Ivancevic_Applied-Diff-Geom

826 Applied Differential Geometry: A Modern IntroductionAs a consequence of (5.64), every connection Γ on a fibre bundle Q → R,Γ : Q → J 1 (R, Q), locally given by Γ = dt ⊗ (∂ t + Γ i ∂ i ), (5.65)is identified with the nowhere vanishing vector–field on Q [Mangiarotti andSardanashvily (1998); Mangiarotti et. al. (1999)],Γ : Q → J 1 (R, Q) ⊂ T Q, locally given by Γ = ∂ t + Γ i ∂ i . (5.66)This is the horizontal lift of the standard vector–field ∂ t on R by means ofthe connection (5.65). Conversely, any vector–field Γ on Q such that dt⌋Γ =1 defines a connection on Q → R. Therefore, the covariant differentialassociated with a connection Γ on Q → R readsD G : J 1 (R, Q) → V Q, locally given by ˙q i ◦ D G = q i t − Γ i .Let J 1 (R, J 1 (R, Q)) denote the (repeated) 1–jet space of the jet bundleJ 1 (R, Q) → R, coordinated by (t, q i , q i t, q i (t) , qi tt). The corresponding 2–jetspace J 2 (R, Q) of the fibre bundle Q → R is the holonomic subbundleq i t = q i (t) of J 1 (R, J 1 (R, Q)), coordinated by (t, q i , q i t, q i tt). There are theimbeddings˚λJ 2 (R, Q) −→ T J 1 (R, Q) −→ T λT T Q, with˚λ : (t, q i , qt, i qtt) i ↦→ (t, q i , qt, i ṫ = 1, ˙q i = qt, i ˙q t i = qtt). i (5.67)T λ ◦ ˚λ : (t, q i , q i t, q i tt) ↦→ (t, q i , ṫ = 1, ˙q i = q i t, ẗ = 0, ¨q i = q i tt), (5.68)where (t, q i , ˙q i , ¨q i ) are holonomic coordinates on the second tangent bundleT T Q. This global geometrical structure of time–dependent mechanics isdepicted in Figure 5.3.Therefore, a dynamical equation ξ on a configuration bundle Q → R,given in local coordinates by (5.62), is defined as the geodesic equationKer D ξ ⊂ J 2 (R, Q) for a holonomic connection ξ on the jet bundleJ 1 (R, Q) → R. Due to the map (5.67), a holonomic connection ξ is representedby the horizontal vector–field on J 1 (R, Q),ξ = ∂ t + q i t∂ i + ξ i (q µ , q i t)∂ t i. (5.69)A dynamical equation ξ is said to be conservative if there exists a trivializationQ ∼ = R×M such that the vector–field ξ (5.69) on J 1 (R, Q) ∼ = R×T Mis projectable onto T M. Then this projectionΞ ξ = ˙q i ∂ i + ξ i (q j , ˙q j ) ˙∂ i