Ivancevic_Applied-Diff-Geom

834 Applied Differential Geometry: A Modern Introductionare the Christoffel symbols of the first–kind of the metric γ αµ given incomponents by (5.88). The corresponding geodesic equation (5.82) on T Qreads¨q i = (µ −1 ) ik Γ λkν ˙q α ˙q ν , ṫ = 1, ẗ = 0, (5.90)such that ˜K (5.63) is a linear connection with the following components˜K 0 αν = 0, ˜Ki αν = (µ −1 ) ik Γ λkν .We have the relation˙q α (∂ α µ ij + K i αν ˙q ν ) = 0. (5.91)One can show that an arbitrary Lagrangian system on a configurationbundle Q → R is a particular Newtonian system on Q → R. The latteris defined as a pair (ξ, µ) of a dynamical equation ξ and a (degenerate)fibre metric µ in the fibre bundle V Q J 1 (R, Q) → J 1 (R, Q) which satisfythe symmetry condition ∂ t k µ ij = ∂ t j µ ik and the compatibility condition[Mangiarotti and Sardanashvily (1998); Mangiarotti et. al. (1999)]ξ⌋dµ ij + µ ik γ k j + µ jk γ k i = 0, (5.92)where γ ξ is the dynamical connection (5.72), i.e.,γ ξ = dt ⊗ [∂ t + (ξ i − 1 2 qj t ∂ t jξ i )∂ t i] + dq j ⊗ [∂ j + 1 2 ∂t jξ i ∂ t i].We restrict our consideration to non–degenerate quadratic Newtoniansystems when ξ is a quadratic dynamical equation (5.80) and µ is a Riemannianfibre metric in V Q, i.e., µ is independent of q i t and the symmetrycondition becomes trivial. In this case, the dynamical equation (5.81) isequivalent to the geodesic equation (5.82) with respect a symmetric linearconnection ˜K (5.83), while the compatibility condition (5.92) takes theform (5.91).Given a symmetric linear connection ˜K (5.83) on the tangent bundleT Q → Q, one can consider the equation for Jacobi vector–fields alonggeodesics of this connection, i.e., along solutions of the dynamical equation(5.81). If Q admits a Riemannian metric, the conjugate points of thesegeodesic can be investigated.