Ivancevic_Applied-Diff-Geom

378 Applied Differential Geometry: A Modern IntroductionTheorem on the existence and uniqueness of the solution of a system ofODEs with smooth r.h.s, that if the manifold (P, ω) is Hausdorff, then forany point x = (q i , p i ) ∈ U, U open in P , there exists a maximal integralcurve of X H = J∇H, passing for t = 0, through point x. In case X H iscomplete, i.e., X H is C p and (P, ω) is compact, the maximal integral curveof X H is the Hamiltonian phase–flow φ t : U → U.The phase–flow φ t is symplectic if ω is constant along φ t , i.e., φ ∗ t ω = ω(φ ∗ t ω denotes the pull–back of ω by φ t ),iff L XH ω = 0(L XH ω denotes the Lie derivative of ω upon X H ).Symplectic phase–flow φ t consists of canonical transformations on(P, ω), i.e., diffeomorphisms in canonical coordinates q i , p i ∈ U, U openon all (P, ω) which leave ω invariant. In this case the Liouville Theoremis valid: φ t preserves the phase volume on (P, ω). Also, the system’s totalenergy H is conserved along φ t , i.e., H ◦ φ t = φ t .Recall that the Riemannian metrics g = on the configuration manifoldM is a positive–definite quadratic form g : T M → R, in local coordinatesq i ∈ U, U open in M, given by (3.139–3.140) above. Given themetrics g ij , the system’s Hamiltonian function represents a momentump−-dependent quadratic form H : T ∗ M → R – the system’s kinetic energyH(p) = T (p) = 1 2 < p, p >, in local canonical coordinates qi , p i ∈ U p ,U p open in T ∗ M, given byH(p) = 1 2 gij (q, m) p i p j , (3.187)where g ij (q, m) = g −1ij (q, m) denotes the inverse (contravariant) materialmetric tensorg ij (q, m) =n∑χ=1m χ δ rs∂q i∂x r ∂q j∂x s .T ∗ M is an orientable manifold, admitting the standard volume formΩ ωH=N(N+1)(−1) 2N!ω N H.For Hamiltonian vector–field, X H on M, there is a base integral curveγ 0 (t) = ( q i (t), p i (t) ) iff γ 0 (t) is a geodesic, given by the one–form forceequation˙¯p i ≡ ṗ i + Γ i jk g jl g km p l p m = 0, with ˙q k = g ki p i , (3.188)