Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 831affine, while the connection ˜K (5.63) on T Q → Q is linear. Then theequation for Jacobi vector–fields along the geodesics of the connection˜K can be considered. This equation coincides with the existent equationfor Jacobi fields of a Lagrangian system [Dittrich and Reuter (1994);Mangiarotti and Sardanashvily (1998)] in the case of non–degeneratequadratic Lagrangians, when they can be compared. We will considermore general case of quadratic Newtonian systems characterized by apair (ξ, µ) of a quadratic dynamical equation ξ and a Riemannian inertiatensor µ which satisfy a certain compatibility condition. Given areference frame, a Riemannian inertia tensor µ is extended to a Riemannianmetric on the configuration space Q. Then conjugate points of solutionsof the dynamical equation ξ can be examined in accordance withthe well–known geometrical criteria [Mangiarotti and Sardanashvily (1998);Sardanashvily (1998)].5.6.2 Quadratic Dynamical EquationsFrom the physical viewpoint, the most interesting dynamical equations arethe quadratic ones, i.e.,ξ i = a i jk(q µ )q j t q k t + b i j(q µ )q j t + f i (q µ ). (5.80)This property is coordinate–independent due to the affine transformationlaw of coordinates q i t. Then, it is clear that the corresponding dynamicalconnection γ ξ (5.72) is affine:γ = dq α ⊗ [∂ α + (γ i λ0(q ν ) + γ i λj(q ν )q j t )∂ t i],and vice versa. This connection is symmetric iff γ i λµ = γi µλ .There is 1–1 correspondence between the affine connections γ on theaffine jet bundle J 1 (R, Q) → Q and the linear connections ˜K (5.76) on thetangent bundle T Q → Q.This correspondence is given by the relationγ i µ = γ i µ0 + γ i µjq j t , γ i µλ = K µiα .In particular, if an affine dynamical connection γ is symmetric, so is thecorresponding linear connection ˜K.Any quadratic dynamical equationq i tt = a i jk(q µ )q j t q k t + b i j(q µ )q j t + f i (q µ ) (5.81)