Ivancevic_Applied-Diff-Geom

290 Applied Differential Geometry: A Modern Introductionby the second–order Lagrangian equationsd ∂Ldt ∂v i= ∂L∂q i . (3.141)For a Lagrangian vector–field X L on M, there is a base integral curveγ 0 (t) = (q i (t), v i (t)) iff γ 0 (t) is a geodesic. This is given by the contravariantvelocity equation˙q i = v i , ˙v i + Γ i jk v j v k = 0. (3.142)Here Γ i jkdenote the Christoffel symbols of the Levi–Civita connection ∇in an open chart U on M, defined on the Riemannian metric g = by(see section 3.10.1.1 above)Γ i jk = g il Γ jkl , Γ ijk = 1 2 (∂ x ig jk + ∂ x j g ki + ∂ x kg ij ). (3.143)The l.h.s ˙¯v i = ˙v i + Γ i jk vj v k in the second part of (3.142) representsthe Bianchi covariant derivative of the velocity with respect to t. Paralleltransport on M is defined by ˙¯v i = 0. When this applies, X L is called thegeodesic spray and its flow the geodesic flow.For the dynamics in the gravitational potential field V : M → R, theLagrangian L : T M → R has an extended formL(v, q) = 1 2 g ijv i v j − V (q),A Lagrangian vector–field X L is still defined by the second–order Lagrangianequations (3.141, 3.142).A general form of the forced, non–conservative Lagrangian equations isgiven asd ∂Ldt ∂v i− ∂L∂q i = F i (t, q i , v i )).Here the F i (t, q i , v i ) represent any kind of covariant forces as a functionsof time, coordinates and momenta. In covariant form we have˙q i = v i , g ij ( ˙v i + Γ i jk v j v k ) = F j (t, q i , v i )).