Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 827Fig. 5.3 Hierarchical geometrical structure of time–dependent mechanics. Note that(for simplicity) intermediate jet spaces, J 1 (R, J 1 (R, Q)) and T J 1 (R, Q), are not shown.is a second–order dynamical equation on a typical fibre M of Q → R,¨q i = Ξ i ξ. (5.70)Conversely, every second–order dynamical equation Ξ (5.70) on a manifoldM can be seen as a conservative dynamical equationξ Ξ = ∂ t + ˙q i ∂ i + u i ˙∂ion the trivial fibre bundle R × M → R.Now we can explore the fundamental relationship between the holonomicconnections ξ (5.69) on the 1−jet bundle J 1 (R, Q) → R and thedynamical connections γ on the affine 1−jet bundle J 1 (R, Q) → Q, givenbyγ = dq α ⊗ (∂ α + γ i α∂ t i), (q α ≡ (t, q i ), ∂ α ≡ (∂ t , ∂ i )). (5.71)Any dynamical connection γ (5.71) defines the holonomic connection ξ γon J 1 (R, Q) → R [Mangiarotti and Sardanashvily (1998)]ξ γ = ∂ t + q i t∂ i + (γ i 0 + q j t γ i j)∂ t i.Conversely, any holonomic connection ξ (5.69) on J 1 (R, Q) → R definesthe dynamical connectionγ ξ = dt ⊗ [∂ t + (ξ i − 1 2 qj t ∂ t jξ i )∂ t i] + dq j ⊗ [∂ j + 1 2 ∂t jξ i ∂ t i]. (5.72)