Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 8355.6.5 Jacobi FieldsLet us consider the quadratic dynamical equation (5.81) and the equivalentgeodesic equation (5.82) with respect to the symmetric linear connection˜K (5.83). Its Riemann curvature tensorR λµαβ = ∂ λ K µαβ − ∂ µ K α λ β + K γ λ βK µαγ − K µγβ K α λ γhas the temporal componentR λµ0β = 0. (5.93)Then the equation for a Jacobi vector–field u along a geodesic c reads˙q β ˙q µ (∇ β (∇ µ u α ) − R λµαβ u λ ) = 0, ∇ β ˙q α = 0, (5.94)where ∇ µ denotes the covariant derivative relative to the connection ˜K (see[Kobayashi and Nomizu (1963/9)]). Due to the relation (5.93), the equation(5.94) for the temporal component u 0 of a Jacobi field takes the formWe chose its solution˙q β ˙q µ (∂ µ ∂ β u 0 + K µγβ ∂ γ u 0 ) = 0.u 0 = 0, (5.95)because all geodesics obey the constraint ṫ = 0.Note that, in the case of a quadratic Lagrangian L, the equation (5.94)coincides with the Jacobi equationu j d 0 (∂ j ˙∂i L) + d 0 ( ˙u j ˙∂i ˙∂j L) − u j ∂ i ∂ j L = 0for a Jacobi field on solutions of the Lagrangian equations for L. Thisequation is the Lagrangian equation for the vertical extension L V of the LagrangianL (see [Dittrich and Reuter (1994); Mangiarotti and Sardanashvily(1998); Mangiarotti et. al. (1999)]).Let us consider a quadratic Newtonian system with a Riemannian inertiatensor µ ij . Given a reference frame (t, q i ) ≡ q α , this inertia tensor isextended to the following Riemannian metric on Qg 00 = 1, g 0i = 0, g ij = µ ij .However, its covariant derivative with respect to the connection ˜K (5.83)does not vanish in general. Nevertheless, due to the relations (5.91) and