal and vertical projections

${\displaystyle h:T(TM\setminus 0)\to T(TM\setminus 0)\quad ;\quad h={\tfrac {1}{2}}{\big (}I-{\mathcal {L}}_{H}J{\big )},}$ 

${\displaystyle v:T(TM\setminus 0)\to T(TM\setminus 0)\quad ;\quad v={\tfrac {1}{2}}{\big (}I+{\mathcal {L}}_{H}J{\big )}.}$ 

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

${\displaystyle \displaystyle T(X,Y)=J[hX,hY]-v[JX,hY]-v[hX,JY].}$ 

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. The vertical part ε=vH of the semispray is known as the first spray invariant, and the semispray H itself decomposes into

${\displaystyle \displaystyle H=\Theta V+\epsilon .}$ 

The first spray invariant is related to the tension

${\displaystyle \tau ={\mathcal {L}}_{V}v={\tfrac {1}{2}}{\mathcal {L}}_{[V,H]-H}J}$ 

of the induced non-linear connection through the ordinary differential equation

${\displaystyle {\mathcal {L}}_{V}\epsilon +\epsilon =\tau \Theta V.}$ 

Therefore, the first spray invariant ε (and hence the whole semi-spray H) can be recovered from the non-linear connection by

${\displaystyle \epsilon |_{\xi }=\int \limits _{-\infty }^{0}e^{-s}(\Phi _{V}^{-s})_{*}(\tau \Theta V)|_{\Phi _{V}^{s}(\xi )}ds.}$ 

From this relation one also sees that the induced connection is homogeneous if and only if H is a full spray.

A good source for Jacobi fields of semisprays is Section 4.4, Jacobi equations of a semi-spray of the publicly available book Finsler-Lagrange Geometry by Bucătaru and Miron. Of particular note is their concept of a dynamic covariant derivative. In another paper Bucătaru, Constantinescu and Dahl relate this concept to that of the Kosambi bi-derivative operator.

For a good introduction to Kosambi's methods, see the article, What is Kosambi-Cartan-Chern theory?.

• Sternberg, Shlomo (1964), Lectures on Differential Geometry, Prentice-Hall.
• Lang, Serge (1999), Fundamentals of Differential Geometry, Springer-Verlag.