Ivancevic_Applied-Diff-Geom

828 Applied Differential Geometry: A Modern IntroductionIt follows that every dynamical connection γ (5.71) induces the dynamicalequation (5.62) on the configuration bundle Q → R, rewritten hereasq i tt = γ i 0 + q j t γ i j. (5.73)Different dynamical connections may lead to the same dynamical equation(5.73). The dynamical connection γ ξ (5.72), associated with a dynamicalequation, possesses the propertyγ k i = ∂ t iγ k 0 + q j t ∂ t iγ k j ,which implies the relation ∂j tγk i = ∂t i γk j . Such a dynamical connection iscalled symmetric. Let γ be a dynamical connection (5.71) and ξ γ the correspondingdynamical equation (5.6). Then the connection (5.72), associatedwith ξ γ , takes the formγ ξγki = 1 2 (γk i + ∂ t iγ k 0 + q j t ∂ t iγ k j ), γ ξγk0 = ξ k − q i tγ ξγki .Note that γ = γ ξγiff γ is symmetric.To explore the relation between the connections γ (5.71) on the affinejet bundle J 1 (R, Q) → Q and the connectionsK = dq α ⊗ (∂ α + K β α ˙∂ β ) (5.74)on the tangent bundle T Q → Q, consider the diagramJ 1 (R, J 1 j 1 λ(R, Q)) ✲ J 1 (Q, T Q)γ❄J 1 (R, Q)λK❄✲ T Q(5.75)where J 1 (Q, T Q) is the 1–jet space of the tangent bundle T Q → Q, coordinatedby (q α , ˙q α , ˙q α µ). The jet prolongation j 1 λ of the canonical imbeddingλ (5.64) over Q readsWe havej 1 λ : (t, q i , q i t, q i µt) ↦→ (t, q i , ṫ = 1, ˙q i = q i t, ṫ µ = 0, ˙q i µ = q i µt).j 1 λ ◦ γ : (t, q i , q i t) ↦→ (t, q i , ṫ = 1, ˙q i = q i t, ṫ µ = 0, ˙q i µ = γ i µ),K ◦ λ : (t, q i , q i t) ↦→ (t, q i , ṫ = 1, ˙q i = q i 0, ṫ µ = K 0 µ, ˙q i µ = K i µ).