Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 379where Γ i jkdenote Christoffel symbols of an affine Levi–Civita connectionon M, defined upon the Riemannian metric g = by (3.143).The l.h.s ˙¯p i of the covariant momentum equation (3.188) represents theintrinsic or Bianchi covariant derivative of the momentum with respectto time t. Basic relation ˙¯p i = 0 defines the parallel transport on T N ,the simplest form of human–motion dynamics. In that case Hamiltonianvector–field X H is called the geodesic spray and its phase–flow is called thegeodesic flow.For Earthly dynamics in the gravitational potential field V : M → R,the Hamiltonian H : T ∗ M → R (3.187) extends into potential formH(p, q) = 1 2 gij p i p j + V (q),with Hamiltonian vector–field X H = J∇H still defined by canonical equations(3.164).A general form of a driven, non–conservative Hamiltonian equationsreads:˙q i = ∂ pi H, ṗ i = F i − ∂ q iH, (3.189)where F i = F i (t, q, p) represent any kind of joint–driving covariant torques,including active neuro–muscular–like controls, as functions of time, anglesand momenta, as well as passive dissipative and elastic joint torques. Inthe covariant momentum formulation (3.188), the non–conservative Hamiltonianequations (3.189) become˙¯p i ≡ ṗ i + Γ i jk g jl g km p l p m = F i , with ˙q k = g ki p i .3.13.2 Hamiltonian–Poisson Biodynamical SystemsRecall from subsection 3.12.3.3 above that Hamiltonian–Poisson mechanicsis a generalized form of classical Hamiltonian mechanics. Let (P, {}) be aPoisson manifold and H ∈ C ∞ (P, R) a smooth real valued function on P .The vector–field X H defined byX H (F ) = {F, H},is the Hamiltonian vector–field with energy function H. The triple(P, {}, H) we call the Hamiltonian–Poisson biodynamical system (HPBS)[Marsden and Ratiu (1999); Puta (1993); Ivancevic and Pearce (2001a)].