Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 829It follows that the diagram (5.75) can be commutative only if the componentsK 0 µ of the connection K on T Q → Q vanish. Since the transitionfunctions t → t ′ are independent of q i , a connection˜K = dq α ⊗ (∂ α + K i α ˙∂ i ) (5.76)with the components K 0 µ = 0 can exist on the tangent bundle T Q → Q. Itobeys the transformation lawK ′ iα = (∂ j x ′i Kµ j + ∂ µ ẋ ′i ) ∂qµ . (5.77)∂x′αNow the diagram (5.75) becomes commutative if the connections γ and ˜Kfulfill the relationγ i µ = K i µ(t, q i , ṫ = 1, ˙q i = q i t),which holds globally since the substitution of ˙q i = q i t into (5.77) restatesthe coordinate transformation law of γ.Every dynamical equation (5.62) on the configuration bundle Q → Rcan be written in the formq i tt = K i 0 ◦ λ + q j t K i j ◦ λ, (5.78)where ˜K is a connection (5.76) on the tangent bundle T Q → Q. Conversely,each connection ˜K (5.76) on T Q → Q defines the dynamical equation (5.78)on Q → R.Consider the geodesic equation (5.6) on T Q with respect to the connection˜K. Its solution is a geodesic curve c(t) which also satisfies thedynamical equation (5.62), and vice versa.From the physical viewpoint, a reference frame in mechanics on a configurationbundle Q → R sets a tangent vector at each point of Q whichcharacterizes the velocity of an ‘observer’ at this point. Then any connectionΓ on Q → R is said to be such a reference frame [Echeverría et. al.(1995); Mangiarotti and Sardanashvily (1998); Massa and Pagani (1994);Sardanashvily (1998)].Each connection Γ on a fibre bundle Q → R defines an atlas of localconstant trivializations of Q → R whose transition functions are independentof t, and vice versa. One finds Γ = ∂ t with respect to this atlas. Inparticular, there is 1–1 correspondence between the complete connectionsΓ (5.66) on Q → R and the trivializations of this bundle.