Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 423As a Lie group, the configuration manifold M is Hausdorff. Thereforefor x = (q i , p i ) ∈ U p (U p open in T ∗ M) there exists a uniqueone–parameter group of diffeomorphisms φ t : T ∗ M → T ∗ M such thatddt | t=0 φ t x = J∇H(x). This is termed Hamiltonian phase–flow and representsthe maximal integral curve t ↦→ (q i (t), p i (t)) of the Hamiltonianvector–field X H passing through the point x for t = 0.The flow φ t is symplectic if ω H is constant along it (that is, φ ∗ t ω H = ω H )iff its Lie derivative vanishes (that is, L XH ω H = 0). A symplectic flow consistsof canonical transformations on T ∗ M, that is, local diffeomorphismsthat leave ω H invariant. By Liouville Theorem, a symplectic flow φ t preservesthe phase volume on T ∗ M. Also, the total energy H = E of thesystem is conserved along φ t , that is, H ◦ φ t = φ t .Lagrangian flow can be defined analogously (see [Abraham and Marsden(1978); Marsden and Ratiu (1999)]).For a Lagrangian (resp. a Hamiltonian) vector–field X L (resp. X H )on M, there is a base integral curve γ 0 (t) = (q i (t), v i (t)) (resp. γ 0 (t) =(q i (t), p i (t))) iffγ 0 (t) is a geodesic. This is given by the contravariant velocityequation˙q i = v i , ˙v i + Γ i jk v j v k = 0, (3.235)in the former case, and by the covariant momentum equation˙q k = g ki p i , ṗ i + Γ i jk g jl g km p l p m = 0, (3.236)in the latter. As before, Γ i jkdenote the Christoffel symbols of an affineconnection ∇ in an open chart U on M, defined by the Riemannian metricg = as: Γ i jk = ( ) gil Γ jkl , Γ jkl = 1 2 ∂q j g kl + ∂ q kg jl − ∂ q lg jk .The l.h.s ˙¯v i = ˙v i + Γ i jk vj v k (resp. ˙¯p i = ṗ i + Γ i jk gjl g km p l p m ) inthe second parts of (3.235) and (3.236) represent the Bianchi covariantderivative of the velocity (resp. momentum) with respect to t. Paralleltransport on M is defined by ˙¯v i = 0, (resp. ˙¯p i = 0). When this applies,X L (resp. X H ) is called the geodesic spray and its flow the geodesic flow.For the dynamics in the gravitational potential field V : M → R, theLagrangian L : T M → R (resp. the Hamiltonian H : T ∗ M → R) has anextended formL(v, q) = 1 2 g ijv i v j − V (q),(resp.H(p, q) = 1 2 gij p i p j + V (q)).