Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 427real vector spaces of cycles and boundaries of degree N. Since ∂ N−1 ∂ N =∂ 2 = 0, then B N (T ∗ M) ⊂ Z N (T ∗ M) (resp. B N (T M) ⊂ Z N (T M)). Thequotient vector space(resp.H C N(T ∗ M) = Z N (T ∗ M)/B N (T ∗ M)H C N(T M) = Z N (T M)/B N (T M))represents an ND biodynamics homology group (vector space). The elementsof HN C(T ∗ M) (resp. HN C (T M)) are equivalence sets of cycles. Twocycles C 1 and C 2 are homologous, or belong to the same equivalence set(written C 1 ∼ C 2 ) iff they differ by a boundary C 1 − C 2 = ∂B. The homologyclass of a finite chain C ∈ C N (T ∗ M) (resp. C ∈ C N (T M)) is[C] ∈ HN C(T ∗ M) (resp. [C] ∈ HN C (T M)). Lagrangian Versus Hamiltonian DualityIn this way, we have proved a commutativity of a triangle:Lag ✠TanBundDiffManMFB∼ =Dual THam❅❅❅❅❅ ❅❘✲ CotBundwhich implies the existence of the unique natural topological equivalencein the rotational biodynamics.Dual T : Lag ∼ = HamGlobally Dual Structure of Rotational BiodynamicsTheorem. Global dual structure of the rotational biodynamics is definedby the unique natural equivalenceDyn : Dual G∼ = DualT .