In _{H}^{t}(ξ)∈''TM'' obey the rule Φ_{H}^{t}(λξ)=Φ_{H}^{λt}(ξ) in positive reparameterizations. If this requirement is dropped, ''H'' is called a semispray.
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.I. Bucataru, R. Miron, ''Finsler-Lagrange Geometry'', Editura Academiei Române, 2007.

_{''TM''},''M'') its tangent bundle. Then a vector field ''H'' on ''TM'' (that is, a Section (fiber bundle), section of the double tangent bundle ''TTM'') is a semispray 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 Double tangent bundle#Secondary vector bundle structure and canonical flip, 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_{*}(λ''H''_{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 (''x''^{''i''},ξ^{''i''}) be the local coordinates on ''TM'' associated with the local coordinates (''x''^{''i''}) on ''M'' using the coordinate basis on each tangent space. Then ''H'' is a semispray on ''M'' if and only if it has a local representation of the form
:$H\_\backslash xi\; =\; \backslash xi^i\backslash frac\backslash Big,\; \_\; -\; 2G^i(x,\backslash xi)\backslash frac\backslash Big,\; \_.$
on each associated coordinate system on ''TM''. The semispray ''H'' is a (full) spray, if and only if the spray coefficients ''G''^{''i''} satisfy
:$G^i(x,\backslash lambda\backslash xi)\; =\; \backslash lambda^2G^i(x,\backslash xi),\backslash quad\; \backslash lambda>0.\backslash ,$

_{''s''}:[''a'',''b'']→''M'' around γ(''t'') = γ_{0}(''t''). This first variation formula can be recast in a more informative form by introducing the following concepts:
* The covector $\backslash alpha\_\backslash xi\; =\; \backslash alpha\_i(x,\backslash xi)\; dx^i,\; \_x\backslash in\; T\_x^*M$ with $\backslash alpha\_i(x,\backslash xi)\; =\; \backslash tfrac(x,\backslash xi)$ is the conjugate momentum of $\backslash xi\; \backslash in\; T\_xM$.
* The corresponding one-form $\backslash alpha\backslash in\backslash Omega^1(TM)$ with $\backslash alpha\_\backslash xi\; =\; \backslash alpha\_i(x,\backslash xi)\; dx^i,\; \_\backslash in\; T^*\_\backslash xi\; TM$ is the Hilbert-form associated with the Lagrangian.
* The bilinear form $g\_\backslash xi\; =\; g\_(x,\backslash xi)(dx^i\backslash otimes\; dx^j),\; \_x$ with $g\_(x,\backslash xi)\; =\; \backslash tfrac(x,\backslash xi)$ is the fundamental tensor of the Lagrangian at $\backslash xi\; \backslash in\; T\_xM$.
* The Lagrangian satisfies the Legendre condition if the fundamental tensor $\backslash displaystyle\; g\_\backslash xi$ is non-degenerate at every $\backslash xi\; \backslash in\; T\_xM$. Then the inverse matrix of $\backslash displaystyle\; g\_(x,\backslash xi)$ is denoted by $\backslash displaystyle\; g^(x,\backslash xi)$.
* The Energy associated with the Lagrangian is $\backslash displaystyle\; E(\backslash xi)\; =\; \backslash alpha\_\backslash xi(\backslash xi)\; -\; L(\backslash xi)$.
If the Legendre condition is satisfied, then ''d''α∈Ω^{2}(''TM'') is a symplectic form, and there exists a unique Hamiltonian vector field ''H'' on ''TM'' corresponding to the Hamiltonian function ''E'' such that
:$\backslash displaystyle\; dE\; =\; -\; \backslash iota\_H\; d\backslash alpha$.
Let (''X''^{''i''},''Y''^{''i''}) be the components of the Hamiltonian vector field ''H'' in the associated coordinates on ''TM''. Then
:$\backslash iota\_H\; d\backslash alpha\; =\; Y^i\; \backslash frac\; dx^j\; -\; X^i\; \backslash frac\; d\backslash xi^j$
and
:$dE\; =\; \backslash Big(\backslash frac\backslash xi^j\; -\; \backslash frac\backslash Big)dx^i\; +\; \backslash xi^j\; \backslash frac\; d\backslash xi^i$
so we see that the Hamiltonian vector field ''H'' is a semispray on the configuration space ''M'' with the spray coefficients
:$G^k(x,\backslash xi)\; =\; \backslash frac\backslash Big(\backslash frac\backslash xi^j\; -\; \backslash frac\backslash Big).$
Now the first variational formula can be rewritten as
:$\backslash frac\backslash Big,\; \_\backslash mathcal\; S(\backslash gamma\_s)\; =\; \backslash Big,\; \_a^b\; \backslash alpha\_i\; X^i\; -\; \backslash int\_a^b\; g\_(\backslash ddot\backslash gamma^k+2G^k)X^i\; dt,$
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.

^{2}(''x'',ξ) = ''g''_{''ij''}(''x'')ξ^{''i''}ξ^{''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
:$F(x,\backslash lambda\backslash xi)\; =\; \backslash lambda\; F(x,\backslash xi),\; \backslash quad\; \backslash lambda>0$
of the Finsler-function implies the following formulae:
:$\backslash alpha\_i=g\_\backslash xi^i,\; \backslash quad\; F^2=g\_\backslash xi^i\backslash xi^j,\; \backslash quad\; E\; =\; \backslash alpha\_i\backslash xi^i\; -\; L\; =\; \backslash tfracF^2.$
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
:$g\_(\backslash lambda\backslash xi)\; =\; g\_(\backslash xi),\; \backslash quad\; \backslash alpha\_i(x,\backslash lambda\backslash xi)\; =\; \backslash lambda\; \backslash alpha\_i(x,\backslash xi),\; \backslash quad\; G^i(x,\backslash lambda\backslash xi)\; =\; \backslash lambda^2\; G^i(x,\backslash xi),$
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 $F(\backslash gamma(t),\backslash dot\backslash gamma(t))=\backslash lambda$, since the energy is automatically a constant of motion.
* For any curve $\backslash gamma:[a,b]\backslash to\; M$ of constant speed the action integral and the length functional are related by
:$\backslash mathcal\; S(\backslash gamma)\; =\; \backslash frac\; =\; \backslash frac.$
Therefore, a curve $\backslash gamma:[a,b]\backslash to\; M$ 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''.

another paper

Bucătaru, Constantinescu and Dahl relate this concept to that of the Kosambi biderivative operator. For a good introduction to Damodar Dharmananda Kosambi, Kosambi's methods, see the article, '

What is Kosambi-Cartan-Chern theory?

''.

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{{DEFAULTSORT:Spray (Mathematics)
Differential geometry
Finsler geometry

differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential geometry of curves, theor ...

, a spray is a vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each att ...

''H'' on the tangent bundle
Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).
In differen ...

''TM'' that encodes a 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''→ΦFormal definitions

Let ''M'' be a differentiable manifold and (''TM'',πSemisprays 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 :$\backslash mathcal\; S(\backslash gamma)\; :=\; \backslash int\_a^b\; L(\backslash gamma(t),\backslash dot\backslash gamma(t))dt$. In the associated coordinates on ''TM'' the first variation of the action integral reads as :$\backslash frac\backslash Big,\; \_\backslash mathcal\; S(\backslash gamma\_s)\; =\; \backslash Big,\; \_a^b\; \backslash fracX^i\; -\; \backslash int\_a^b\; \backslash Big(\backslash frac\; \backslash ddot\backslash gamma^j\; +\; \backslash frac\; \backslash dot\backslash gamma^j\; -\; \backslash frac\; \backslash Big)\; X^i\; dt,$ where ''X'':[''a'',''b'']→R is the variation vector field associated with the variation γGeodesic spray

The locally length minimizing curves of Riemannian manifold, 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 :$L(x,\backslash xi)\; =\; \backslash tfracF^2(x,\backslash xi),$ where ''F'':''TM''→R is the Finsler manifold, Finsler function. In the Riemannian case one uses ''F''Correspondence with nonlinear connections

A semispray ''H'' on a smooth manifold ''M'' defines an Ehresmann-connection ''T''(''TM''\0) = ''H''(''TM''\0) ⊕ ''V''(''TM''\0) on the slit tangent bundle through its horizontal and vertical projections :$h:T(TM\backslash setminus\; 0)\backslash to\; T(TM\backslash setminus\; 0)\; \backslash quad\; ;\; \backslash quad\; h\; =\; \backslash tfrac\backslash big(\; I\; -\; \backslash mathcal\; L\_H\; J\; \backslash big),$ :$v:T(TM\backslash setminus\; 0)\backslash to\; T(TM\backslash setminus\; 0)\; \backslash quad\; ;\; \backslash quad\; v\; =\; \backslash tfrac\backslash big(\; I\; +\; \backslash mathcal\; L\_H\; J\; \backslash big).$ 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 :$\backslash displaystyle\; T(X,Y)\; =\; J[hX,hY]\; -\; v[JX,hY)\; -\; v[hX,JY].$ Introducing the canonical vector field ''V'' on ''TM''\0 and the adjoint structure Θ of the induced connection the horizontal part of the semispray can be written as ''hH''=Θ''V''. The vertical part ε=''vH'' of the semispray is known as the first spray invariant, and the semispray ''H'' itself decomposes into :$\backslash displaystyle\; H\; =\; \backslash Theta\; V\; +\; \backslash epsilon.$ The first spray invariant is related to the tension :$\backslash tau\; =\; \backslash mathcal\; L\_Vv\; =\; \backslash tfrac\backslash mathcal\; L\_\; J$ of the induced non-linear connection through the ordinary differential equation :$\backslash mathcal\; L\_V\backslash epsilon+\backslash epsilon\; =\; \backslash tau\backslash Theta\; V.$ Therefore, the first spray invariant ε (and hence the whole semi-spray ''H'') can be recovered from the non-linear connection by :$\backslash epsilon,\; \_\backslash xi\; =\; \backslash int\backslash limits\_^0\; e^(\backslash Phi\_V^)\_*(\backslash tau\backslash Theta\; V),\; \_\; ds.$ From this relation one also sees that the induced connection is homogeneous if and only if ''H'' is a full spray.Jacobi fields of sprays and semisprays

A good source for Jacobi fields of semisprays is Section 4.4, ''Jacobi equations of a semispray'' of the publicly available book ''Finsler-Lagrange Geometry'' by Bucătaru and Miron. Of particular note is their concept of a dynamical covariant derivative. Ianother paper

Bucătaru, Constantinescu and Dahl relate this concept to that of the Kosambi biderivative operator. For a good introduction to Damodar Dharmananda Kosambi, Kosambi's methods, see the article, '

What is Kosambi-Cartan-Chern theory?

''.

References