Ivancevic_Applied-Diff-Geom

830 Applied Differential Geometry: A Modern IntroductionGiven a reference frame Γ, any connection K (5.74) on the tangentbundle T Q → Q defines the dynamical equationξ i = (K i α − Γ i K 0 α) ˙q α |ṫ=1, ˙q i =q i t . (5.79)Given a connection Γ on the fibre bundle Q → R and a connection Kon the tangent bundle T Q → Q, there is the connection ˜K on T Q → Qwith the components˜K 0 α = 0, ˜Ki α = K i α − Γ i K 0 α.5.6.1 GeodesicsIn this subsection we continue our study of non–autonomous, time–dependent mechanics on a configuration bundle Q → R, that we started insubsection 5.6 above. Recall that R is the time axis, while the correspondingvelocity phase–space manifold is the 1–jet space J 1 (R, Q) of sectionss : R −→ Q of the bundle Q → R. Also, recall that second–order dynamicalequation (dynamical equation, for short) on a fibre bundle Q → R is definedas a first–order dynamical equation on the jet bundle J 1 (R, Q) → R,given by a holonomic connection ξ on J 1 (R, Q) → R which takes its valuesin the 2–jet space J 2 (R, Q) ⊂ J 1 (R, J 1 (R, Q)) (see [León et. al. (1996);Mangiarotti and Sardanashvily (1998); Massa and Pagani (1994); Mangiarottiand Sardanashvily (1999)]). The global geometrical structure oftime–dependent mechanics is depicted in Figure 5.3 above.Since a configuration bundle Q → R is trivial, the existent formulationsof mechanics often imply its preliminary splitting Q = R × M [Cariñenaand F.Núñez (1993); Echeverría et. al. (1991); León et. al. (1996);Morandi et. al. (1990)]. This is not the case of mechanical systems subjectto time–dependent transformations, including inertial frame transformations.Recall that different trivializations of Q → R differ from eachother in projections Q → M. Since a configuration bundle Q → R hasno canonical trivialization in general, mechanics on Q → R is not a repetitionof mechanics on R × M, but implies additionally a connection onQ → R which is a reference frame [Mangiarotti and Sardanashvily (1998);Sardanashvily (1998)]. Considered independently on a trivialization ofQ → R, mechanical equations make the geometrical sense of geodesic equations.We now examine quadratic dynamical equations in details. In thiscase, the corresponding dynamical connection γ on J 1 (R, Q) → Q is