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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 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.〕 == Formal definitions == 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 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 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 ()=''H''. * The integral curves ''t''→ΦHt(ξ)∈''TM''\0 of ''H'' satisfy ΦHt(λξ)=ΦHλt(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 : on each associated coordinate system on ''TM''. The semispray ''H'' is a (full) spray, if and only if the spray coefficients ''G''''i'' satisfy : 抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)』 ■ウィキペディアで「Spray (mathematics)」の詳細全文を読む スポンサード リンク
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