Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 825its tangent bundle T M. However, in modern geometrical settings of non–autonomous (see [Giachetta et. al. (1997); Mangiarotti et. al. (1999); Mangiarottiand Sardanashvily (2000a); Saunders (1989); Sardanashvily (1993);Sardanashvily (1995); Sardanashvily (2002a)]), the configuration manifoldof time–dependent mechanics is a fibre bundle Q → R, called the configurationbundle, coordinated by (t, q i ), where t ∈ R is a Cartesian coordinateon the time axis R with the transition functions t ′ = t+const. The correspondingvelocity phase–space is the 1–jet space J 1 (R, Q), which admitsthe adapted coordinates (t, q i , qt). i It was proved in [Giachetta (1992);León et. al. (1996); Mangiarotti and Sardanashvily (1998)] that everydynamical equation ξ defines a connection on the affine jet bundleJ 1 (R, Q) → Q, and vice versa.Due to the canonical imbedding J 1 (R, Q) → T Q, every dynamical connectioninduces a nonlinear connection on the tangent bundle T Q → Q, andvice versa. As a consequence, every dynamical equation on Q induces anequivalent geodesic equation on the tangent bundle T Q → Q in accordancewith the following proposition. Given a configuration bundle Q → R, coordinatedby (t, q i ), and its 2–jet space J 2 (R, Q), coordinated by (t, q i , qt, i qtt),iany dynamical equation ξ on the configuration bundle Q → R,q i tt = ξ i (t, q i , q i t) (5.62)is equivalent to the geodesic equation with respect to a connection ˜K onthe tangent bundle T Q → Q,which fulfills the conditionsṫ = 1, ẗ = 0, ¨q i = ˜K i 0 + ˜K i j ˙q j ,˜K 0 α = 0, ξ i = ˜K i 0 + q j t ˜K i j |ṫ=1, ˙qi =q i t . (5.63)Recall that the 1–jet space J 1 (R, Q) is defined as the set of equivalenceclasses j 1 t c of sections c i : R → Q of the fibre bundle Q → R, which areidentified by their values c i (t) and the values of their partial derivatives∂ t c i = ∂ t c i (t) at time points t ∈ R. Also recall that there is the canonicalimbeddingλ : J 1 (R, Q) ↩→ T Q, locally given by λ = d t = ∂ t + q i t∂ i , (5.64)where d t denotes the total time derivative. From now on, we will identifyJ 1 (R, Q) with its image in the tangent bundle T Q. This is an affine bundlemodelled over the vertical tangent bundle V Q of Q → R.