Ivancevic_Applied-Diff-Geom

380 Applied Differential Geometry: A Modern IntroductionThe map F ↦→ {F, H} is a derivation on the space C ∞ (P, R), hence it definesa vector–field on P . The map F ∈ C ∞ (P, R) ↦→ X F ∈ X (P ) is a Liealgebra anti–homomorphism, i.e., [X F , X g ] = −X {F,g} .Let (P, {}, H) be a HPBS and φ t the flow of X H . Then for all F ∈C ∞ (P, R) we have the conservation of energy:H ◦ φ t = H,and the equations of motion in Poisson bracket form,ddt (F ◦ φ t) = {F, H} ◦ φ t = {F ◦ φ t , H},that is, the above Poisson evolution equation (3.179) holds. Now, the functionF is constant along the integral curves of the Hamiltonian vector–fieldX H iff{F, H} = 0.φ t preserves the Poisson structure.Next we present two main examples of HPBS.‘Ball–and–Socket’ Joint Dynamics in Euler Vector FormThe dynamics of human body–segments, classically modelled via Lagrangianformalism (see [Hatze (1977b); Ivancevic (1991); Ivancevic et al.(1995); Ivancevic and Ivancevic (2006)]), may be also prescribed by Euler’sequations of rigid body dynamics. The equations of motion for a free rigidbody, described by an observer fixed on the moving body, are usually givenby Euler’s vector equationṗ = p × w. (3.190)Here p, w ∈ R 3 , p i = I i w i and I i (i = 1, 2, 3) are the principal momentsof inertia, the coordinate system in the segment is chosen so that the axesare principal axes, w is the angular velocity of the body and p is the correspondingangular momentum.The kinetic energy of the segment is the Hamiltonian function H : R 3 →R given by [Ivancevic and Pearce (2001a)]H(p) = 1 2 p · wand is a conserved quantity for (3.190).