Ivancevic_Applied-Diff-Geom

426 Applied Differential Geometry: A Modern Introductiona closed 2−form or two–cocycle and an exact 2−form or two–coboundary.Therefore the 2D–de Rham cohomology group of human motion is definedas a quotient vector space(resp.H 2 DR(T ∗ M) = Z 2 (T ∗ M)/B 2 (T ∗ M)H 2 DR(T M) = Z 2 (T M)/B 2 (T M)).As T ∗ M (resp. T M) is a compact Hamiltonian symplectic (resp. Lagrangiansymplectic) manifold of dimension 2N, it follows that ω N H (resp.ω N L ) is a volume element on T ∗ M (resp. T M), and the 2ND de Rham’s cohomologyclass [ ωH] N ∈ H2NDR (T ∗ M) (resp. [ ]ω N L ∈ H2NDR (T M)) is nonzero.Since [ ωH] N = [ωH ] N [ ](resp. ωNL = [ωL ] N ), then [ω H ] ∈ HDR 2 (T ∗ M)(resp. [ω L ] ∈ HDR 2 (T M) ) and all of its powers up to the N−-th must bezero as well. The existence of such an element is a necessary condition forT ∗ M (resp. T M) to admit a Hamiltonian symplectic structure ω H (resp.Lagrangian symplectic structure ω L ).The de Rham complex A • DR on T ∗ M (resp. T M) can be consideredas a system of second–order ODEs d 2 θ H = 0, θ H ∈ Ω N (T ∗ M) (resp.d 2 θ L = 0, θ L ∈ Ω N (T M)) having a solution represented by Z N (T ∗ M)(resp. Z N (T M)). In local coordinates q i , p i ∈ U p (U p open in T ∗ M) (resp.q i , v i ∈ U v (U v open in T M)) we have d 2 θ H = d 2 (p i dq i ) = d(dp i ∧ dq i ) =0, (resp. d 2 θ L = d 2 (L v idq i ) = d(dL v i ∧ dq i ) = 0). Homological Proof. If C = H • M, (resp. C = L • M) represents an Abeliancategory of chains on T ∗ M (resp. T M), we have a category S • (H • M) (resp.S • (L • M)) of generalized chain complexes A • in H • M (resp. L • M), andif A = 0 for n < 0 we have a subcategory S• C (H • M) (resp. S• C (L • M)) ofchain complexes in H • M (resp. L • M)A • : 0 ← C 0 (T ∗ M)· · ·(resp. A • : 0 ← C 0 (T M)∂←− C 1 (T ∗ M)∂←− C n (T ∗ ∂M) ←− · · ·∂←− C 1 (T M)∂· · · ←− C n ∂(T M) ←− · · · ).∂←− C 2 (T ∗ M)∂←− C 2 (T M)∂←− · · ·∂←− · · ·Here A N = C N (T ∗ M) (resp. A N = C N (T M)) is the vector space of allfinite chains C on T ∗ M (resp. T M) over R, and ∂ N = ∂ : C N+1 (T ∗ M) →C N (T ∗ M) (resp. ∂ N = ∂ : C N+1 (T M) → C N (T M)). A finite chain Csuch that ∂C = 0 is an N−cycle. A finite chain C such that C = ∂Bis an N−boundary. Let Z N (T ∗ M) = Ker(∂) (resp. Z N (T M) = Ker(∂))and B N (T ∗ M) = Im(∂) (resp. B N (T M) = Im(∂)) denote respectively