Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 419This thermal reflection on the biodynamics topology complies with thebasic biophysics of human muscles (see [Hill (1938)]).3.13.5.4 Lagrangian–Hamiltonian Duality in BiodynamicsThe present section uncovers the underlying dual geometro–topologicalstructure beneath the general biodynamics. It presents a parallel developmentof Hamiltonian and Lagrangian formulations of biodynamics (see[Ivancevic and Snoswell (2001); Ivancevic (2002); Ivancevic and Pearce(2001b); Ivancevic and Pearce (2001b); Ivancevic (2005)]), proves bothdifferential–geometrical and algebraic–topo-logical dualities between thesetwo formulations, and finally establishes a unique functorial relation betweenbiodynamics geometry and biodynamics topology.Lagrangian formulation of biodynamics is performed on the tangentbundle T M, while Hamiltonian formulation is performed on the cotangentbundle T ∗ M. Both Riemannian and symplectic geometry are used. Thegeometrical duality (see [Kolar et al. (1993); Choquet-Bruhat and DeWitt-Morete (1982)]) of Lie groups and algebras between these two biodynamicsformulations is proved as an existence of natural equivalence between Lieand canonical functors. The topological duality (see [Dodson and Parker(1997)]) between these two biodynamics formulations is proved as an existenceof natural equivalence between Lagrangian and Hamiltonian functorsin both homology and cohomology categories. In the case of reduced configurationmanifold, the Betti numbers and Euler–Poincaré characteristicare given.Geometrical Duality Theorem for MTheorem. There is a geometrical duality between rotational Lagrangianand Hamiltonian biodynamical formulations on M (as given by Figure 3.6).In categorical terms, there is a unique natural geometrical equivalenceDual G : Lie ∼ = Canin biodynamics (symbols are described in the next subsection).Proof. The proof has two parts: Lie–functorial and geometrical.Lie–Functorial Proof. If we apply the functor Lie on the category • [SO(n) i ](for n = 2, 3 and i = 1, . . . , N) of rotational Lie groups SO(n) i (and theirhomomorphisms) we get the category • [so(n) i ] of corresponding tangentLie algebras so(n) i (and their homomorphisms). If we further apply the