Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 223product law(a 1 , b 1 )(a 2 , b 2 ) = (a 1 a 2 , b 1 b 2 ), (a 1,2 ∈ A, b 1,2 ∈ B).Generalizing the direct product to N rotational joint groups, we candraw an anthropomorphic product–tree (see Figure 3.6) using a line segment‘–’ to represent direct products of human SO(n)−-joints. This is our basicmodel of the biodynamical configuration manifold M.Fig. 3.6 Purely rotational, whole–body biodynamical manifold, with a singleSO(3)−joint representing the whole spinal movability.Let T q M be a tangent space to M at the point q. The tangent bundleT M represents a union ∪ q∈M T q M, together with the standard topologyon T M and a natural smooth manifold structure, the dimension of whichis twice the dimension of M. A vector–field X on M represents a sectionX : M → T M of the tangent bundle T M.Analogously let Tq ∗ M be a cotangent space to M at q, the dual toits tangent space T q M. The cotangent bundle T ∗ M represents a union∪ q∈M Tq ∗ M, together with the standard topology on T ∗ M and a naturalsmooth manifold structure, the dimension of which is twice the dimensionof M. A 1−form θ on M represents a section θ : M → T ∗ M of thecotangent bundle T ∗ M.We refer to the tangent bundle T M of biodynamical configuration manifoldM as the velocity phase–space manifold, and to its cotangent bundleT ∗ M as the momentum phase–space manifold.