Ivancevic_Applied-Diff-Geom

832 Applied Differential Geometry: A Modern Introductionis equivalent to the geodesic equationṫ = 1, ẗ = 0,for the symmetric linear connection¨q i = a i jk(q µ ) ˙q i ˙q j + b i j(q µ ) ˙q j ṫ + f i (q µ )ṫṫ. (5.82)˜K = dq α ⊗ (∂ α + K µ αν(t, q i ) ˙q ν ˙∂µ )on T Q → Q, given by the componentsK 0 αν = 0, K 0i0 = f i , K 0ij = K ji0 = 1 2 bi j, K jik = a i jk. (5.83)Conversely, any linear connection K on the tangent bundle T Q → Qdefines the quadratic dynamical equationq i tt = K jik q j t q k t + (K 0ij + K ji0 )q j t + K 0i0 ,written with respect to a given reference frame (t, q i ) ≡ q µ .However, the geodesic equation (5.82) is not unique for the dynamicalequation (5.81). Any quadratic dynamical equation (5.80), being equivalentto the geodesic equation with respect to the linear connection ˜K (5.83), isalso equivalent to the geodesic equation with respect to an affine connectionK ′ on T Q → Q which differs from ˜K (5.83) in a soldering form σ onT Q → Q with the local componentsσ 0 α = 0, σ i k = h i k + (s − 1)h i k ˙q 0 , σ i 0 = −sh i k ˙q k − h i 0 ˙q 0 + h i 0,where s and h i α are local functions on Q.5.6.3 Equation of Free–MotionWe say that the dynamical equation (5.62), that is: qtt i = ξ i (t, q i , qt),iis a free motion equation iff there exists a reference frame (t, q i ) on theconfiguration bundle Q → R such that this equation readsq i tt = 0. (5.84)With respect to arbitrary bundle coordinates (t, q i ), a free motion equationtakes the formq i tt = d t Γ i + ∂ j Γ i (q j t − Γ j ) − ∂qi∂q µ ∂q µ∂q j ∂q k (qj t − Γ j )(q k t − Γ k ), (5.85)